二元多项式矩阵Smith型等价问题

二元多项式矩阵Smith型等价问题 On the Equivalence to the Smith Form of Bivariate Polynomial Matrices 1 first-author 李冬梅，女，副教授，现从事代数与符号计算方面的研究。E-mail:dmli@hnust.edu.cn 李冬梅，女，副教授，现从事代数与符号计算方面的研究。E-mail:dmli@hnust.edu.cn dmli@hnust.edu.cn 1 湖南科技大学 数学与计算科学学院 湖南 湘潭 411201 湖南科技大学 数学与计算科学学院 湖南 湘潭 411201 School of Mathematics and Computational Science, Hunan University of Science and Technology,Xiangtan 411201,Hunan,China School of Mathematics and Computational Science, Hunan University of Science and Technology,Xiangtan 411201,Hunan,China 二维系统等价是多维系统研究中的重要问题，它常被转化为二元多项式矩阵的等价问题来研究。二元多项式矩阵与其Smith型等价是矩阵等价研究中的重要问题。本文主要研究几类二元多项式矩阵与其Smith型等价问题，给出二元多项式矩阵约化到其Smith型的一些结果和判别条件。这些条件可以通过计算矩阵既约子式的约化Gröbner基进行检验。 The equivalence of two-dimensional systems is an important content of multidimensional systems, which is usually represented by the bivariate polynomial matrices. Reducing a matrix to its Smith form is a very important study in the equivalence of matrices. In this paper we mainly investigate the reduction of several kinds of bivariate polynomial matrices to their Smith forms. Some new results and criteria are presented. These criteria can be verified easily by computing the Gröbner basis of the associated ideals. 二维系统 二元多项式矩阵 矩阵等价 Smith型 two-dimensional system bivariate polynomial matrix equivalence of matrices the Smith form 0　 引 言 1 多维系统在数字滤波器、图像、信号处理等领域有广泛应用，多维系统等价是多维系统的一个重要内容，在实际应用中，常常需要将一个给定的复杂系统转化为形式和结构更简单的等价系统 [ 1 1 , 2 2 , 3 3 , 4 4 , 5 5 , 6 6 , 7 7 , 8 8 ] 。由系统理论中的多项式表示方法，人们常将多维系统等价问题转化为多项式矩阵等价问题来研究。矩阵的Smith型在矩阵等价中占非常重要的地位，将一个多项式矩阵化简为其Smith型，可以降低对应多维系统的未知量数或方程数。1970年，Rosenbrock [ 5 5 ] 利用主理想整环 <math id="M1"><mi>K</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423308&type= 8.97466660 3.55599999 的算法性质，得到 <math id="M2"><mi>K</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423310&type= 8.97466660 3.55599999 上任意一元多项式矩阵都与其Smith型等价，完全解决了一维系统的等价问题。但多元多项式环不是欧几里德环，其环上多元多项式矩阵的Smith型等价问题等价于两个有限生成投射模的同构问题，这类同构问题在代数上几乎是不可能被完全解决的，因此多元多项式矩阵等价问题有待继续深入研究。 科学家们对多元多项式矩阵与其Smith型等价的研究起源于1978年Frost和Storey [ 9 9 ] 对实数域上的二元多项式矩阵的研究，他们给出了一个二元多项式矩阵与其Smith型等价的充分必要条件。但是1983年Lee和Zak [ 10 10 ] 通过举例证明了上述条件只是必要的不是充分的，并给出了一类形如 <math id="M3"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">A</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423312&type= 18.54199982 4.48733330 的二元多项式矩阵与其Smith型等价的一个充分必要条件是存在一个循环向量，但在实际中很难找到程序判别这种循环向量。1986年，Frost和Boudellioua [ 11 11 ] 证明了二元行满秩多项式方阵 <math id="M4"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423314&type= 2.62466669 2.62466669 与其Smith型等价当且仅当存在一个幺模列向量 <math id="M5"><mi mathvariant="bold-italic">U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423316&type= 2.96333337 2.62466669 使得 <math id="M6"><mfenced separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">T</mi></mtd><mtd><mi mathvariant="bold-italic">U</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423318&type= 11.85333347 3.72533321 存在右逆，但这种向量的寻找同样不容易且 <math id="M7"><mfenced separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">T</mi></mtd><mtd><mi mathvariant="bold-italic">U</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423320&type= 11.85333347 3.72533321 的右逆存在性不易判别。所以，2006年Lin等 [ 12 12 ] 用构造性的方法对一类特殊的多项式矩阵进行讨论，这类矩阵 <math id="M8"><mi mathvariant="bold-italic">M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423322&type= 3.47133350 2.79399991 的行列式为 <math id="M9"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">M</mi><mo>=</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423324&type= 40.04733276 4.65666676 ，并得到这类矩阵与其Smith型 <math id="M10"><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">M</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423326&type= 23.53733444 8.55133343 等价。本文作者在文献[ 13 13 ]拓展了Lin等的结论，证明了如果多项式矩阵 <math id="M11"><mi mathvariant="bold-italic">M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423328&type= 3.47133350 2.79399991 所有的 <math id="M12"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423330&type= 25.31533432 3.55599999 阶子式生成单位理想且 <math id="M13"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">M</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423332&type= 13.37733269 2.96333337 <math id="M14"><mo stretchy="false">[</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mi>q</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423334&type= 31.07266808 4.91066647 ,则 <math id="M15"><mi mathvariant="bold-italic">M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423336&type= 3.47133350 2.79399991 也与其Smith型 <math id="M16"><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">M</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423338&type= 23.53733444 8.55133343 等价，并给出一种找到等价变换中幺模矩阵的构造性方法。 二元多项式矩阵的等价在多元多项式矩阵等价问题中占重要地位，本文在文献[ 12 12 ]和[13] 的基础上，结合二元多项式环的性质以及Gröbner基理论，考虑二元多项式环上几类二元多项式矩阵的等价问题，希望找到一些行列式为其他特殊形式的矩阵，这些矩阵也与其Smith型等价。 1　 预备知识 1 <math id="M17"><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423341&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423340&type= 13.63133430 3.55599999 表示实数域 <math id="M18"><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423342&type= 2.45533323 2.70933342 上含两个变量的多项式环; <math id="M19"><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423344&type= 17.35666656 3.81000018 表示多项式环 <math id="M20"><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423347&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423346&type= 13.63133430 3.55599999 上的所有 <math id="M21"><mi>l</mi><mo>×</mo><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423349&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423348&type= 7.28133297 2.79399991 级矩阵的集合; <math id="M22"><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423351&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423350&type= 22.26733398 4.48733330 表示在多项式环 <math id="M23"><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423352&type= 13.63133430 3.55599999 上的所有 <math id="M24"><mi>l</mi><mo>×</mo><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423354&type= 7.28133297 2.79399991 级可逆矩阵的集合; <math id="M25"><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>r</mi><mo>,</mo><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423356&type= 4.40266657 3.64066648 表示 <math id="M26"><mi>r</mi><mo>×</mo><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423359&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423358&type= 7.62000036 2.62466669 级零矩阵， <math id="M27"><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423361&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423360&type= 2.45533323 3.55599999 表示 <math id="M28"><mi>r</mi><mo>×</mo><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423363&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423362&type= 7.95866680 2.62466669 级单位矩阵;令 <math id="M29"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423365&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423364&type= 33.95133591 3.81000018 ， <math id="M30"><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423366&type= 33.69733429 4.48733330 表示矩阵 <math id="M31"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423369&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423368&type= 12.95399952 3.55599999 的所有 <math id="M32"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423370&type= 25.31533432 3.55599999 级子式, <math id="M33"><mi>t</mi><mo>=</mo><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423373&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423372&type= 8.55133343 3.04799986 。 定义1 令 <math id="M34"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423374&type= 33.95133591 3.81000018 ，则 <math id="M35"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423376&type= 12.95399952 3.55599999 的Smith型定义为 <math id="M36"><mi>S</mi><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mo stretchy="false">{</mo><msub><mrow><mi>Φ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423378&type= 23.87599945 4.65666676 ，其中 <math display="block" id="M37"><msub><mrow><mi>Φ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>,</mo></mtd><mtd><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>r</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo></mtd><mtd><mi>r</mi><mo><</mo><mi>i</mi><mo>≤</mo><mi>l</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423381&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423380&type= 36.91466904 14.30866623 <math id="M38"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423382&type= 1.52399993 2.62466669 是 <math id="M39"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423384&type= 12.95399952 3.55599999 的秩， <math id="M40"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>≡</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423386&type= 10.07533360 3.64066648 , <math id="M41"><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423388&type= 2.70933342 3.55599999 是 <math id="M42"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423390&type= 12.95399952 3.55599999 的 <math id="M43"><mi>i</mi><mo>×</mo><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423393&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423392&type= 7.02733326 2.62466669 级子式的最大公因子且 <math id="M44"><msub><mrow><mi>Φ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423394&type= 3.64066648 3.55599999 满足整除性 <math id="M45"><msub><mrow><mi>Φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>Φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">|</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">|</mo><msub><mrow><mi>Φ</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423396&type= 17.35666656 4.40266657 。 