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一类华罗庚域与复欧氏空间的不相关性
数学 | 更新时间：2023-07-11
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一类华罗庚域与复欧氏空间的不相关性
Non-Relativity of a Class of Hua Domains and Complex Euclidean Spaces
- 武汉大学学报（理学版）
- 作者机构：
1.吉林师范大学 数学与计算机学院，吉林 四平 136000
- 作者简介：
程晓亮，男，教授，现从事多复变与复几何方面的研究。E-mail：chengxiaoliang92@163.com
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- 基金信息：国家自然科学基金(12026420)
DOI：10.14188/j.1671-8836.2022.0013
中图分类号： O174.56纸质出版日期：2023-06-24，
收稿日期：2022-01-07，
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引用本文
程晓亮，马会波.一类华罗庚域与复欧氏空间的不相关性［J］.武汉大学学报（理学版），2023，69（3）：409-416. DOI：10.14188/j.1671-8836.2022.0013.
CHENG Xiaoliang，MA Huibo.Non-Relativity of a Class of Hua Domains and Complex Euclidean Spaces ［J］.J Wuhan Univ （Nat Sci Ed），2023，69（3）：409-416. DOI：10.14188/j.1671-8836.2022.0013（Ch）.
一类华罗庚域与复欧氏空间的不相关性
一类华罗庚域与复欧氏空间的不相关性 Non-Relativity of a Class of Hua Domains and Complex Euclidean Spaces first-author 程晓亮，男，教授，现从事多复变与复几何方面的研究。E-mail：chengxiaoliang92@163.com 程晓亮，男，教授，现从事多复变与复几何方面的研究。E-mail：chengxiaoliang92@163.com chengxiaoliang92@163.com 吉林师范大学 数学与计算机学院，吉林 四平 136000 吉林师范大学 数学与计算机学院，吉林 四平 136000 College of Mathematics and Computer， Jilin Normal University， Siping 136000， Jilin， China College of Mathematics and Computer， Jilin Normal University， Siping 136000， Jilin， China 华罗庚域是Cartan-Egg域的推广，其矩阵形式要复杂得多。本文主要考虑 <math id="M1"><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mrow></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526486&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526484&type= 20.57399940 3.89466691 都是正整数，而 <math id="M2"><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526500&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526491&type= 2.79399991 3.89466691 是正实数时的华罗庚域，证明了这类具有Bergman度量的华罗庚域与具有平坦度量的复欧氏空间是不相关的。 Hua domain is a generalization of Cartan-Egg domain. Its matrix form is much more complex. We mainly consider the Hua domain , in which <math id="M3"><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mrow></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526507&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526504&type= 18.79599953 3.89466691 are the positive integers, <math id="M4"><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526512&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526497&type= 2.79399991 3.89466691 is the positive real number. It is proved that Hua domain equipped with Bergman metrics is not related to complex Euclidean spaces equipped with canonical metrics. 华罗庚域 全纯等距嵌入 Nash函数 Bergman度量 Hua domain holomorphic isometric embedding Nash function Bergman metric 0 引 言 1 全纯等距嵌入是多复变与复几何领域的经典问题。1953年,Calabi [ 1 1 ] 得到了复Kähler流形与Fubini-Study空间之间的局部全纯等距的存在性、唯一性和全局延拓,证明了在典则Kähler度量下具有不同曲率符号的任何两个Fubini-Study空间不能局部全纯等距嵌入。如果两个Kähler流形具有公共的Kähler子流形,则称它们是相关的,否则称为不相关的。1987年,Umehara [ 2 2 ] 推广了Calabi的结果,证明了两个具有不同曲率符号的复空间型不能在各自的诱导度量下具有公共Kähler子流形,并且具有Bergman度量的有界域和具有正则度量的紧型Hermite对称空间是不相关的。2007年,Di Scala和Loi [ 3 3 ] 证明了不同紧型的Hermite对称空间是不相关的。2010年,Di Scala和Loi [ 4 4 ] 证明了具有Bergman度量的有界域与具有Fubini-Study度量的复射影流形是不相关的。2015年,Huang和Yuan [ 5 5 ] 证明了复欧氏空间和非紧型Hermite对称空间也是不相关的。2017年,Cheng,Di Scala和Yuan [ 6 6 ] 证明了具有平坦度量的不定复空间型与具有非平坦度量的不定复空间型是不相关的,并给出了两个有限维Fubini-Study空间是相关的充分条件和必要条件,这是Umehara结论的非平凡推广。同年,Cheng和Niu [ 7 7 ] 证明了具有Bergman度量的Cartan-Hartogs域与具有平坦度量的复欧氏空间是不相关的。2018年,Su,Tang和Tu [ 8 8 ] 证明了具有正则度量的对称多圆盘与具有平坦度量的复欧氏空间是不相关的。2021年,Cheng和Hao [ 9 9 ] 给出了实解析Kähler流形与具有典则度量的复空间型不相关的充分条件。作为直接应用,可以得出具有Bergman度量的最小球、有界齐性域与复欧氏空间均是不相关的。 Cartan对不可约有界对称域在双全纯等价的条件下进行分类,得到了四类对称典型域和两个复维数分别为16和27的例外情况。殷慰萍等 [ 10 10 ] 将四类对称典型域依次推广为Cartan-Hartogs域,Cartan-Egg域,华罗庚域,广义华罗庚域和华罗庚结构。华罗庚域是Cartan-Egg域的推广,本文主要研究 <math id="M5"><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mrow></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526517&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526514&type= 21.84399986 4.57200003 都是正整数,而 <math id="M6"><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528455&type= 3.21733332 4.57200003 是正实数时,这类具有Bergman度量的华罗庚域与具有平坦度量的复欧氏空间的相关性。 下面首先介绍华罗庚域的定义。 1 华罗庚域 1 华罗庚域是在对称典型域和Egg域的基础上建立起来的,其形式定义如下： <math display="block" id="M7"><mi>H</mi><mi>E</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo>,</mo><mi mathvariant="bold-italic">Z</mi><mo>∈</mo><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi>D</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo><</mo><msub><mrow><mi>N</mi></mrow><mrow><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi>D</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">Z</mi><mo>,</mo><mover><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mo>¯</mo></mover></mrow></mfenced><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>r</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526537&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526535&type= 104.56333923 11.26066685 其中 <math id="M8"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><msup><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></munderover></mstyle><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>j</mi><mi>k</mi></mrow></msub></mrow></mfenced></mrow></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526527&type= 25.90800095 9.82133293 , <math id="M9"><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526548&type= 3.72533321 3.80999994 是正整数, <math id="M10"><mi>j</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526555&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526552&type= 21.08200073 3.64066648 , <math id="M11"><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526565&type= 4.48733330 3.72533321 是四类典型域, <math id="M12"><mi>D</mi><mo>=</mo><mi mathvariant="normal">I</mi><mo>,</mo><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mo>,</mo><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mo>,</mo><mi mathvariant="normal">I</mi><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526579&type= 25.06133461 3.21733332 , <math id="M13"><msub><mrow><mi>N</mi></mrow><mrow><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi>D</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">Z</mi><mo>,</mo><mover accent="true"><mi mathvariant="bold-italic">Z</mi><mo>¯</mo></mover></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526603&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526591&type= 15.15533447 4.91066647 表示典型域上的一般模, <math id="M14"><mover accent="true"><mi mathvariant="bold-italic">Z</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526600&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526605&type= 2.79399991 3.30200005 表示 <math id="M15"><mi mathvariant="bold-italic">Z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526630&type= 2.79399991 2.79399991 的共轭, <math id="M16"><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526612&type= 4.40266657 3.04800010 表示 <math id="M17"><mi mathvariant="bold-italic">Z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526630&type= 2.79399991 2.79399991 的转置。 殷慰萍等 [ 10 10 ] 给出了5种情况下华罗庚域的Bergman核函数表达式： 1) <math id="M18"><mi>r</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526638&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526636&type= 27.26266861 4.57200003 时,此时即为Cartan-Hartogs域 [ 11 11 ] 。 2) <math id="M19"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo>⋯</mo><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526653&type= 49.19132996 4.57200003 时,此时即为Cartan-Egg域 [ 12 12 ] 。 3) <math id="M20"><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mrow></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526665&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526645&type= 18.96533203 4.57200003 都是正整数。 4) <math id="M21"><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mrow></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526660&type= 22.52133369 4.57200003 都是正整数,而 <math id="M22"><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528455&type= 3.21733332 4.57200003 是正实数。 5) <math id="M23"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526677&type= 17.86466599 4.57200003 都是正整数。 下面给出 <math id="M24"><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mrow></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526691&type= 22.52133369 4.57200003 都是正整数,而 <math id="M25"><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528455&type= 3.21733332 4.57200003 是正实数时,这类华罗庚域的Bergman核函数表达式 [ 10 10 ] 。 2 一类华罗庚域的Bergman核函数 1 情形1:当 <math id="M26"><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526716&type= 13.03866673 3.72533321 时,华罗庚域为： <math display="block" id="M27"><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo>,</mo><mi mathvariant="bold-italic">Z</mi><mo>∈</mo><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub><mfenced separators="|"><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></mfenced><mfenced open="|" close="" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo><</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">Z</mi><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">Z</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526721&type= 132.24932861 11.26066685 其Bergman核函数表达式为 [ 10 10 ] ： <math display="block" id="M28"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi><mo>,</mo><mi mathvariant="bold-italic">z</mi><mo>;</mo><mover accent="true"><mi mathvariant="bold-italic">w</mi><mo>¯</mo></mover><mo>,</mo><mover accent="true"><mi mathvariant="bold-italic">z</mi><mo>¯</mo></mover></mrow></mfenced><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mi>m</mi><mi>n</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mrow></msup></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><mo>∂</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mo>∂</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msubsup><mo>⋯</mo><mo>∂</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mo>⋯</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mi mathvariant="normal">Γ</mi><mfenced separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></mrow></mrow><mo>⋅</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>l</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msubsup><msubsup><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></msubsup></mrow></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mo>-</mo><mfenced separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msup><mo>⋅</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi><mi>n</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>h</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>h</mi><mi>k</mi></mrow></msub></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>l</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msubsup><msubsup><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></msubsup></mrow></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>h</mi><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></msup></mrow></mfrac></mrow><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">z</mi><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">z</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>-</mo><mfenced separators="|"><mrow><mi>m</mi><mo>+</mo><mi>n</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></mrow></mfenced></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526741&type= 157.73399353 47.24399948 情形2:当 <math id="M29"><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526732&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526744&type= 13.71599960 3.72533321 时,华罗庚域为： <math display="block" id="M30"><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>p</mi><mo>;</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo>,</mo><mi mathvariant="bold-italic">Z</mi><mo>∈</mo><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub><mfenced separators="|"><mrow><mi>p</mi></mrow></mfenced><mfenced open="|" close="" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo><</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">Z</mi><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">Z</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526737&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526747&type= 124.71399689 11.26066685 其Bergman核函数表达式为 [ 13 13 ] ： <math display="block" id="M31"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi><mo>,</mo><mi mathvariant="bold-italic">z</mi><mo>;</mo><mover accent="true"><mi mathvariant="bold-italic">w</mi><mo>¯</mo></mover><mo>,</mo><mover accent="true"><mi mathvariant="bold-italic">z</mi><mo>¯</mo></mover></mrow></mfenced><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>+</mo><mfrac><mrow><mi>p</mi><mfenced separators="|"><mrow><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mfenced separators="|"><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mi>p</mi></mrow></mfenced></mrow></msup></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mfrac><mrow><mi>p</mi><mfenced separators="|"><mrow><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></munderover></mstyle><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mfrac><mrow><mo>∂</mo><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><mi>r</mi></mrow></mmultiscripts></mrow><mrow><mo>∂</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>∂</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>⋯</mo><mo>∂</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow></mfrac><mo>⋅</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><msub><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mo>⋯</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><msub><mrow><mi>j</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>k</mi></mrow></mfenced><mi>j</mi></mrow></mrow></mrow><mo>!</mo><mfrac><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>l</mi></mrow><mrow><msub><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></msubsup></mrow></mrow></mfenced></mrow><mrow><mfrac><mrow><mi>j</mi></mrow><mrow><mi>q</mi><mo>-</mo><mfenced separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></mfrac></mrow></msup></mrow><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>l</mi></mrow><mrow><msub><mrow><mi>j</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msubsup><msubsup><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></msubsup></mrow></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi></mrow></mfrac></mrow></msup><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">z</mi><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">z</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>-</mo><mfenced separators="|"><mrow><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mo>⋯</mo><mo>+</mo><mfrac><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526756&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526749&type= 129.03199768 46.39733505 情形3:当 <math id="M32"><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526769&type= 14.39333248 3.72533321 时,华罗庚域为： <math display="block" id="M33"><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>q</mi><mo>;</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo>,</mo><mi mathvariant="bold-italic">Z</mi><mo>∈</mo><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub><mfenced separators="|"><mrow><mi>q</mi></mrow></mfenced><mfenced open="|" close="" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo><</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">Z</mi><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">Z</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526776&type= 125.56066132 11.26066685 其Bergman核函数表达式为 [ 14 14 ] ： <math display="block" id="M34"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi><mo>,</mo><mi mathvariant="bold-italic">z</mi><mo>;</mo><mover accent="true"><mi mathvariant="bold-italic">w</mi><mo>¯</mo></mover><mo>,</mo><mover accent="true"><mi mathvariant="bold-italic">z</mi><mo>¯</mo></mover></mrow></mfenced><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>+</mo><mfrac><mrow><mi>q</mi><mfenced separators="|"><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><mo>∂</mo><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo>⋯</mo><mo>∂</mo><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo>∂</mo><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac><mo>⋅</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mfrac><mrow><mi>q</mi><mfenced separators="|"><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></munderover></mstyle><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><msub><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mo>⋯</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><msub><mrow><mi>j</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></mrow><mi>j</mi><mo>!</mo><mo>⋅</mo><mfrac><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msup><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><mrow><msub><mrow><mi>j</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msup><msubsup><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></msubsup></mrow></mrow></mfenced></mrow><mrow><mfrac><mrow><mi>j</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac><mo>-</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>k</mi></mrow></mfenced></mrow></msup></mrow><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msup><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><mrow><msub><mrow><mi>j</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msup><msubsup><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></msubsup></mrow></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></msup><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>+</mo><mi mathvariant="bold-italic">z</mi><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">z</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>-</mo><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac><mo>+</mo><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mrow></mfenced></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526763&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526760&type= 140.03866577 46.90533447 上述三种情形中 <math id="M35"><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">z</mi><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">z</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526799&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526765&type= 40.30133438 6.85799980 , <math id="M36"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>l</mi><mi>k</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526788&type= 27.68600082 9.14400005 <math id="M37"><mi>l</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526792&type= 20.65866661 3.21733332 ； <math id="M38"><msub><mrow><mi>p</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>,</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526810&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526795&type= 18.45733452 4.57200003 皆为常数。 情形4:当 <math id="M39"><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">V</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526829&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526812&type= 14.56266594 3.72533321 时,华罗庚域为： <math display="block" id="M40"><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">V</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>;</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo>,</mo><mi mathvariant="bold-italic">Z</mi><mo>∈</mo><msub><mrow><mi mathvariant="fraktur">R</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">V</mi></mrow></msub><mfenced separators="|"><mrow><mi>n</mi></mrow></mfenced><mfenced open="|" close="" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo><</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mn mathvariant="normal">2</mn><mi mathvariant="bold-italic">Z</mi><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">Z</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>+</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="bold-italic">Z</mi><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526831&type= 134.45066833 11.26066685 其Bergman核函数表达式为 [ 15 15 ] ： <math display="block" id="M41"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">V</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi><mo>,</mo><mi mathvariant="bold-italic">z</mi><mo>,</mo><mover accent="true"><mi mathvariant="bold-italic">w</mi><mo>¯</mo></mover><mo>,</mo><mover accent="true"><mi mathvariant="bold-italic">z</mi><mo>¯</mo></mover></mrow></mfenced><mo>=</mo><mfrac><mrow><msubsup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>⋯</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mi>r</mi><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msup></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><mi>r</mi></mrow></msup></mrow><mrow><mo>∂</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>∂</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>⋯</mo><mo>∂</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow></mfrac><mo>⋅</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></munderover></mstyle><mrow><mo>⋯</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></munderover></mstyle><mrow><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow><msub><mrow><mi mathvariant="normal">e</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>r</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mi mathvariant="normal">Γ</mi><mfenced separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>⋅</mo><msup><mrow><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mi>f</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></msup><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced></mrow><mrow><mo>-</mo><mfenced separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msup></mrow><mrow><mo>-</mo><mfenced separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526815&type= 184.82733154 11.09133339 <math display="block" id="M42"><mo>⋅</mo><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mi>f</mi></mrow><mrow><mo>-</mo><mi>r</mi><mo>-</mo><mi>n</mi><mo>+</mo><mfrac><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></msup><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>r</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mo>+</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mi mathvariant="normal">Γ</mi><mfenced separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced><msup><mrow><mi>f</mi></mrow><mrow><mo>-</mo><mi>r</mi><mo>-</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></msup></mrow></mfenced><mo>⋅</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="bold-italic">z</mi><msup><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><mi mathvariant="bold-italic">z</mi><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">z</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>-</mo><mfenced separators="|"><mrow><mi>n</mi><mo>+</mo><mfrac><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mo>⋯</mo><mo>+</mo><mfrac><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526841&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526819&type= 118.11000061 10.66800022 其中 <math id="M43"><mi>f</mi><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><msub><mrow><mi mathvariant="normal">e</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>-</mo><mo>⋯</mo><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><msub><mrow><mi mathvariant="normal">e</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526844&type= 53.84799957 6.94266701 <math id="M44"><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="bold-italic">z</mi><msup><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><mi mathvariant="bold-italic">z</mi><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">z</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow><mrow><mfrac><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac></mrow></msup></mrow></mfrac><mo>,</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>j</mi><mi>k</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526823&type= 71.37400055 15.83266735 <math id="M45"><msub><mrow><mi mathvariant="normal">e</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526827&type= 3.38666677 3.89466691 是全部 <math id="M46"><msub><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526863&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526861&type= 2.53999996 4.57200003 次单位根, <math id="M47"><msub><mrow><mi>α</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526866&type= 41.48666382 4.57200003 , <math id="M48"><mi>b</mi><mo>,</mo><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526854&type= 5.16466665 3.21733332 是常数。 