定义2 称矩阵 <math id="M46"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423398&type= 33.95133591 3.81000018 具有性质DM（determinant minor），如果 <math id="M47"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423401&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423400&type= 18.28800011 3.55599999 与 <math id="M48"><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423402&type= 12.78466702 3.55599999 的所有 <math id="M49"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423405&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423404&type= 25.31533432 3.55599999 级子式没有共同零点。 定义3 设 <math id="M50"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423406&type= 48.76799774 3.81000018 ，如果存在 <math id="M51"><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423408&type= 54.77933502 4.48733330 ，使得 <math id="M52"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423411&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423410&type= 16.93333435 3.55599999 <math id="M53"><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423412&type= 14.81666660 3.55599999 <math id="M54"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi>V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423414&type= 26.75466537 3.55599999 ，则称 <math id="M55"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423416&type= 12.95399952 3.55599999 与 <math id="M56"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423418&type= 13.12333298 3.55599999 等价。 定义4 设 <math id="M57"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>m</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>l</mi><mo>≤</mo><mi>m</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423421&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423420&type= 47.41333389 3.89466667 是行满秩的，称 <math id="M58"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423422&type= 12.95399952 3.55599999 是零互素（ZLP，zero left prime），如果 <math id="M59"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423425&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423424&type= 12.95399952 3.55599999 的所有 <math id="M60"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423426&type= 25.31533432 3.55599999 级子式没有共同零点。 引理1 [ 13 13 ] 令 <math id="M61"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423429&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423428&type= 33.95133591 3.81000018 ， <math id="M62"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423430&type= 33.27399826 4.65666676 是 <math id="M63"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423432&type= 18.28800011 3.55599999 的一个因子，则 <math id="M64"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423434&type= 13.71600056 4.48733330 ， <math id="M65"><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423436&type= 15.32466698 4.48733330 … <math id="M66"><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>r</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423438&type= 14.73199940 4.48733330 没有共同零点(在复数域 <math id="M67"><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423440&type= 17.44133377 3.81000018 )，当且仅当 <math id="M68"><mo>∀</mo><mi>w</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423442&type= 16.17133331 3.38666677 ，都有 <math id="M69"><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">k</mi><mtext> </mtext><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423445&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423444&type= 40.72466660 3.81000018 。 推论1 令 <math id="M70"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423447&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423446&type= 33.02000046 3.81000018 ， <math id="M71"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>w</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423448&type= 32.08866501 4.65666676 是 <math id="M72"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423450&type= 18.28800011 3.55599999 的一个因子，那么 <math id="M73"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423453&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423452&type= 31.24200058 4.48733330 … <math id="M74"><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>r</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423454&type= 14.73199940 4.48733330 没有共同零点(在复数域 <math id="M75"><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423457&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423456&type= 17.44133377 3.81000018 )，当且仅当 <math id="M76"><mo>∀</mo><mi>z</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423459&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423458&type= 14.90133286 3.38666677 ，都有 <math id="M77"><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">k</mi><mtext> </mtext><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423461&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423460&type= 37.33800125 3.81000018 。 引理2 [ 13 13 ] 令 <math id="M78"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423463&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423462&type= 65.78599548 4.65666676 , <math id="M79"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423465&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423464&type= 45.97399902 4.48733330 。若 <math id="M80"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423467&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423466&type= 18.28800011 3.55599999 与 <math id="M81"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423469&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423468&type= 12.95399952 3.55599999 所有的 <math id="M82"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423471&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423470&type= 25.31533432 3.55599999 级子式没有共同零点，则 <math id="M83"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423472&type= 19.04999924 4.48733330 和 <math id="M84"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423474&type= 13.80066681 4.48733330 所有的 <math id="M85"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423476&type= 25.31533432 3.55599999 级子式没有共同零点，其中 <math id="M86"><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423478&type= 11.51466751 3.13266659 。 推论2 令 <math id="M87"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423480&type= 33.95133591 3.81000018 , <math id="M88"><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423482&type= 55.03333664 4.48733330 ,那么 <math id="M89"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423485&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423484&type= 12.95399952 3.55599999 具有性质DM当且仅当 <math id="M90"><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423487&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423486&type= 40.63999939 3.55599999 具有性质DM。 引理3 [ 13 13 ] 令 <math id="M91"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423488&type= 33.95133591 3.81000018 , <math id="M92"><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">k</mi><mtext> </mtext><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423491&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423490&type= 39.20066452 3.81000018 对每个 <math id="M93"><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423493&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423492&type= 16.93333435 3.81000018 ,那么可以构造一个幺模向量 <math id="M94"><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423494&type= 25.23066711 3.81000018 ,使得 <math display="block" id="M95"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423497&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423496&type= 47.15933228 4.65666676 2　 主要结果 1 引理4 令 <math id="M96"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423499&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423498&type= 33.95133591 3.81000018 ，且 <math id="M97"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423500&type= 56.64200211 3.97933340 , <math id="M98"><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423503&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423502&type= 8.04333305 3.04799986 , <math id="M99"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423505&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423504&type= 1.86266661 3.47133350 是一个正整数。如果 <math id="M100"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423507&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423506&type= 12.95399952 3.55599999 具有性质DM，那么 <math id="M101"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423508&type= 12.95399952 3.55599999 等价于矩阵 <math display="block" id="M102"><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⋯</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423511&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423510&type= 118.11000061 4.48733330 其中， <math id="M103"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423513&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423512&type= 50.46133423 9.39799976 ， <math id="M104"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mi>w</mi><mo>-</mo><mi>a</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423515&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423514&type= 45.12733459 8.29733276 ， <math id="M105"><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423517&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423516&type= 40.13199997 4.48733330 ， <math id="M106"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423519&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423518&type= 14.05466652 4.48733330 <math display="block" id="M107"><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423521&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423520&type= 69.84999847 4.65666676 。 