本文借助Huang和Yuan [ 5 5 ] 的方法证明 <math id="M49"><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mrow></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526872&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526873&type= 22.52133369 4.57200003 都是正整数,而 <math id="M50"><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528455&type= 3.21733332 4.57200003 是正实数时这类具有Bergman度量的华罗庚域与具有平坦度量的复欧氏空间是不相关的。 3 一类华罗庚域与复欧氏空间不相关 1 Nash函数是一类具有代数性的特殊的全纯函数,它是复空间中非零多项式的根。全纯多项式和全纯有理函数都是Nash函数 [ 9 9 ] 。本文借助Nash函数的性质得到具有Bergman度量的一类华罗庚域与具有平坦度量的复欧氏空间不相关的充分条件。 下面给出域扩张的相关定义和Nash函数的性质。 若域 <math id="M51"><mi mathvariant="normal">ℑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528382&type= 2.45533323 2.79399991 包含域 <math id="M52"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526984&type= 2.79399991 2.79399991 作为它的子域,则称 <math id="M53"><mi mathvariant="normal">ℑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528382&type= 2.45533323 2.79399991 是 <math id="M54"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526984&type= 2.79399991 2.79399991 的一个扩域。若 <math id="M55"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527257&type= 2.20133328 2.79399991 是 <math id="M56"><mi mathvariant="normal">ℑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528382&type= 2.45533323 2.79399991 的子集,则称 <math id="M57"><mi>R</mi><mfenced separators="|"><mrow><mi>S</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526927&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526925&type= 8.72066593 3.97933316 为 <math id="M58"><mi mathvariant="normal">ℑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528382&type= 2.45533323 2.79399991 的包含 <math id="M59"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526984&type= 2.79399991 2.79399991 与 <math id="M60"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527257&type= 2.20133328 2.79399991 的最小子域。设 <math id="M61"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526956&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526953&type= 24.29933357 4.14866638 ,多项式 <math id="M62"><mi>P</mi><mo>∈</mo><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526945&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526942&type= 9.05933285 3.13266683 ,若 <math id="M63"><mi>P</mi><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526959&type= 31.58066559 3.97933316 当且仅当 <math id="M64"><mi>P</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526961&type= 9.22866726 2.79399991 ,则称 <math id="M65"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527257&type= 2.20133328 2.79399991 在 <math id="M66"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45526984&type= 2.79399991 2.79399991 上是代数无关的。扩域的极大代数无关子集称为超越基,其元素个数称为超越次数 [ 16 16 ] 。 引理1 [ 16 16 ] 设 <math id="M67"><mi mathvariant="bold-italic">ℚ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528374&type= 2.87866688 3.30200005 是有理函数域, <math id="M68"><mi mathvariant="normal">ℑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528382&type= 2.45533323 2.79399991 是 <math id="M69"><mi mathvariant="bold-italic">ℚ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528374&type= 2.87866688 3.30200005 的一个扩域,若 <math id="M70"><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527016&type= 2.53999996 2.79399991 是 <math id="M71"><mi mathvariant="normal">ℑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528382&type= 2.45533323 2.79399991 的超越基,则 <math id="M72"><mi mathvariant="normal">ℑ</mi><mo>\</mo><mi mathvariant="bold-italic">ℚ</mi><mfenced separators="|"><mrow><mi>B</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527039&type= 13.37733269 3.97933316 中的每一个元素都是Nash函数,且 <math id="M73"><mi mathvariant="normal">ℑ</mi><mo>\</mo><mi mathvariant="bold-italic">ℚ</mi><mfenced separators="|"><mrow><mi>B</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527039&type= 13.37733269 3.97933316 的超越次数为0。 引理2 [ 9 9 ] 设 <math id="M74"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527782&type= 2.96333337 2.79399991 是 <math id="M75"><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>n</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527968&type= 3.72533321 2.96333337 的连通开集， <math id="M76"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527782&type= 2.96333337 2.79399991 上的Nash函数族记作 <math id="M77"><mi>N</mi><mfenced separators="|"><mrow><mi>D</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527071&type= 10.07533360 3.97933316 ,设 <math id="M78"><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>N</mi><mfenced separators="|"><mrow><mi>D</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527074&type= 18.37266731 4.06400013 ,则有以下性质： 1） <math id="M79"><mi>f</mi><mo>±</mo><mi>g</mi><mo>,</mo><mi>f</mi><mi>g</mi><mo>,</mo><mfrac><mrow><mi>f</mi></mrow><mrow><mi>g</mi></mrow></mfrac><mo>∈</mo><mi>N</mi><mfenced separators="|"><mrow><mi>D</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527091&type= 31.58066559 8.80533314 ,且对于 <math id="M80"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528216&type= 14.56266594 3.30200005 ,都有 <math id="M81"><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo>∈</mo><mi>N</mi><mfenced separators="|"><mrow><mi>D</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527116&type= 18.62666702 8.38199997 。 2） 固定 <math id="M82"><mfenced separators="|"><mrow><msubsup><mrow><mi>z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup><mo>,</mo><mo>⋯</mo><mo>,</mo><msubsup><mrow><mi>z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527108&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527119&type= 19.55800056 4.48733330 ,则 <math id="M83"><mi>f</mi><mfenced separators="|"><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi>z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup><mo>,</mo><mo>⋯</mo><mo>,</mo><msubsup><mrow><mi>z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527128&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527110&type= 36.83000183 4.48733330 是关于 <math id="M84"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527132&type= 12.78466606 4.14866638 的Nash函数。 引理3 设 <math id="M85"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527782&type= 2.96333337 2.79399991 是 <math id="M86"><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>n</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527968&type= 3.72533321 2.96333337 的连通开集, <math id="M87"><mi>f</mi><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527149&type= 8.12800026 4.06400013 是 <math id="M88"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527782&type= 2.96333337 2.79399991 上的Nash函数, <math id="M89"><mi>Q</mi><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>p</mi><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow><mrow><mi>q</mi><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527157&type= 21.50533485 9.48266602 是 <math id="M90"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527782&type= 2.96333337 2.79399991 上的有理式,则 <math id="M91"><mi>Q</mi><mfenced separators="|"><mrow><mi>f</mi><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>p</mi><mfenced separators="|"><mrow><mi>f</mi><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>q</mi><mfenced separators="|"><mrow><mi>f</mi><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527198&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527196&type= 34.20533371 11.09133339 是 <math id="M92"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527782&type= 2.96333337 2.79399991 上的Nash函数。 引理4 [ 5 5 ] 设 <math id="M93"><mi>V</mi><mo>⊂</mo><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>κ</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527205&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527204&type= 11.43000031 3.13266683 是连通开集, <math id="M94"><mi>ξ</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>κ</mi></mrow></msub></mrow></mfenced><mo>∈</mo><mi>V</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>κ</mi></mrow></msub></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>S</mi></mrow></msub><mfenced close="" separators="|"><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo></mrow></mfenced><mfenced open="" separators="|"><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>κ</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527206&type= 79.16333008 4.74133301 和 <math id="M95"><mi>H</mi><mfenced separators="|"><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>κ</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527232&type= 19.04999924 4.74133301 是 <math id="M96"><mi>V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527287&type= 3.21733332 2.87866688 上的全纯Nash函数,其中 <math id="M97"><mi>κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527254&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527252&type= 1.94733346 2.79399991 和 <math id="M98"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527257&type= 2.20133328 2.79399991 是正整数。如果对于某些实数 <math id="M99"><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527264&type= 10.49866676 4.57200003 <math id="M100"><msub><mrow><mi>μ</mi></mrow><mrow><mi>S</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527272&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527268&type= 3.47133350 4.57200003 有 <math id="M101"><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mi>H</mi><mfenced separators="|"><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>κ</mi></mrow></msub></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>S</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>κ</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527278&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527275&type= 59.68999863 8.55133343 ,那么 <math id="M102"><mi>H</mi><mfenced separators="|"><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>κ</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527293&type= 20.15066719 4.74133301 是 <math id="M103"><mi>V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527287&type= 3.21733332 2.87866688 上的常值函数。 设 <math id="M104"><mi>H</mi><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527585&type= 5.67266655 2.79399991 为四类华罗庚域, <math id="M105"><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow/><mrow><mi>D</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527324&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527307&type= 6.85799980 3.47133350 为华罗庚域的子集。由于 <math id="M106"><mi>H</mi><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527585&type= 5.67266655 2.79399991 是圆型域,则对于任意的 <math id="M107"><mi>θ</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527322&type= 8.29733276 3.13266683 , <math id="M108"><mi>ξ</mi><mo>,</mo><mi>η</mi><mo>∈</mo><mi>H</mi><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527340&type= 14.73200035 3.64066648 , <math id="M109"><mi>H</mi><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527585&type= 5.67266655 2.79399991 的Bergman核函数满足 <math id="M110"><mi>K</mi><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mroot><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></mroot><mi>θ</mi></mrow></msup><mi>ξ</mi><mo>,</mo><mover><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mroot><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></mroot><mi>θ</mi></mrow></msup><mi>η</mi></mrow><mo>¯</mo></mover></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527358&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527352&type= 32.00400162 6.94266701 <math id="M111"><mi>K</mi><mfenced separators="|"><mrow><mi>ξ</mi><mo>,</mo><mover accent="true"><mi>η</mi><mo>¯</mo></mover></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527372&type= 12.19200039 4.82600021 ,即 <math id="M112"><mi>K</mi><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mroot><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></mroot><mi>θ</mi></mrow></msup><mi>ξ</mi><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527365&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527362&type= 23.19866753 5.67266655 <math id="M113"><mi>K</mi><mfenced separators="|"><mrow><mi>ξ</mi><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527383&type= 11.34533310 4.74133301 。