证 因为 <math id="M108"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423522&type= 18.28800011 3.55599999 和 <math id="M109"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423525&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423524&type= 12.95399952 3.55599999 的所有 <math id="M110"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423526&type= 25.31533432 3.55599999 级子式没有共同零点，所以 <math id="M111"><mi>w</mi><mo>-</mo><mi>a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423529&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423528&type= 9.65199947 2.62466669 和 <math id="M112"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423531&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423530&type= 12.95399952 3.55599999 的所有 <math id="M113"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423532&type= 25.31533432 3.55599999 级子式没有共同零点。因为 <math id="M114"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423534&type= 56.64200211 3.97933340 , <math id="M115"><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423537&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423536&type= 19.30399895 3.55599999 , <math id="M116"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423538&type= 1.86266661 3.47133350 是一个正整数，那么 <math id="M117"><mi>w</mi><mo>-</mo><mi>a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423541&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423540&type= 9.65199947 2.62466669 是 <math id="M118"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423543&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423542&type= 18.28800011 3.55599999 的一个因子。根据推论1，对每个 <math id="M119"><mi>z</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423544&type= 8.12800026 3.13266659 ，有 <math id="M120"><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">k</mi><mtext> </mtext><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423547&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423546&type= 31.83466721 3.55599999 。由引理3，存在一个幺模向量 <math id="M121"><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423549&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423548&type= 26.41599846 4.65666676 ，使得 <math display="block" id="M122"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423550&type= 42.07933426 4.65666676 根据Quillen-Suslin定理 [ 14 14 ] ，可以构造一个幺模矩阵 <math id="M123"><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423552&type= 26.83933258 4.31799984 ，使得 <math id="M124"><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423555&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423554&type= 9.73666668 4.48733330 是它的最后一列且 <math id="M125"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423557&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423556&type= 22.35199928 4.06400013 。由(1)式得，存在 <math id="M126"><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423558&type= 32.93533325 4.65666676 <math display="block" id="M127"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>,</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr></mtable><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423561&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423560&type= 43.18000031 4.06400013 由(2)式计算得 <math display="block" id="M128"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423562&type= 57.31933212 4.99533367 其中， <math id="M129"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423564&type= 35.13666534 4.65666676 , <math id="M130"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mi>w</mi><mo>-</mo><mi>a</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423566&type= 43.26466751 8.29733276 。那么有 <math display="block" id="M131"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423569&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423568&type= 56.55733490 4.48733330 其中， <math id="M132"><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423570&type= 26.66999817 4.99533367 。注意到 <math id="M133"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423572&type= 22.35199928 4.06400013 ，所以 <math id="M134"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423574&type= 21.59000015 4.48733330 。由于 <math id="M135"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423576&type= 56.64200211 3.97933340 ,那么 <math id="M136"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423579&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423578&type= 61.38333511 4.91066647 。由引理2有， <math id="M137"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423581&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423580&type= 14.22399998 4.48733330 也具有性质DM。所以 <math id="M138"><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423583&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423582&type= 14.90133286 3.81000018 和 <math id="M139"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423584&type= 14.05466652 4.48733330 的所有 <math id="M140"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423586&type= 25.31533432 3.55599999 级子式没有共同零点。再根据引理1，得到： <math id="M141"><mo>∀</mo><mi>w</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423588&type= 13.46199894 3.30199981 <math id="M142"><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">k</mi><mtext> </mtext><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423591&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423590&type= 40.47066879 3.81000018 。结合引理3，存在一个幺模列向量 <math id="M143"><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423593&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423592&type= 28.61733246 4.65666676 使得 <math display="block" id="M144"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423595&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423594&type= 51.64666748 4.91066647 由Quillen-Suslin定理，可以构造一个幺模矩阵 <math id="M145"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423597&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423596&type= 29.12533188 4.31799984 ,使得 <math id="M146"><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423598&type= 10.92199993 4.48733330 是它的最后一列且 <math id="M147"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423601&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423600&type= 23.53733444 4.06400013 。由(5)式得，存在 <math id="M148"><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423603&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423602&type= 35.22133255 4.65666676 使得 <math display="block" id="M149"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>,</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr></mtable><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423605&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423604&type= 51.39266586 4.06400013 由(6)式,容易得 <math display="block" id="M150"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423607&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423606&type= 58.67399979 4.48733330 其中， <math id="M151"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423608&type= 26.75466537 4.99533367 。注意到 <math id="M152"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423611&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423610&type= 22.43666649 4.06400013 ,所以 <math id="M153"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423612&type= 21.59000015 4.48733330 。由于 <math id="M154"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423614&type= 61.38333511 4.91066647 ，则 <math id="M155"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423616&type= 64.85466766 4.91066647 。根据引理2， <math id="M156"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423618&type= 14.22399998 4.48733330 也具有性质DM，重复以上过程，可以得到我们的结论。证毕。 定理1 令 <math id="M157"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423621&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423620&type= 51.56200027 4.65666676 ,且 <math id="M158"><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423622&type= 8.38199997 3.04799986 , <math id="M159"><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423624&type= 31.41133308 4.48733330 , <math id="M160"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423627&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423626&type= 41.90999985 4.48733330 <math id="M161"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423628&type= 13.97000027 4.48733330 。如果 <math id="M162"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423630&type= 12.95399952 3.55599999 具有性质DM，则 <math id="M163"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423633&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423632&type= 12.95399952 3.55599999 等价于矩阵 <math display="block" id="M164"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423634&type= 63.75400162 9.39799976 证 对 <math id="M165"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423637&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423636&type= 1.18533337 2.79399991 进行归纳证明，当 <math id="M166"><mi>l</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423638&type= 8.04333305 2.79399991 时，令 <math id="M167"><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423640&type= 72.22066498 9.99066734 则 <math display="block" id="M168"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423642&type= 96.60466766 10.24466705 且 <math id="M169"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423644&type= 52.23933029 3.81000018 。