因为 <math id="M114"><mi>K</mi><mfenced separators="|"><mrow><mi>ξ</mi><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527392&type= 11.34533310 4.74133301 是正常数,所以 <math id="M115"><mi>K</mi><mfenced separators="|"><mrow><mi>ξ</mi><mo>,</mo><mover accent="true"><mi>η</mi><mo>¯</mo></mover></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527403&type= 12.19200039 4.82600021 不包含 <math id="M116"><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527462&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527460&type= 1.86266661 3.64066648 的任何非常值全纯项, <math id="M117"><msub><mrow><mi>K</mi></mrow><mrow><mi>H</mi><mi>E</mi></mrow></msub><mfenced separators="|"><mrow><mi>ξ</mi><mo>,</mo><mover accent="true"><mi>η</mi><mo>¯</mo></mover></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527429&type= 14.73200035 4.82600021 也不包含 <math id="M118"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527450&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527436&type= 1.86266661 3.64066648 的任何非常值共轭全纯项,因此 <math id="M119"><msub><mrow><mi>K</mi></mrow><mrow><mi>H</mi><mi>E</mi></mrow></msub><mfenced separators="|"><mrow><mi>ξ</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mo>¯</mo></mover></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527440&type= 14.64733315 5.24933338 不包含 <math id="M120"><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527462&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527460&type= 1.86266661 3.64066648 的非常值的多重调和项,正规化后得 <math id="M121"><msub><mrow><mi>K</mi></mrow><mrow><mi>H</mi><mi>E</mi></mrow></msub><mfenced separators="|"><mrow><mi>ξ</mi><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527476&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527475&type= 21.50533485 4.74133301 。 华罗庚域的Bergman度量记为 <math id="M122"><msub><mrow><mi>ω</mi></mrow><mrow><mi>H</mi><mi>E</mi></mrow></msub><mo>=</mo><mroot><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></mroot><mo>∂</mo><mover accent="true"><mrow><mo>∂</mo></mrow><mo>¯</mo></mover><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>H</mi><mi>E</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527468&type= 33.61266708 4.74133301 。设 <math id="M123"><mi>D</mi><mo>⊂</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527481&type= 6.94266701 2.79399991 <math id="M124"><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>κ</mi></mrow></msup><mfenced separators="|"><mrow><mi>κ</mi><mo>≤</mo><mi>n</mi><mo>,</mo><mi>κ</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">N</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527486&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527483&type= 26.50066566 4.48733330 是连通开集, <math id="M125"><msub><mrow><mi>ω</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527494&type= 4.14866638 3.72533321 是 <math id="M126"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527782&type= 2.96333337 2.79399991 上的Kähler度量。不失一般性,假设 <math id="M127"><mn mathvariant="normal">0</mn><mo>∈</mo><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527513&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527511&type= 8.55133343 3.13266683 。下面将证明不存在全纯等距嵌入既能满足 <math id="M128"><mi>F</mi><mo>:</mo><mfenced separators="|"><mrow><mi>D</mi><mo>,</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>D</mi></mrow></msub></mrow></mfenced><mo>→</mo><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msub><mrow><mi>ω</mi></mrow><mrow><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527534&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527515&type= 33.86666870 4.57200003 又能使 <math id="M129"><mi>L</mi><mo>:</mo><mfenced separators="|"><mrow><mi>D</mi><mo>,</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>D</mi></mrow></msub></mrow></mfenced><mo>→</mo><mfenced separators="|"><mrow><mi>H</mi><mi>E</mi><mo>,</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>H</mi><mi>E</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527538&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527521&type= 36.99933624 3.97933316 成立。 定理5 设 <math id="M130"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527526&type= 2.96333337 2.79399991 是 <math id="M131"><mi mathvariant="double-struck">C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527531&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527530&type= 2.70933342 2.70933342 的连通开集, <math id="M132"><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>n</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527552&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527548&type= 15.07066536 3.04800010 和 <math id="M133"><mi>L</mi><mo>=</mo><mfenced separators="|"><mrow><mi>G</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>=</mo><mfenced close="" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>,</mo></mrow></mfenced><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>:</mo><mi>D</mi><mo>→</mo><mi>H</mi><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527564&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527554&type= 108.79666138 4.74133301 都是全纯映射且满足 <math id="M134"><mi>L</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527557&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527567&type= 14.64733315 3.97933316 。如果在 <math id="M135"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527782&type= 2.96333337 2.79399991 上有 <math id="M136"><msup><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><msub><mrow><mi>ω</mi></mrow><mrow><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><msub><mrow><mi>ω</mi></mrow><mrow><mi>H</mi><mi>E</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527560&type= 22.77533340 4.74133301 ,那么 <math id="M137"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528447&type= 2.62466669 2.79399991 一定是常值映射。 对于四类华罗庚域 <math id="M138"><mi>H</mi><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527585&type= 5.67266655 2.79399991 ，定理的证明方法类似,本文主要对情形一时的华罗庚域 <math id="M139"><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527608&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527589&type= 6.51933336 3.72533321 加以详细证明。 证 设映射 <math id="M140"><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>n</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527604&type= 13.20800018 3.04800010 和 <math id="M141"><mi>L</mi><mo>=</mo><mfenced separators="|"><mrow><mi>G</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfenced close="" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>,</mo></mrow></mfenced><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>:</mo><mi>D</mi><mo>→</mo><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527616&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527615&type= 89.32333374 4.74133301 , <math id="M142"><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527630&type= 34.88266373 4.65666676 满足 <math id="M143"><msup><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><msub><mrow><mi>ω</mi></mrow><mrow><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><msub><mrow><mi>ω</mi></mrow><mrow><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527637&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527634&type= 23.28333473 3.80999994 。利用反证法,假设 <math id="M144"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528447&type= 2.62466669 2.79399991 不是常值映射。不失一般性,可以假设 <math id="M145"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527782&type= 2.96333337 2.79399991 是包含原点的单连通域且 <math id="M146"><mi>F</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527662&type= 14.47799969 3.55599999 。根据 <math id="M147"><msup><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><msub><mrow><mi>ω</mi></mrow><mrow><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><msub><mrow><mi>ω</mi></mrow><mrow><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527664&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527644&type= 23.28333473 3.80999994 ,可得对于任意的 <math id="M148"><mi mathvariant="bold-italic">z</mi><mo>∈</mo><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528077&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528076&type= 8.55133343 3.13266683 都有： <math display="block" id="M149"><mo>∂</mo><mover><mrow><mo>∂</mo></mrow><mo>¯</mo></mover><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow></mfenced><mo>=</mo><mo>∂</mo><mover accent="true"><mrow><mo>∂</mo></mrow><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow><mo>¯</mo></mover><mo>,</mo><mover><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow><mo>¯</mo></mover></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527705&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527679&type= 94.40332794 9.65200043 其中 <math id="M151"><msub><mrow><mi>K</mi></mrow><mrow><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi>ξ</mi><mo>,</mo><mover accent="true"><mi>η</mi><mo>¯</mo></mover></mrow></mfenced><mo>=</mo><msub><mo>∑</mo><mrow><mi>l</mi></mrow></msub><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><mfenced separators="|"><mrow><mi>ξ</mi></mrow></mfenced><mover><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi>η</mi></mrow></mfenced></mrow><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527680&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527708&type= 42.67200089 5.58799982 是 <math id="M152"><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527714&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527681&type= 6.51933336 3.72533321 上的Bergman核函数, <math id="M153"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi>ξ</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527728&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527717&type= 11.26066685 5.50333309 是 <math id="M154"><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527730&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527720&type= 6.51933336 3.72533321 上平方可积的全纯函数的一组标准正交基。通过比较 <math id="M155"><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527724&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527722&type= 4.57200003 3.97933316 式中 <math id="M156"><mi mathvariant="bold-italic">z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528057&type= 1.94733346 2.79399991 的全纯项和共轭全纯项,去掉 <math id="M157"><mo>∂</mo><mover accent="true"><mrow><mo>∂</mo></mrow><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527739&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527738&type= 3.38666677 3.30200005 得： <math display="block" id="M158"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>=</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow><mo>¯</mo></mover><mo>,</mo><mover><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow><mo>¯</mo></mover></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527744&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527742&type= 79.33266449 8.55133343 通过极化, <math id="M160"><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527748&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527746&type= 4.74133301 3.97933316 式等价于： <math display="block" id="M161"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><msub><mrow><mover accent="true"><mi>f</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow><mo>=</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>,</mo><msub><mrow><mover accent="true"><mi>H</mi><mo>¯</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527760&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527750&type= 86.78334045 8.55133343 ， <math display="block" id="M162"><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>∈</mo><mi>D</mi><mo>×</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">j</mi><mfenced separators="|"><mrow><mi>D</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527762&type= 33.27399826 4.14866638 其中, <math id="M164"><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">j</mi><mfenced separators="|"><mrow><mi>D</mi></mrow></mfenced><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mfenced open="" close="|" separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi></mrow></mfenced><mover accent="true"><mi mathvariant="bold-italic">z</mi><mo>¯</mo></mover><mo>∈</mo><mi>D</mi></mrow></mfenced><mo>,</mo><msub><mrow><mover accent="true"><mi>f</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>=</mo><mover><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mover accent="true"><mi mathvariant="bold-italic">w</mi><mo>¯</mo></mover></mrow></mfenced></mrow><mo>¯</mo></mover><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>=</mo><mover><mrow><mi>G</mi><mfenced separators="|"><mrow><mover accent="true"><mi mathvariant="bold-italic">w</mi><mo>¯</mo></mover></mrow></mfenced></mrow><mo>¯</mo></mover><mo>,</mo><msub><mrow><mover accent="true"><mi>H</mi><mo>¯</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>=</mo><mover><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mover accent="true"><mi mathvariant="bold-italic">w</mi><mo>¯</mo></mover></mrow></mfenced></mrow><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527767&type= 118.02532196 4.99533367 。 下面将分三步来证明定理。 步骤I:对于任意的 <math id="M165"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528216&type= 14.56266594 3.30200005 ,可证 <math id="M166"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528218&type= 7.95866632 4.06400013 是 <math id="M167"><mi>L</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527776&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527791&type= 36.32199860 4.74133301 的全纯有理式。即证明存在全纯有理式 <math id="M168"><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527778&type= 32.51200104 3.97933316 ,使得 <math id="M169"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi>L</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527796&type= 31.24200058 4.74133301 。必要时可以将 <math id="M170"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527782&type= 2.96333337 2.79399991 向原点收缩。 等式 <math id="M171"><mfenced separators="|"><mrow><mn mathvariant="normal">3</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528135&type= 4.74133301 3.97933316 两侧同时对 <math id="M172"><mi mathvariant="bold-italic">w</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527932&type= 3.13266683 2.79399991 微分,并令 <math id="M173"><mi mathvariant="bold-italic">w</mi><mo>→</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527979&type= 9.82133293 2.79399991 可得： <math display="block" id="M174"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">w</mi></mrow></mfrac><msub><mrow><mover accent="true"><mi>f</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>A</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><msub><mrow><mover accent="true"><mi>H</mi><mo>¯</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">w</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>A</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><msub><mrow><mover accent="true"><mi>H</mi><mo>¯</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>λ</mi></mrow><mrow><mi>B</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mo>∂</mo><mi>B</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">w</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527807&type= 202.60733032 11.43000031 其中， <math display="block" id="M176"><mi>A</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><msub><mrow><mover accent="true"><mi>H</mi><mo>¯</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><mo>∂</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mo>∂</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msubsup><mo>⋯</mo><mo>∂</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mo>⋯</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mi mathvariant="normal">Γ</mi><mfenced separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>l</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msubsup><msubsup><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></msubsup></mrow></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mo>-</mo><mfenced separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msup></mrow></mrow></mrow><mo>⋅</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi><mi>n</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>h</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>h</mi><mi>k</mi></mrow></msub></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>l</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msubsup><msubsup><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></msubsup></mrow></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>h</mi><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></msup></mrow></mfrac></mrow><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527810&type= 215.39199829 27.26266861 <math display="block" id="M177"><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><msub><mrow><mover accent="true"><mi>H</mi><mo>¯</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mover accent="true"><mi>G</mi><mo>¯</mo></mover><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></msup><mo>,</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>l</mi><mi>k</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac><mo>,</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>i</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></msup><mo>,</mo><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>r</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527813&type= 151.80732727 10.07533360 <math display="block" id="M178"><mi>B</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mover accent="true"><mi>G</mi><mo>¯</mo></mover><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527817&type= 69.25733185 6.18066692 <math display="block" id="M179"><mi>λ</mi><mo>=</mo><mo>-</mo><mfenced separators="|"><mrow><mi>m</mi><mo>+</mo><mi>n</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfrac></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527836&type= 36.83000183 9.82133293 由于 <math id="M180"><mi>L</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527840&type= 14.64733315 3.97933316 ,即 <math id="M181"><mi>G</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527819&type= 33.61266708 3.97933316 ,因此 <math id="M182"><mi>B</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></mrow></mfenced><mo>≡</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527849&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527847&type= 31.91933441 4.99533367 。因为 <math id="M183"><mi>B</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527822&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527821&type= 26.50066566 4.99533367 是关于 <math id="M184"><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527864&type= 8.63599968 3.97933316 和 <math id="M185"><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527826&type= 9.82133293 4.14866638 的多项式,因此 <math id="M186"><msub><mrow><mfenced open="" close="|" separators="|"><mrow><mfrac><mrow><mo>∂</mo><mi>B</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">w</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527830&type= 35.72933197 12.36133385 是关于 <math id="M187"><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527864&type= 8.63599968 3.97933316 的全纯多项式, <math id="M188"><msub><mrow><mfenced open="" close="|" separators="|"><mrow><mfrac><mrow><mi>λ</mi></mrow><mrow><mi>B</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mo>∂</mo><mi>B</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">w</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527860&type= 65.02400208 13.12333298 是关于 <math id="M189"><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527864&type= 8.63599968 3.97933316 的全纯有理式,也是关于 <math id="M190"><mi>L</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527898&type= 36.32199860 3.97933316 的Nash函数。由于 <math id="M191"><mi>A</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><msub><mrow><mover accent="true"><mi>H</mi><mo>¯</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced><mfenced open="|" close="" separators="|"><mrow><msub><mrow/><mrow><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527873&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527871&type= 57.91199875 4.99533367 和 <math id="M192"><msub><mrow><mfenced open="" close="|" separators="|"><mrow><mfrac><mrow><mo>∂</mo><mi>A</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><msub><mrow><mover accent="true"><mi>H</mi><mo>¯</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">w</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527875&type= 59.77466202 12.36133385 都是关于 <math id="M193"><mi>L</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527877&type= 31.15733528 3.97933316 的全纯有理式,因此 <math id="M194"><mfrac><mrow><mo>∂</mo><mi>A</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><msub><mrow><mover accent="true"><mi>H</mi><mo>¯</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">w</mi></mrow></mfrac><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527881&type= 53.00133133 9.22866726 <math id="M195"><msub><mrow><mfenced open="" close="|" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>A</mi><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><msub><mrow><mover accent="true"><mi>H</mi><mo>¯</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac></mrow></mfenced></mrow><mrow><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527897&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527894&type= 55.88000107 13.12333298 是关于 <math id="M196"><mi>L</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527898&type= 36.32199860 3.97933316 的全纯有理式,也是关于 <math id="M197"><mi>L</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527884&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527902&type= 36.32199860 3.97933316 的Nash函数。 令 <math id="M198"><msup><mrow><mi>D</mi></mrow><mrow><mi>δ</mi></mrow></msup><mo>=</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mi>δ</mi></mrow></msup></mrow><mrow><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>δ</mi></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527905&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527886&type= 15.66333294 8.38199997 ,当 <math id="M199"><mi>δ</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527906&type= 8.63599968 2.79399991 时, <math id="M200"><mfenced separators="|"><mrow><mn mathvariant="normal">4</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528207&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528206&type= 4.91066647 3.97933316 式可以改写成： <math display="block" id="M201"><mi>F</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>⋅</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced close="" separators="|"><mrow><mi mathvariant="bold-italic">w</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>,</mo></mrow></mfenced><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mo>⋯</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527889&type= 98.12866211 4.91066647 其中 <math id="M203"><mi>F</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527913&type= 32.59666443 4.82600021 。令 <math id="M204"><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528035&type= 9.82133293 2.79399991 ,则 <math id="M205"><msub><mrow><mi>ϕ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced close="" separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>,</mo></mrow></mfenced><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527919&type= 66.20932770 4.74133301 是 <math display="block" id="M206"><mi>L</mi><mo>=</mo><mfenced close="" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo></mrow></mfenced><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527927&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527921&type= 100.66867065 4.74133301 的全纯有理式。等式 <math id="M207"><mfenced separators="|"><mrow><mn mathvariant="normal">5</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528224&type= 4.74133301 3.97933316 两侧对 <math id="M208"><mi mathvariant="bold-italic">w</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527932&type= 3.13266683 2.79399991 进行微分,当 <math id="M209"><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528035&type= 9.82133293 2.79399991 , <math id="M210"><mi>δ</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527939&type= 8.63599968 2.79399991 为任意阶时： <math display="block" id="M211"><mi>F</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>⋅</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>δ</mi></mrow></msup><mfenced separators="|"><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>δ</mi></mrow></msub><mfenced close="" separators="|"><mrow><mi mathvariant="bold-italic">w</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>,</mo></mrow></mfenced><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527942&type= 99.99134064 4.91066647 则 <math id="M213"><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>δ</mi></mrow></msub><mfenced close="" separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>,</mo></mrow></mfenced><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527946&type= 66.20932770 4.74133301 是 <math display="block" id="M214"><mi>L</mi><mo>=</mo><mfenced close="" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo></mrow></mfenced><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527955&type= 100.66867065 4.74133301 的全纯有理式。 