首先， <math id="M170"><mi>w</mi><mo>-</mo><mi>a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423647&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423646&type= 9.05933285 2.62466669 不是 <math id="M171"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423649&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423648&type= 10.83733368 4.48733330 的零点，否则 <math id="M172"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423650&type= 18.28800011 3.55599999 和 <math id="M173"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423652&type= 13.03866673 4.48733330 <math id="M174"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423655&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423654&type= 25.73866653 4.48733330 <math id="M175"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423657&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423656&type= 74.42200470 4.65666676 有一个公共零点 <math id="M176"><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423659&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423658&type= 15.15533447 3.81000018 ，与 <math id="M177"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423661&type= 12.95399952 3.55599999 具有性质DM矛盾。注意到 <math id="M178"><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423665&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423663&type= 32.59666443 4.48733330 ， <math id="M179"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423666&type= 23.79133415 4.48733330 没有公共零点。所以 <math id="M180"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423670&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423669&type= 36.99933624 4.48733330 没有公共零点， <math id="M181"><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo></mtd><mtd><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423672&type= 43.34933472 4.31799984 是一个幺模行。根据Quillen-Suslin定理，可以构造一个幺模矩阵 <math id="M182"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423676&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423674&type= 15.07066631 4.48733330 <math id="M183"><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423678&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423677&type= 25.90799904 4.48733330 ，使得 <math id="M184"><mfenced separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423681&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423680&type= 70.95066833 4.65666676 ，即 <math display="block" id="M185"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423684&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423683&type= 66.97133636 9.39799976 那么存在 <math id="M186"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423685&type= 45.63533401 9.39799976 ，使得 <math id="M187"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423689&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423688&type= 71.11999512 9.39799976 。 因为 <math id="M188"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423691&type= 59.09733200 4.48733330 ，所以 <math id="M189"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>u</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423693&type= 45.55066681 4.65666676 <math id="M190"><mi>u</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423696&type= 38.69266510 3.81000018 <math id="M191"><mn mathvariant="normal">0</mn><mo>≠</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423700&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423699&type= 6.09599972 3.04799986 <math id="M192"><mi>u</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423702&type= 8.04333305 3.04799986 。因此， <math id="M193"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423706&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423704&type= 12.95399952 3.55599999 等价于矩阵 <math display="block" id="M194"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423707&type= 57.48866653 9.39799976 假设 <math id="M195"><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423711&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423710&type= 7.95866680 2.79399991 时结论是成立的，考虑 <math id="M196"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423714&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423712&type= 1.18533337 2.79399991 的情形。 令 <math id="M197"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423716&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423715&type= 69.42666626 9.99066734 ，其中， <math id="M198"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423718&type= 38.86199951 4.65666676 ， <math id="M199"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423722&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423720&type= 34.45933533 4.65666676 , <math id="M200"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423724&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423723&type= 15.07066631 4.48733330 <math id="M201"><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423726&type= 18.71133423 3.81000018 ， <math id="M202"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423730&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423728&type= 29.88733292 4.65666676 。那么 <math display="block" id="M203"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423732&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423731&type= 98.89067078 10.24466705 设 <math id="M204"><mfenced separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423735&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423734&type= 49.19132996 4.74133348 是矩阵 <math id="M205"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423736&type= 11.00666714 4.48733330 的第一行。由 <math id="M206"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423739&type= 35.98333359 4.48733330 得 <math id="M207"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⋯</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423742&type= 18.79599953 4.48733330 <math id="M208"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423746&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423744&type= 17.27199936 4.48733330 <math id="M209"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423748&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423747&type= 10.41399956 4.48733330 没有共同零点。进一步可得， <math id="M210"><mfenced separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423750&type= 62.22999954 4.74133348 是 <math id="M211"><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423754&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423752&type= 10.07533360 3.55599999 上的幺模行（否则，假设 <math id="M212"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423756&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423755&type= 59.60533142 4.48733330 有一个共同零点，这个共同零点一定是 <math id="M213"><mi>w</mi><mo>=</mo><mi>a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423759&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423758&type= 9.65199947 2.62466669 ,那么 <math id="M214"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423762&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423760&type= 18.28800011 3.55599999 和 <math id="M215"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423763&type= 12.95399952 3.55599999 的所有 <math id="M216"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423766&type= 25.31533432 3.55599999 级子式有一个共同零点 <math id="M217"><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423768&type= 15.15533447 3.81000018 ，与 <math id="M218"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423771&type= 12.95399952 3.55599999 具有性质DM矛盾）。因此，根据Quillen-Suslin定理，可以构造一个幺模矩阵 <math id="M219"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423774&type= 15.07066631 4.48733330 <math id="M220"><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423776&type= 16.93333435 4.48733330 <math id="M221"><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423779&type= 25.56933403 4.48733330 使得 <math display="block" id="M222"><mi mathvariant="bold-italic">N</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>≜</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423781&type= 96.09666443 9.22866726 显然， <math id="M223"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">N</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>u</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo>≠</mo><mi>u</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423783&type= 99.48333740 4.65666676 。由于 <math id="M224"><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423786&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423785&type= 12.78466702 3.55599999 具有性质DM，由推论2，知 <math id="M225"><mi mathvariant="bold-italic">N</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423788&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423787&type= 13.46199894 3.55599999 也具有性质DM，即 <math id="M226"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">N</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423789&type= 18.71133423 3.55599999 与 <math id="M227"><mi mathvariant="bold-italic">N</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423792&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423791&type= 13.46199894 3.55599999 的所有 <math id="M228"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423793&type= 25.31533432 3.55599999 级子式没有共同零点。由于 <math id="M229"><mi mathvariant="bold-italic">N</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423796&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423795&type= 13.46199894 3.55599999 的一个 <math id="M230"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423797&type= 25.31533432 3.55599999 级子式或是 <math id="M231"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423799&type= 14.22399998 4.48733330 的一个 <math id="M232"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423802&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423801&type= 25.48466682 3.55599999 级子式或是 <math id="M233"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423803&type= 19.47333336 4.48733330 或是0,则 <math id="M234"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423805&type= 19.47333336 4.48733330 与 <math id="M235"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423808&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423807&type= 14.22399998 4.48733330 的所有 <math id="M236"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423810&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423809&type= 25.48466682 3.55599999 级子式没有共同零点。由归纳假设，存在 <math id="M237"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423812&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423811&type= 60.79066849 4.48733330 使得 <math display="block" id="M238"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">2,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msub></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423813&type= 95.67333221 9.39799976 所以 <math display="block" id="M239"><mtable columnalign="left"><mtr><mtd><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mi mathvariant="bold-italic">N</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423816&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423815&type= 118.27933502 20.40466690 因此， <math id="M240"><mi mathvariant="bold-italic">N</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423817&type= 13.46199894 3.55599999 等价于矩阵 <math display="block" id="M241"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423820&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423819&type= 63.75400162 9.39799976 结合 <math id="M242"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423822&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423821&type= 12.95399952 3.55599999 等价于矩阵 <math id="M243"><mi mathvariant="bold-italic">N</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423823&type= 13.46199894 3.55599999 ，由等价关系的传递性，得到 <math id="M244"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423825&type= 12.95399952 3.55599999 等价于矩阵 <math id="M245"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423827&type= 12.69999981 3.55599999 。证毕。 定理2 令 <math id="M246"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423830&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423829&type= 33.95133591 3.81000018 且 <math id="M247"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423832&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423831&type= 54.10200119 3.81000018 ,其中 <math id="M248"><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423833&type= 8.04333305 3.04799986 。则 <math id="M249"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423835&type= 12.95399952 3.55599999 等价于矩阵 <math display="block" id="M250"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423837&type= 63.75400162 9.39799976 当且仅当 <math id="M251"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423839&type= 12.95399952 3.55599999 具有性质DM。 证 充分性：由 <math id="M252"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423841&type= 54.10200119 3.81000018 且 <math id="M253"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423843&type= 54.10200119 3.81000018 及引理4， <math id="M254"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423846&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423845&type= 12.95399952 3.55599999 等价于 <math id="M255"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423847&type= 40.97866821 4.48733330 其中， <math id="M256"><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423849&type= 33.35866928 4.48733330 <math id="M257"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423851&type= 48.42933273 9.39799976 ， <math id="M258"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423853&type= 18.03400040 4.48733330 <math id="M259"><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mi>w</mi><mo>-</mo><mi>a</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423855&type= 24.12999916 8.29733276 。根据定理1，结论成立。 必要性：由 <math id="M260"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423858&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423857&type= 12.69999981 3.55599999 的矩阵形式知， <math id="M261"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423859&type= 18.03400040 3.55599999 与 <math id="M262"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423861&type= 12.69999981 3.55599999 的所有 <math id="M263"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423863&type= 25.31533432 3.55599999 级子式没有共同零点。由推论2， <math id="M264"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423865&type= 18.28800011 3.55599999 与 <math id="M265"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423867&type= 12.95399952 3.55599999 的所有 <math id="M266"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423869&type= 25.31533432 3.55599999 级子式也没有共同零点。因此 <math id="M267"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423872&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423871&type= 12.95399952 3.55599999 具有性质DM。证毕。 推论3 令 <math id="M268"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423873&type= 33.95133591 3.81000018 且 <math id="M269"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423876&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423875&type= 53.00133133 3.81000018 ，其中 <math id="M270"><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423878&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423877&type= 8.04333305 3.04799986 。则 <math id="M271"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423879&type= 12.95399952 3.55599999 等价于矩阵 <math display="block" id="M272"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423881&type= 62.65333557 9.39799976 当且仅当 <math id="M273"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423884&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423883&type= 12.95399952 3.55599999 具有性质DM。 引理5 令 <math id="M274"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423886&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423885&type= 33.69733429 3.81000018 且 <math id="M275"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi>g</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423887&type= 42.07933426 9.39799976 ，其中 <math id="M276"><mi>g</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423889&type= 29.54866600 3.72533321 。令 <math id="M277"><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423891&type= 16.59466553 3.55599999 <math id="M278"><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423894&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423893&type= 22.26733398 4.48733330 。如果 <math id="M279"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>s</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423895&type= 58.92799759 3.81000018 具有性质DM，那么 <math id="M280"><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423898&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423897&type= 12.78466702 3.55599999 等价于矩阵 <math id="M281"><msup><mrow><mi>P</mi></mrow><mrow><mi>s</mi><mo>+</mo><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423899&type= 16.59466553 3.81000018 , 其中， <math id="M282"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423902&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423901&type= 1.26999998 2.62466669 和 <math id="M283"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423903&type= 1.10066664 2.62466669 是正整数。 证 令 <math id="M284"><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423906&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423905&type= 57.48866653 9.99066734 其中， <math id="M285"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423908&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423907&type= 43.68799973 4.65666676 , <math id="M286"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423909&type= 39.79333496 4.65666676 , <math display="block" id="M287"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423912&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423911&type= 41.48666763 4.65666676 , <math display="block" id="M288"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423913&type= 37.08399963 4.65666676 。则 <math display="block" id="M289"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>s</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>s</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>s</mi><mo>+</mo><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423915&type= 132.84199524 10.41399956 所以 <math id="M290"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>s</mi><mo>+</mo><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423917&type= 39.28533173 3.97933340 。 首先，证明 <math id="M291"><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423920&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423919&type= 49.78400040 4.74133348 是一个ZLP矩阵。注意到 <math id="M292"><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423922&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423921&type= 39.36999893 4.48733330 ，所以 <math id="M293"><mfenced separators="|"><mrow><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423923&type= 35.89866638 4.65666676 是一个ZLP矩阵。