设 <math id="M215"><mi>L</mi><mo>=</mo><mi mathvariant="normal">S</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">a</mi><msub><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="double-struck">C</mi></mrow></msub><msub><mrow><mfenced open="{" close="}" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>δ</mi></mrow></msup><mfenced separators="|"><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced><mfenced open="|" close="" separators="|"><mrow><msub><mrow/><mrow><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>δ</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527957&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527952&type= 49.52999878 6.43466663 是 <math id="M216"><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>n</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527968&type= 3.72533321 2.96333337 的一个向量子空间, <math id="M217"><msubsup><mrow><mfenced open="{" close="}" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mfenced separators="|"><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced><mfenced open="|" close="" separators="|"><mrow><msub><mrow/><mrow><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>τ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527973&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527972&type= 32.68133163 7.36600018 是 <math id="M218"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528002&type= 2.45533323 2.79399991 的一组基。当 <math id="M219"><mi mathvariant="bold-italic">w</mi><mo>→</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527979&type= 9.82133293 2.79399991 时, <math id="M220"><mover accent="true"><mi>F</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527983&type= 9.56733322 4.06400013 的泰勒展式为 <math id="M221"><mover accent="true"><mi>F</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>δ</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></munder><mrow><mfrac><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>δ</mi></mrow></msup><mfenced separators="|"><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover></mrow></mfenced><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></mrow><mrow><mi>δ</mi><mo>!</mo></mrow></mfrac></mrow><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>δ</mi></mrow></msup><mo>∈</mo><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527988&type= 47.07466888 10.15999985 。用 <math id="M222"><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527992&type= 3.64066648 3.64066648 表示 <math id="M223"><mi mathvariant="double-struck">C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527996&type= 2.70933342 2.70933342 中以0为圆心的开圆盘,则有 <math id="M224"><mover accent="true"><mi>F</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>⊂</mo><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527997&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527962&type= 17.18733406 4.06400013 。 设 <math id="M225"><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo>-</mo><mi>τ</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528001&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527999&type= 30.47999954 4.74133301 是 <math id="M226"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528002&type= 2.45533323 2.79399991 的欧氏正交补的一组基,则有: <math display="block" id="M227"><mi>F</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527965&type= 20.06599998 4.06400013 当 <math id="M229"><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528035&type= 9.82133293 2.79399991 , <math id="M230"><mi>δ</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi mathvariant="normal">τ</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528009&type= 23.19866753 3.97933316 时将 <math id="M231"><mfenced separators="|"><mrow><mn mathvariant="normal">6</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45527967&type= 4.65666676 3.97933316 式和 <math id="M232"><mfenced separators="|"><mrow><mn mathvariant="normal">7</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528031&type= 4.74133301 3.97933316 式整理成下列矩阵形式： <math display="block" id="M233"><mfenced separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><mtable><mtr><mtd><msup><mrow><mi>D</mi></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mfenced separators="|"><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced><mfenced open="|" close="" separators="|"><mrow><msub><mrow/><mrow><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msup><mrow><mi>D</mi></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi>τ</mi></mrow></msub></mrow></msup><mfenced separators="|"><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced><mfenced open="|" close="" separators="|"><mrow><msub><mrow/><mrow><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>n</mi><mo>-</mo><mi>τ</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>ϕ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced close="" separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>,</mo></mrow></mfenced><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>τ</mi></mrow></msub><mfenced close="" separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>,</mo></mrow></mfenced><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528015&type= 127.93133545 34.45933533 当 <math id="M235"><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528035&type= 9.82133293 2.79399991 时,等式 <math id="M236"><mfenced separators="|"><mrow><mn mathvariant="normal">8</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528018&type= 4.82600021 3.97933316 左侧的系数矩阵 <math id="M237"><msup><mrow><mfenced separators="|"><mrow><mtable><mtr><mtd><msup><mrow><mi>D</mi></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mfenced separators="|"><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced><mfenced open="|" close="" separators="|"><mrow><msub><mrow/><mrow><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msup><mrow><mi>D</mi></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi>τ</mi></mrow></msub></mrow></msup><mfenced separators="|"><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced><mfenced open="|" close="" separators="|"><mrow><msub><mrow/><mrow><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>n</mi><mo>-</mo><mi>τ</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528047&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528045&type= 30.90333366 34.45933533 是非奇异阵。等式 <math id="M238"><mfenced separators="|"><mrow><mn mathvariant="normal">8</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528018&type= 4.82600021 3.97933316 右侧矩阵的每一行元素是 <math id="M239"><mi>L</mi><mo>=</mo><mfenced separators="|"><mrow><mi>G</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528021&type= 19.55800056 4.06400013 的全纯有理式。应用克莱姆法则解上述线性方程组,可知 <math id="M240"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528218&type= 7.95866632 4.06400013 是关于 <math id="M241"><mfenced close="" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo></mrow></mfenced><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528026&type= 93.38733673 4.74133301 的全纯有理式,也是Nash函数。 设 <math id="M242"><mi mathvariant="bold-italic">ℚ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528374&type= 2.87866688 3.30200005 是由变量 <math id="M243"><mi mathvariant="bold-italic">z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528057&type= 1.94733346 2.79399991 形成的有理函数域,其中 <math id="M244"><mi mathvariant="bold-italic">z</mi><mo>∈</mo><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528077&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528076&type= 8.55133343 3.13266683 。考虑域的扩张 <math id="M245"><mi mathvariant="normal">ℑ</mi><mo>=</mo><mi mathvariant="bold-italic">ℚ</mi><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528060&type= 61.55266953 4.82600021 <math id="M246"><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528084&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528082&type= 12.27666664 3.89466691 ,即 <math id="M247"><mi mathvariant="normal">ℑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528382&type= 2.45533323 2.79399991 是包含 <math id="M248"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528124&type= 59.18199921 4.74133301 和Nash函数的有理函数域的最小子域,记 <math id="M249"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528061&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528093&type= 1.26999998 2.79399991 是扩域 <math id="M250"><mi mathvariant="normal">ℑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528382&type= 2.45533323 2.79399991 的超越次数。 步骤Ⅱ:假设 <math id="M251"><mi>l</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528097&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528065&type= 7.87400007 2.79399991 ,则 <math id="M252"><mi mathvariant="normal">ℑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528382&type= 2.45533323 2.79399991 的超越基是空集,即 <math id="M253"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528124&type= 59.18199921 4.74133301 中每个元素都是Nash函数,根据步骤I, <math id="M254"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528218&type= 7.95866632 4.06400013 可以写成如下形式： <math display="block" id="M255"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>=</mo><mover accent="true"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>^</mo></mover><mfenced close="" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo></mrow></mfenced><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528105&type= 108.71199799 4.91066647 其中 <math id="M256"><mover accent="true"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>^</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">X</mi></mrow></mfenced><mo>=</mo><mover accent="true"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>^</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>m</mi><mi>n</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528110&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528108&type= 48.68333054 10.49866676 是全纯有理函数,则 <math id="M257"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528218&type= 7.95866632 4.06400013 是全纯Nash函数。考虑下列等式 <math display="block" id="M258"><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><msub><mrow><mover accent="true"><mi>f</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mrow></mfenced><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>,</mo><msub><mrow><mover accent="true"><mi>H</mi><mo>¯</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528120&type= 90.59333038 9.65200043 该式等价于式 <math id="M260"><mfenced separators="|"><mrow><mn mathvariant="normal">3</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528135&type= 4.74133301 3.97933316 。由引理4知, <math id="M261"><mi>F</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528123&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528137&type= 26.75466537 4.82600021 是常值映射。 步骤Ⅲ:假设 <math id="M262"><mi>l</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528141&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528139&type= 7.87400007 2.79399991 ,即 <math id="M263"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528124&type= 59.18199921 4.74133301 中存在一些元素不是Nash函数。简记 <math id="M264"><mi>L</mi><mo>=</mo><mfenced separators="|"><mrow><mi>G</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528126&type= 49.86866760 10.49866676 。设 <math id="M265"><mi mathvariant="normal">℘</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced><mfenced separators="|"><mrow><mi>l</mi><mo>≤</mo><mi>m</mi><mi>n</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528146&type= 62.14533615 9.65200043 是 <math id="M266"><mi mathvariant="normal">ℑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528382&type= 2.45533323 2.79399991 的极大代数无关子集,即 <math id="M267"><mi mathvariant="normal">℘</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528129&type= 2.28600001 3.21733332 是 <math id="M268"><mi mathvariant="normal">ℑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528382&type= 2.45533323 2.79399991 的超越基, <math id="M269"><mi mathvariant="normal">ℑ</mi><mo>\</mo><mi mathvariant="bold-italic">ℚ</mi><mfenced separators="|"><mrow><mi mathvariant="normal">℘</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528157&type= 12.95400047 3.97933316 是由Nash函数生成的集合,所以 <math id="M270"><mi mathvariant="normal">ℑ</mi><mo>\</mo><mi mathvariant="bold-italic">ℚ</mi><mfenced separators="|"><mrow><mi mathvariant="normal">℘</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528157&type= 12.95400047 3.97933316 的超越次数为0。 