令 <math id="M294"><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423925&type= 33.69733429 4.48733330 表示矩阵 <math id="M295"><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423928&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423927&type= 35.89866638 4.65666676 的所有 <math id="M296"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423929&type= 14.56266594 3.55599999 <math id="M297"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423931&type= 10.83733368 3.55599999 级子式，则矩阵 <math id="M298"><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423933&type= 49.78400040 4.74133348 的所有 <math id="M299"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423935&type= 25.31533432 3.55599999 级子式为 <math id="M300"><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⋯</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423937&type= 21.08199883 4.48733330 <math id="M301"><msub><mrow><mi>c</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423939&type= 26.66999817 4.91066647 。下面用反证法证明 <math id="M302"><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423941&type= 47.66733170 4.91066647 没有共同零点。假设 <math id="M303"><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423943&type= 47.66733170 4.91066647 有一个共同零点 <math id="M304"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423945&type= 3.13266659 3.55599999 ，注意到 <math id="M305"><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423947&type= 33.69733429 4.48733330 没有公共零点，那么 <math id="M306"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423949&type= 3.13266659 3.55599999 一定是 <math id="M307"><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423951&type= 13.03866673 4.48733330 和 <math id="M308"><msup><mrow><mi>g</mi></mrow><mrow><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423954&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423953&type= 13.20799923 3.97933340 的公共零点。注意到 <math id="M309"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423956&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423955&type= 12.95399952 3.55599999 的最后一行都有一个公因子 <math id="M310"><mi>g</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423958&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423957&type= 12.27666664 3.72533321 ，结合 <math id="M311"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423959&type= 12.95399952 3.55599999 的所有 <math id="M312"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423962&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423961&type= 25.31533432 3.55599999 级子式要么为 <math id="M313"><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423963&type= 15.32466698 4.48733330 要么包含因子 <math id="M314"><mi>g</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423966&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423965&type= 12.27666664 3.72533321 ，以及 <math id="M315"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423967&type= 18.28800011 3.55599999 包含因子 <math id="M316"><mi>g</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423969&type= 12.27666664 3.72533321 ,那么 <math id="M317"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423971&type= 18.28800011 3.55599999 和 <math id="M318"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423973&type= 12.95399952 3.55599999 的所有 <math id="M319"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423975&type= 25.31533432 3.55599999 级子式有一个共同零点 <math id="M320"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423977&type= 3.13266659 3.55599999 ，与 <math id="M321"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423980&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423979&type= 12.95399952 3.55599999 具有性质DM矛盾，那么 <math id="M322"><mfenced separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423982&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423981&type= 40.30133438 4.74133348 是 <math id="M323"><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423983&type= 13.63133430 3.55599999 上的幺模行。根据Quillen-Suslin定理，存在矩阵 <math id="M324"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423985&type= 18.20333290 4.48733330 <math id="M325"><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423987&type= 22.26733398 4.48733330 使得 <math id="M326"><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>s</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr></mtable><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423989&type= 84.24333191 4.74133348 。这样， <math display="block" id="M327"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423991&type= 66.97133636 9.39799976 其中， <math id="M328"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423993&type= 76.79266357 4.91066647 。从而存在 <math id="M329"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423995&type= 45.63533401 9.39799976 ，使得 <math display="block" id="M330"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423997&type= 71.11999512 9.39799976 注意到 <math id="M331"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3423999&type= 58.25066376 4.48733330 ，所以 <math id="M332"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>u</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>u</mi><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424001&type= 80.94133759 4.65666676 <math id="M333"><mn mathvariant="normal">0</mn><mo>≠</mo><mi>u</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424003&type= 14.64733315 3.21733332 。因此， <math id="M334"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424005&type= 12.95399952 3.55599999 等价于矩阵 <math display="block" id="M335"><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>s</mi><mo>+</mo><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msup><mrow><mi>g</mi></mrow><mrow><mi>s</mi><mo>+</mo><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424007&type= 52.83199692 9.39799976 证毕。 定理3 令 <math id="M336"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424010&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424009&type= 33.95133591 3.81000018 且 <math id="M337"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424011&type= 56.64200211 3.97933340 ，其中 <math id="M338"><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424013&type= 8.04333305 3.04799986 。如果 <math id="M339"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424015&type= 12.95399952 3.55599999 具有性质DM，则 <math id="M340"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424017&type= 12.95399952 3.55599999 等价于矩阵 <math display="block" id="M341"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424020&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424019&type= 66.63265991 9.39799976 证 由引理4， <math id="M342"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424021&type= 12.95399952 3.55599999 可以分解为 <math display="block" id="M343"><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⋯</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424023&type= 119.88799286 4.48733330 其中， <math display="block" id="M344"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mi>w</mi><mo>-</mo><mi>a</mi></mtd></mtr></mtable></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424025&type= 95.33466339 9.39799976 <math id="M345"><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424027&type= 126.74599457 4.65666676 。根据定理1，我们有 <math display="block" id="M346"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424030&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424029&type= 89.49266815 4.48733330 其中， <math id="M347"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>q</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424031&type= 80.68733215 4.65666676 <math id="M348"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424034&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424033&type= 63.83866501 9.39799976 。那么 <math display="block" id="M349"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⋯</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>q</mi><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>q</mi><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>≜</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424035&type= 152.73866272 4.74133348 <math display="block" id="M350"><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>⋯</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424037&type= 101.68466949 4.48733330 其中， <math id="M351"><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424039&type= 47.58266830 4.48733330 ， <math id="M352"><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn mathvariant="normal">1,1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424041&type= 67.64866638 4.48733330 ， <math id="M353"><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424043&type= 17.35666656 4.48733330 <math id="M354"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>q</mi><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424045&type= 26.16199875 4.48733330 ， <math id="M355"><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424047&type= 23.87599945 3.55599999 。显然， <math id="M356"><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424049&type= 39.45466614 4.48733330 ， <math id="M357"><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424051&type= 17.18733406 3.55599999 。 由引理5， <math id="M358"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424053&type= 40.97866821 4.48733330 等价于 <math id="M359"><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424055&type= 14.05466652 3.81000018 ,即存在 <math id="M360"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424057&type= 57.15000153 4.48733330 使得 <math id="M361"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424059&type= 90.08533478 4.65666676 。因此， <math display="block" id="M362"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>⋯</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>⋯</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424062&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424061&type= 123.27467346 9.82133293 若 <math id="M363"><mi>q</mi><mo>≥</mo><mn mathvariant="normal">3</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424063&type= 10.32933331 3.81000018 令 <math id="M364"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424065&type= 48.00600052 4.48733330 ，则 <math id="M365"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424067&type= 40.13199997 4.48733330 。由引理5， <math id="M366"><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424070&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424069&type= 14.05466652 3.81000018 <math id="M367"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424071&type= 14.22399998 4.48733330 · <math id="M368"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424073&type= 12.69999981 3.55599999 等价于矩阵 <math id="M369"><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424075&type= 14.05466652 3.81000018 ,即存在 <math id="M370"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424077&type= 57.15000153 4.48733330 使得 <math display="block" id="M371"><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424079&type= 91.35533142 4.65666676 因此， <math display="block" id="M372"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>⋯</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>⋯</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>⋯</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424081&type= 127.08467102 15.40933418 重复以上过程，得到 <math display="block" id="M373"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>⋯</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424084&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424083&type= 130.80999756 4.65666676 显然， <math id="M374"><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>⋯</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424085&type= 125.98400116 4.48733330 。因此，可以得到 <math id="M375"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424088&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424087&type= 12.95399952 3.55599999 等价于矩阵 <math display="block" id="M376"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424089&type= 66.63265991 9.39799976 证毕。 定理4 令 <math id="M377"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424091&type= 33.95133591 3.81000018 且 <math id="M378"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424093&type= 56.64200211 3.97933340 ，其中 <math id="M379"><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424095&type= 8.04333305 3.04799986 。则 <math id="M380"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424097&type= 12.95399952 3.55599999 等价于矩阵 <math display="block" id="M381"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424099&type= 85.25933075 9.39799976 当且仅当 <math id="M382"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424102&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424101&type= 12.95399952 3.55599999 具有性质DM。 证 充分性：根据定理3，结论成立。 必要性：由 <math id="M383"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424104&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424103&type= 13.12333298 3.55599999 的矩阵表达形式知， <math id="M384"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424105&type= 18.37266731 3.55599999 与 <math id="M385"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424108&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424107&type= 13.12333298 3.55599999 的所有 <math id="M386"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424110&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424109&type= 25.31533432 3.55599999 级子式没有共同零点。由推论2， <math id="M387"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424111&type= 18.28800011 3.55599999 与 <math id="M388"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424113&type= 12.95399952 3.55599999 的所有 <math id="M389"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424116&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424115&type= 25.31533432 3.55599999 级子式没有共同零点。因此， <math id="M390"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424117&type= 12.95399952 3.55599999 具有性质DM。证毕。 推论4 令 <math id="M391"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424119&type= 33.95133591 3.81000018 且 <math id="M392"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424121&type= 55.45666885 3.97933340 ，其中 <math id="M393"><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424124&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424123&type= 8.04333305 3.04799986 。则 <math id="M394"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424125&type= 12.95399952 3.55599999 等价于矩阵 <math display="block" id="M395"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424128&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424127&type= 65.44733429 9.39799976 当且仅当 <math id="M396"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424129&type= 12.95399952 3.55599999 具有性质DM。 令 <math id="M397"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>l</mi><mo>×</mo><mi>l</mi></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424131&type= 33.95133591 3.81000018 且 <math id="M398"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mi>f</mi><msup><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424134&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424133&type= 55.45666885 3.97933340 ，其中 <math id="M399"><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424136&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424135&type= 9.90599918 3.21733332 <math id="M400"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424137&type= 1.86266661 3.47133350 是一个正整数。定理4给了一个判别 <math id="M401"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424140&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424139&type= 12.95399952 3.55599999 与其Smith型等价的充分必要条件。 <math id="M402"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424141&type= 12.95399952 3.55599999 具有性质DM意味着 <math id="M403"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424143&type= 18.28800011 3.55599999 与 <math id="M404"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424146&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424145&type= 12.95399952 3.55599999 的所有 <math id="M405"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424147&type= 25.31533432 3.55599999 级子式生成单位理想。我们知道一个理想是单位理想 <math id="M406"><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424150&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424149&type= 2.45533323 2.70933342 当且仅当该理想的既约Gröbner基是 <math id="M407"><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424151&type= 4.48733330 3.89466667 。因此，这个条件可以通过计算 <math id="M408"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424153&type= 18.28800011 3.55599999 与 <math id="M409"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424155&type= 12.95399952 3.55599999 的所有 <math id="M410"><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424158&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424157&type= 25.31533432 3.55599999 级子式所生成理想的既约Gröbner基进行检验。 3　 例 子 1 本节通过一个例子来说明我们的构造性方法。应用该方法可以直接得到矩阵等价变换过程中的幺模矩阵。 例 考虑一个在 <math id="M411"><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn mathvariant="normal">3</mn><mo>×</mo><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424159&type= 17.94933319 3.81000018 上的二元多项式矩阵 <math display="block" id="M412"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd><mtd><mi>w</mi></mtd><mtd><mn mathvariant="normal">2</mn><mi>w</mi><mi>z</mi><mo>-</mo><mn mathvariant="normal">4</mn><mi>z</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">2</mn><mi>w</mi><mi>z</mi><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424161&type= 61.63733673 14.30866623 经计算， <math id="M413"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424164&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424163&type= 51.81599808 3.81000018 且 <math id="M414"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424165&type= 12.95399952 3.55599999 具有性质DM。考虑 <math display="block" id="M415"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">2</mn><mi>z</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">2</mn><mi>z</mi><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424167&type= 49.02199936 13.46199894 解方程： <math id="M416"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424169&type= 29.97199821 4.48733330 得到 <math id="M417"><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1,2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mi>z</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424171&type= 43.18000031 4.65666676 。