存在连通开集 <math id="M271"><mi>U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528171&type= 3.30200005 2.79399991 ,使得对于任意的 <math id="M272"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528159&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528174&type= 1.60866666 3.64066648 有 <math id="M273"><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∉</mo><mi mathvariant="normal">℘</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528178&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528161&type= 8.80533314 4.65666676 ，不失一般性，假设 <math id="M274"><mn mathvariant="normal">0</mn><mo>∈</mo><mi>U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528180&type= 8.97466660 3.13266683 。设 <math id="M275"><mover accent="true"><mi>U</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528222&type= 3.30200005 3.72533321 是 <math id="M276"><mfenced open="{" close="}" separators="|"><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfenced><mfenced open="|" close="" separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>∈</mo><mi>U</mi></mrow></mfenced></mrow></mfenced><mo>⊆</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528193&type= 37.84600067 5.50333309 <math id="M277"><mi mathvariant="double-struck">C</mi><mo>×</mo><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mrow><mi>l</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528196&type= 10.92199993 3.04800010 的邻域,则存在 <math id="M278"><mover accent="true"><mi>U</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528222&type= 3.30200005 3.72533321 上的Nash函数 <math id="M279"><msubsup><mrow><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mover accent="true"><mi>g</mi><mo>^</mo></mover></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>j</mi><mo>=</mo><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi><mi>n</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></msubsup><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528192&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528190&type= 51.22333527 8.21266651 ,使得对于 <math id="M280"><mo>∀</mo><mi mathvariant="bold-italic">z</mi><mo>∈</mo><mi>U</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528209&type= 11.85333347 3.38666677 有 <math id="M281"><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>=</mo><msub><mrow><mover accent="true"><mi>g</mi><mo>^</mo></mover></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfenced><mo>,</mo><mi>j</mi><mo>=</mo><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>m</mi><mi>n</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528202&type= 72.81333160 8.55133343 。 根据 <math id="M282"><mfenced separators="|"><mrow><mn mathvariant="normal">4</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528207&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528206&type= 4.91066647 3.97933316 式和 <math id="M283"><mfenced separators="|"><mrow><mn mathvariant="normal">5</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528224&type= 4.74133301 3.97933316 式知, <math id="M284"><msub><mrow><mfenced open="" close="|" separators="|"><mrow><mfrac><mrow><mo>∂</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi>G</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>,</mo><msub><mrow><mover accent="true"><mi>H</mi><mo>¯</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">w</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528227&type= 68.32599640 12.36133385 是关于 <math id="M285"><mfenced separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528220&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528233&type= 24.46866608 10.49866676 的全纯有理式,由于对于任意的 <math id="M286"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528216&type= 14.56266594 3.30200005 , <math id="M287"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528218&type= 7.95866632 4.06400013 是关于 <math id="M288"><mfenced separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi><mi>n</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528220&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528233&type= 24.46866608 10.49866676 的全纯有理式,因此存在 <math id="M289"><mover accent="true"><mi>U</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528222&type= 3.30200005 3.72533321 上的全纯Nash函数 <math id="M290"><mover accent="true"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>^</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528240&type= 23.02933502 4.65666676 <math id="M291"><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>∈</mo><mi>U</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528244&type= 12.19200039 3.97933316 ,使得 <math id="M292"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>=</mo><msub><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528248&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528236&type= 45.97399902 4.99533367 。令： <math display="block" id="M293"><mi mathvariant="bold-italic">Ψ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced><mover><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>-</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mi>K</mi><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub><mfenced separators="|"><mrow><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mover accent="true"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo>^</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><mover><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo></mrow></mfenced><mo>,</mo><msub><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msup></mrow><mrow><mo>∂</mo><msup><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac><mi mathvariant="bold-italic">Ψ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">w</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528250&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528253&type= 165.77734375 8.89000034 其中 <math id="M294"><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>∈</mo><mover accent="true"><mi>U</mi><mo>^</mo></mover><mo>×</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">j</mi><mfenced separators="|"><mrow><mi>U</mi></mrow></mfenced><mo>,</mo><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528270&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528252&type= 60.19800186 4.57200003 <math id="M295"><mi>X</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528255&type= 6.77333355 2.79399991 <math id="M296"><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo> </mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528273&type= 18.20333290 3.97933316 。下证 <math id="M297"><mi mathvariant="bold-italic">Ψ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">w</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528258&type= 18.37266731 3.97933316 在 <math id="M298"><mover accent="true"><mi>U</mi><mo>^</mo></mover><mo>×</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">j</mi><mfenced separators="|"><mrow><mi>U</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528278&type= 20.65866661 4.57200003 上恒为零。 对于任意的 <math id="M299"><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced><mo>≈</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mo>,</mo><mi mathvariant="bold-italic">w</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528281&type= 33.86666870 4.14866638 ,易证 <math id="M300"><msub><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">w</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528264&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528262&type= 18.62666702 3.97933316 对于任意的 <math id="M301"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528303&type= 1.69333339 2.79399991 都是 <math id="M302"><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528287&type= 9.82133293 3.97933316 的Nash函数。假设 <math id="M303"><msub><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528301&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528289&type= 24.04533195 3.97933316 ,则存在某些 <math id="M304"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528303&type= 1.69333339 2.79399991 依赖于 <math id="M305"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528442&type= 3.13266683 2.79399991 ,即当 <math id="M306"><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="bold-italic">Ψ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">w</mi></mrow></mfenced></mrow><mrow><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528310&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528308&type= 28.19400024 8.55133343 时,存在 <math id="M307"><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>→</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528296&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528294&type= 11.00666618 3.72533321 ,使得 <math id="M308"><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="bold-italic">Ψ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528313&type= 29.37933159 8.55133343 。由 <math id="M309"><mi mathvariant="bold-italic">Ψ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">w</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528316&type= 17.94933319 3.97933316 的定义可知 <math id="M310"><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="bold-italic">Ψ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528318&type= 22.69066620 8.55133343 是 <math id="M311"><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528335&type= 9.82133293 3.97933316 的Nash函数,则存在 <math id="M312"><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">t</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced><msup><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528326&type= 68.41066742 3.97933316 其中， <math id="M313"><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced><mo>≠</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528331&type= 22.35199928 3.97933316 <math id="M314"><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528344&type= 12.78466606 3.97933316 是 <math id="M315"><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528335&type= 9.82133293 3.97933316 有理式,使得 <math id="M316"><mi>P</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="bold-italic">Ψ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac></mrow></mfenced><mo>≡</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528341&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528339&type= 44.02666473 9.90600014 。对 <math id="M317"><mo>∀</mo><mi mathvariant="bold-italic">z</mi><mo>∈</mo><mi>U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528358&type= 11.17599964 3.13266683 , <math id="M318"><mi mathvariant="bold-italic">Ψ</mi><mfenced close="" separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced><mo>,</mo><mfenced open="" separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>≡</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528350&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528361&type= 46.05867004 4.74133301 ,因此 <math display="block" id="M319"><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>≡</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528365&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528363&type= 52.83200073 8.46666718 <math display="block" id="M320"><mi>P</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528367&type= 122.00466156 9.65200043 <math display="block" id="M321"><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced><mo>≡</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528370&type= 139.86933899 8.46666718 所以 <math id="M322"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528354&type= 26.41600037 4.74133301 在 <math id="M323"><mi mathvariant="bold-italic">ℚ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528374&type= 2.87866688 3.30200005 上是代数相关的,与假设 <math id="M324"><mi mathvariant="normal">℘</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528378&type= 34.54399872 4.74133301 是 <math id="M325"><mi mathvariant="normal">ℑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528382&type= 2.45533323 2.79399991 的极大代数无关子集矛盾,故 <math id="M326"><msub><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mo>≡</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528356&type= 22.26733208 4.14866638 。令 <math id="M327"><mi>Φ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528387&type= 21.59000015 4.14866638 ,则有 <math id="M328"><mi mathvariant="bold-italic">Ψ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528391&type= 22.94466591 4.14866638 ,即: <math display="block" id="M329"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mover accent="true"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>^</mo></mover></mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mover><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>¯</mo></mover><mo stretchy="false">(</mo><mi mathvariant="bold-italic">w</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo>…</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mover accent="true"><mi>g</mi><mo>^</mo></mover></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mover accent="true"><mi>g</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">j</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">w</mi><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528405&type= 100.58399963 8.55133343 其中 <math id="M331"><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">w</mi></mrow></mfenced><mo>∈</mo><mover accent="true"><mi>U</mi><mo>^</mo></mover><mo>×</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528394&type= 38.69266510 4.57200003 。 若 <math id="M332"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mover accent="true"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>^</mo></mover></mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mover><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>¯</mo></mover><mo stretchy="false">(</mo><mi mathvariant="bold-italic">w</mi><mo stretchy="false">)</mo><mo>≡</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528397&type= 34.88266373 8.55133343 ,令 <math id="M333"><mi mathvariant="bold-italic">w</mi><mo>=</mo><mi mathvariant="bold-italic">z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528412&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528410&type= 9.82133293 2.79399991 , <math id="M334"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo stretchy="false">)</mo></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mover accent="true"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>^</mo></mover></mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">z</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">)</mo><mover><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">z</mi></mrow></mfenced><mo>≡</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528415&type= 63.24599838 8.55133343 ,则 <math id="M335"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528447&type= 2.62466669 2.79399991 是常值映射,与假设矛盾。 若存在 <math id="M336"><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>∈</mo><mi>U</mi><mo>×</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">j</mi><mfenced separators="|"><mrow><mi>U</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528420&type= 36.40666580 3.97933316 ,使得 <math id="M337"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mover accent="true"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>^</mo></mover></mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mover><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>¯</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528436&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528434&type= 36.83000183 8.55133343 ,则 <math id="M338"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mover accent="true"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>^</mo></mover></mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mover><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>¯</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528438&type= 30.39533424 8.55133343 是 <math id="M339"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528442&type= 3.13266683 2.79399991 上非常值的Nash函数。由 <math id="M340"><mo stretchy="false">(</mo><mn mathvariant="normal">10</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528425&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528444&type= 6.60400009 3.55599999 式得: <math display="block" id="M341"><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mover accent="true"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>^</mo></mover></mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mover><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>¯</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>H</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">X</mi><mo>,</mo><msub><mrow><mover accent="true"><mi>g</mi><mo>^</mo></mover></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mover accent="true"><mi>g</mi><mo>^</mo></mover></mrow><mrow><mi>m</mi><mi>n</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mover accent="true"><mi>g</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced open="" separators="|"><mrow><msub><mrow><mover accent="true"><mi>g</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">n</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi mathvariant="normal">i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">r</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi mathvariant="normal">N</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub></mrow></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528429&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528427&type= 149.09800720 11.26066685 由引理4可知, <math id="M342"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528447&type= 2.62466669 2.79399991 是常值映射,与假设矛盾。综上所述，定理5得证。 在上述情形下可得到如下推论。 推论6 一类具有Bergman度量的华罗庚域与具有平坦度量的复欧氏空间不存在包含零点的公共Kähler子流形,即一类具有Bergman度量的华罗庚域与具有平坦度量的复欧氏空间不相关。 4 结 语 1 殷慰萍等 [ 10 10 ] 给出了5种情况下华罗庚域的Bergman核函数表达式,其中包含Cartan-Hartogs域和Cartan-Egg域两种特殊情形,还包含其Bergman核函数是多维超几何函数的情形。本文考虑 <math id="M343"><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mrow></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528453&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528451&type= 22.52133369 4.57200003 都是正整数,而 <math id="M344"><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45528455&type= 3.21733332 4.57200003 是正实数时的华罗庚域,证明了一类具有Bergman度量的华罗庚域与具有平坦度量的复欧氏空间是不相关的。 参考文献 CALABI E . Isometric imbedding of complex manifolds ［J］. The Annals of Mathematics ， 1953 ， 58 （ 1 ）： 1 - 23 . DOI： 10.2307/1969817 http://dx.doi.org/10.2307/1969817 . Isometric imbedding of complex manifolds UMEHARA M . Kähler submanifolds of complex space forms ［J］. Tokyo Journal Mathematics ， 1987 ， 10 （ 1 ）： 203 - 214 . DOI: 10.3836/tjm/1270141804 http://dx.doi.org/10.3836/tjm/1270141804 . Kähler submanifolds of complex space forms DI SCALA A ， LOI A . Kähler maps of Hermitian symmetric spaces into complex space forms ［J］. Geom Dedic ， 2007 ， 125 （ 1 ）： 103 - 113 . DOI： 10.1007/s10711-007-9142-z http://dx.doi.org/10.1007/s10711-007-9142-z . Kähler maps of Hermitian symmetric spaces into complex space forms DI SCALA A ， LOI A . Kähler manifolds and their relatives ［J］. Ann Sc Norm Super Pisa Cl Sci ， 2010 ， 9 （ 3 ）： 495 - 501 . Kähler manifolds and their relatives HUANG X J ， YUAN Y . Submanifolds of Hermitian symmetric spaces ［J］. Analysis and Geometry ， 2015 ， 127 （ 3 ）： 197 - 206 . DOI： 10.1007/978-3-319-17443-3_10 http://dx.doi.org/10.1007/978-3-319-17443-3_10 . Submanifolds of Hermitian symmetric spaces CHENG X L ， DI SCALA A J ， YUAN Y . Kähler submanifolds and the Umehara algebra ［J］. International Journal of Mathematics ， 2017 ， 28 （ 4 ）： 179 - 195 . DOI： 10.1142/s0129167x17500276 http://dx.doi.org/10.1142/s0129167x17500276 . Kähler submanifolds and the Umehara algebra CHENG X L ， NIU Y Y . Submanifolds of Cartan-Hartogs domains and complex Euclidean spaces ［J］. Journal of Mathematical Analysis and Applications ， 2017 ， 452 ： 1262 - 1268 . DOI： 10.1016/j.jmaa.2017.03.043 http://dx.doi.org/10.1016/j.jmaa.2017.03.043 . Submanifolds of Cartan-Hartogs domains and complex Euclidean spaces SU G C ， TANG Y Y ， TU Z H . Kähler submanifolds of the symmetrized polydisc ［J］. Comptes Rendus Mathematique ， 2018 ， 356 ： 387 - 394 . DOI： 10.1016/j.crma.2018.03.009 http://dx.doi.org/10.1016/j.crma.2018.03.009 . Kähler submanifolds of the symmetrized polydisc CHENG X L ， HAO Y H . On the non-existence of common submanifolds of Kähler manifolds and complex space forms ［J］. Annals of Global Analysis and Geometry ， 2021 ， 60 ： 167 - 180 . DOI： 10.1007/s10455-021-09776-3 http://dx.doi.org/10.1007/s10455-021-09776-3 . On the non-existence of common submanifolds of Kähler manifolds and complex space forms 殷慰萍 ， 王安 ， 赵振刚 ， 等 . 华罗庚域的Bergman核函数 ［J］. 中国科学（A辑） ， 2001 ， 31 （ 6 ）： 503 - 516 . 华罗庚域的Bergman核函数 YIN W P ， WANG A ， ZHAO Z G ， et al . The Bergman kernel function of Hua domains ［J］. Science in China（Ser A） ， 2001 ， 31 （ 6 ）： 503 - 516 （Ch） . The Bergman kernel function of Hua domains YIN W P . The Bergman kernels on Cartan-Hartogs domain ［J］. Chinese Science Bulletin ， 1999 ， 44 （ 21 ）： 1947 - 1951 . DOI： 10.1007/BF02887114 http://dx.doi.org/10.1007/BF02887114 . The Bergman kernels on Cartan-Hartogs domain 殷慰萍 ， 王男 ， 赵玲 ， 等 . 四类Cartan-Egg域的Bergman核函数 ［J］. 首都师范大学学报（自然科学版） ， 2001 ， 22 （ 2 ）： 1 - 13 . DOI： 10.3969/j.issn.1004-9398.2001.02.001 http://dx.doi.org/10.3969/j.issn.1004-9398.2001.02.001 . 四类Cartan-Egg域的Bergman核函数 YIN W P ， WANG N ， ZHAO L ， et al . The Bergman kernel functions on four classes of Cartan-Egg domains ［J］. Journal of Capital Normal University ， 2001 ， 22 （ 2 ）： 1 - 13 . DOI： 10.3969/j.issn.1004-9398.2001.02.001（Ch http://dx.doi.org/10.3969/j.issn.1004-9398.2001.02.001（Ch ）. The Bergman kernel functions on four classes of Cartan-Egg domains 殷慰萍 ， 赵振刚 . 第二类华罗庚域的Bergman核函数的计算 ［J］. 厦门大学学报（自然科学版） ， 2001 ， 40 （ 6 ）： 1184 - 1190 . DOI： 10.3321/j.issn： 0438-0479.2001.06.002 http://dx.doi.org/10.3321/j.issn：0438-0479.2001.06.002 . 第二类华罗庚域的Bergman核函数的计算 YIN W P ， ZHAO Z G . Calculation of the Bergman kernel functions for the second class of Hua domain ［J］. Journal of Xiamen University ， 2001 ， 40 （ 6 ）： 1184 - 1190 . DOI： 10.3321/j.issn： 0438-0479.2001.06.002（Ch http://dx.doi.org/10.3321/j.issn：0438-0479.2001.06.002（Ch ）. Calculation of the Bergman kernel functions for the second class of Hua domain 殷慰萍 ， 赵晓霞 . 第三类华罗庚域的Bergman核函数 ［J］. 数学年刊A辑（中文版） ， 2003 ， 24 （ 1 ）： 81 - 90 . DOI： 10.3321/j.issn： 1000-8134.2003.01.010 http://dx.doi.org/10.3321/j.issn：1000-8134.2003.01.010 . 第三类华罗庚域的Bergman核函数 YIN W P ， ZHAO X X . The Bergman kernel functions on Hua domain of the third class ［J］. Chinese Annals of Mathematics ， Series A， 2003 ， 24 （ 1 ）： 81 - 90 . DOI： 10.3321/j.issn：1000-8134.2003.01.010（Ch http://dx.doi.org/10.3321/j.issn：1000-8134.2003.01.010（Ch ）. The Bergman kernel functions on Hua domain of the third class 殷慰萍 ， 管冰辛 . 第四类华罗庚域的Bergman核函数 ［J］. 数学学报 ， 2003 ， 46 （ 1 ）： 85 - 94 . DOI： 10.3321/j.issn： 0583-1431.2003.01.012 http://dx.doi.org/10.3321/j.issn：0583-1431.2003.01.012 . 第四类华罗庚域的Bergman核函数 YIN W P ， GUAN B X . The Bergman kernel functions on Hua domain of the fourth class ［J］. Acta Mathematica Sinica ， 2003 ， 46 （ 1 ）： 85 - 94 . DOI： 10.3321/j.issn： 0583-1431.2003.01.012（Ch http://dx.doi.org/10.3321/j.issn：0583-1431.2003.01.012（Ch ）. The Bergman kernel functions on Hua domain of the fourth class 李文威 . 代数学方法（第一卷）：基础架构 ［M］. 北京 ： 高等教育出版社 ， 2019 . 代数学方法（第一卷）：基础架构 LI W W . Algebraic Methods（Volume I）：Infrastructure ［M］. Beijing ： Higher Education Press ， 2019 （Ch）. Algebraic Methods（Volume I）：Infrastructure 10.14188/j.1671-8836.2022.0013 O174.56 A 1671-8836（2023）03-0409-08 程晓亮，马会波.一类华罗庚域与复欧氏空间的不相关性［J］.武汉大学学报（理学版），2023，69（3）：409-416. DOI： 10.14188/j.1671-8836.2022.0013 http://dx.doi.org/10.14188/j.1671-8836.2022.0013 . 程晓亮，马会波.一类华罗庚域与复欧氏空间的不相关性［J］.武汉大学学报（理学版），2023，69（3）：409-416. DOI：10.14188/j.1671-8836.2022.0013. CHENG Xiaoliang，MA Huibo.Non-Relativity of a Class of Hua Domains and Complex Euclidean Spaces ［J］. J Wuhan Univ （ Nat Sci Ed ），2023，69（3）：409-416. DOI： 10.14188/j.1671-8836.2022.0013（Ch http://dx.doi.org/10.14188/j.1671-8836.2022.0013(Ch ）. CHENG Xiaoliang，MA Huibo.Non-Relativity of a Class of Hua Domains and Complex Euclidean Spaces ［J］.J Wuhan Univ （Nat Sci Ed），2023，69（3）：409-416. DOI：10.14188/j.1671-8836.2022.0013（Ch）. 国家自然科学基金（12026420） 2022-01-07 2023-06-24 2023-07-11 39574499 武汉大学学报（理学版） 3 69
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