将 <math id="M418"><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1,2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>+</mo><mn mathvariant="normal">2</mn><mi>z</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424173&type= 28.53266716 3.81000018 嵌入到一个幺模矩阵 <math id="M419"><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn mathvariant="normal">3</mn><mo>×</mo><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424175&type= 27.60133362 4.65666676 使得 <math display="block" id="M420"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424178&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424177&type= 57.57333374 4.48733330 其中， <math display="block" id="M421"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mi>z</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424180&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424179&type= 91.69400024 13.46199894 那么 <math display="block" id="M422"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424181&type= 61.21400070 4.48733330 这里 <math display="block" id="M423"><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><mi>z</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424184&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424183&type= 71.79733276 13.29266644 再考虑 <math id="M424"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424185&type= 14.22399998 4.48733330 ，重复以上过程，有 <math display="block" id="M425"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424187&type= 58.75866699 4.48733330 其中， <math display="block" id="M426"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mi>w</mi></mtd><mtd><mn mathvariant="normal">4</mn><mi>w</mi><mi>z</mi><mo>-</mo><mn mathvariant="normal">4</mn><mi>z</mi><mo>+</mo><mn mathvariant="normal">2</mn><mi>w</mi><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd><mtd><mi>z</mi><mo>+</mo><mi>w</mi><mi>z</mi><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>z</mi><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>z</mi><mo>+</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">5</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><mi>w</mi><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mi>z</mi><mo>+</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">2</mn><mi>w</mi><mo>+</mo><mn mathvariant="normal">2</mn><mi>z</mi><mo>+</mo><mn mathvariant="normal">2</mn><mi>w</mi><mi>z</mi><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>z</mi><mo>+</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424189&type= 152.23066711 13.71600056 <math display="block" id="M427"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">5</mn><mi>w</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">5</mn><mi>w</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn><mo>-</mo><mi>w</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424192&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424191&type= 130.04800415 14.13933372 <math display="block" id="M428"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi>z</mi><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424193&type= 43.43400192 13.29266644 因此， <math id="M429"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424195&type= 135.72065735 4.48733330 。 重复以上过程，得 <math display="block" id="M430"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424198&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424197&type= 142.40933228 4.48733330 其中， <math display="block" id="M431"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mi>w</mi></mtd><mtd><mn mathvariant="normal">4</mn><mi>w</mi><mi>z</mi><mo>-</mo><mn mathvariant="normal">4</mn><mi>z</mi><mo>+</mo><mn mathvariant="normal">2</mn><mi>w</mi><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">4</mn><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><mi>w</mi><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mi>z</mi><mo>+</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">2</mn><mi>w</mi><mo>+</mo><mn mathvariant="normal">4</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424199&type= 96.51999664 13.71600056 ， <math display="block" id="M432"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">8</mn><mi>w</mi><mo>-</mo><mn mathvariant="normal">10</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">6</mn><mi>z</mi><mo>-</mo><mn mathvariant="normal">10</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424202&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424201&type= 119.54933167 13.29266644 由于 <math id="M433"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424203&type= 27.85533333 4.48733330 下面考虑 <math id="M434"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424205&type= 39.79333496 4.48733330 和 <math id="M435"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424207&type= 39.79333496 4.48733330 ，经过计算可得 <math display="block" id="M436"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424210&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424209&type= 73.82933807 4.48733330 其中， <math display="block" id="M437"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">8</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">8</mn><mi>w</mi><mo>+</mo><mn mathvariant="normal">10</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424211&type= 149.18266296 14.13933372 且 <math display="block" id="M438"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424213&type= 73.82933807 4.48733330 <math display="block" id="M439"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">5</mn><mi>w</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">5</mn><mi>w</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424216&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424215&type= 148.16667175 14.13933372 则 <math display="block" id="M440"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424217&type= 147.40466309 9.82133293 令 <math id="M441"><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424220&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424219&type= 101.09200287 4.48733330 ，则 <math display="block" id="M442"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424221&type= 88.05333710 4.48733330 其中， <math display="block" id="M443"><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><mi>w</mi><mo>-</mo><mn mathvariant="normal">6</mn><mi>z</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">2</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">3</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mi>z</mi></mtd></mtr><mtr><mtd><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424223&type= 157.05667114 13.46199894 令 <math display="block" id="M444"><mi mathvariant="bold-italic">A</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mo stretchy="false">(</mo><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mi>w</mi><mo>+</mo><mn mathvariant="normal">6</mn><mi>z</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo stretchy="false">(</mo><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424225&type= 149.09799194 14.90133286 对 <math id="M445"><mi mathvariant="bold-italic">A</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424228&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424227&type= 12.69999981 3.55599999 进行初等变换可得 <math id="M446"><mi mathvariant="bold-italic">A</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424229&type= 48.00600052 4.48733330 其中 <math display="block" id="M447"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mo stretchy="false">(</mo><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mi>w</mi><mo>+</mo><mn mathvariant="normal">6</mn><mi>z</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424231&type= 168.31733704 14.13933372 因此 <math display="block" id="M448"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424233&type= 111.84466553 9.82133293 其中， <math id="M449"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424235&type= 35.22133255 4.48733330 <math id="M450"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424237&type= 48.00600052 4.48733330 。经计算， <math id="M451"><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mi mathvariant="normal">。</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424239&type= 53.76333237 4.48733330 因此， <math id="M452"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424241&type= 12.95399952 3.55599999 等价于矩阵 <math display="block" id="M453"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><msup><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424244&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424243&type= 61.04467010 14.13933372 4　 结 语 1 本文研究了多项式环 <math id="M454"><mi mathvariant="double-struck">R</mi><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424245&type= 13.63133430 3.55599999 上的几类二元多项式矩阵等价问题，其中 <math id="M455"><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424248&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424247&type= 2.45533323 2.70933342 是实数域，得到了一些有意义的结果，并获得了几类二元多项式矩阵 <math id="M456"><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424250&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424249&type= 12.95399952 3.55599999 与其Smith型等价的判别条件。这些条件可以通过计算 <math id="M457"><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424252&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3424251&type= 12.78466702 3.55599999 的行列式及其低一阶子式所生成的理想的既约Gröbner基进行检验。文中我们给出了一个例子对我们的方法进行说明，该分解方法可以在Singular系统中实现，这些都有利于工程中二维系统的简化。 参考文献 BOSE N K . 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Computation of the Smith form for multivariate polynomial matrices using maple [J]. American Journal of Computational Mathematics , 2012 . 2 : 21 - 26 . DOI: 10.4236/ajcm.2012.21003 http://dx.doi.org/10.4236/ajcm.2012.21003 . Computation of the Smith form for multivariate polynomial matrices using maple FROST M G , STOREY C . Equivalence of a matrix over R [ s , z ] with its Smith form [J]. International Journal of Control , 1978 , 28 ( 5 ): 665 - 671 . DOI: 10.1080/00207177808922487 http://dx.doi.org/10.1080/00207177808922487 . Equivalence of a matrix over R[s, z] with its Smith form LEE E , ZAK S . Smith form over R [ z 1 , z 2 ] [J]. IEEE Trans Autom Control , 1983 , 28 ( 1 ): 115 - 118 . DOI: 10.1109/TAC.1983.1103118 http://dx.doi.org/10.1109/TAC.1983.1103118 . Smith form over R[z1,z2] FROST M G , BOUDELLIOUA M S . Some further results concerning matrices with elements in a polynomial ring [J]. International Journal of Control , 1986 , 43 ( 5 ): 1543 - 1555 . 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DOI:10.14188/j.1671-8836.2020.0031 国家自然科学基金（11871207） 2020-02-17 2020-06-24 2020-09-30 武汉大学学报（理学版） 3 66