P-adic integration and Hartogs-Stawski's theorem

P-adic 积分和 Hartogs-Stawski 定理

• 批准号: 10640045
• 负责人: ENDOU Mikihiko
• 金额: $ 1.02万
• 依托单位: RIKKYO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640045/
• 关键词: Hartogs; Stawski; p-adic analytic function; analyticity; p-adic integration; p-adic complex number; analytic domain; p-進解析関数; 解析性; p-進積分; p-進複素数体; 解析領域; p進積分; 非アルキメデス的Hartogsの定理; Stawskiの定理; 線形外測度; analytic element

The purpose of this reseach is to give a correct proof of Stawski's Theorem (non-archimedean version of Hartogs' Theorem) . The first step of the resarch is to put in order Stawski's outer linear measure theory. This was done by a good advice of Alain Escassut.A correct proof of Hartogs-Stawski's theorem was given when the underling field K is a complete, but not locally compact subfield of the p-adic complex field CィイD2pィエD2. If the value group |KィイD1xィエD1| is discrete, the following revised version of Stawski's theorem holds.Theorem 1 If a function f (x) = f(xィイD21ィエD2, xィイD22ィエD2, ..., xィイD2nィエD2) is analytic for each variable on the domain|xィイD21ィエD2|≦ RィイD21ィエD2,|xィイD22ィエD2|≦ RィイD22ィエD2,..., |xィイD2nィエD2|≦ RィイD2nィエD2then the function f(x) is an analytic function in the whole variables on the domain|xィイD21ィエD2|< RィイD21ィエD2,|xィイD22ィエD2|≦ ィイD2qィエD2RィイD22ィエD2,..., |xィイD2nィエD2|≦ ィイD2qィエD2RィイD2nィエD2where q = |π|< 1 (πis a prime element of K) .By symmetry we slightly extended the domain of analyticity of the functions. We also see that the Theorem holds if the field K is a general non-archimedean field which is complete, but not locally compact.

这项研究的目的是给出了Stawski定理的正确证明（Hartogs'定理的非架构版本）。 RESARCH的第一步是将Stawski的外部线性测量理论列入顺序。这是通过Alain Escassut的良好建议来完成的。当底座k是完整但不是局部紧凑的子场CII D2PIE D2时，给出了Hartogs-Stawski定理的正确证明。如果值组| kii d1xie d1 |是离散的，以下是Stawski定理的修订版。理论1如果函数f（x）= f（xie d21e d2，xie d22e d2，...，xie d2nie d2）是对域上每个变量的分析性的，则在域上进行分析。 d2，...，| xie d2nie d2 |≦rii d2nie d2在函数f（x）中是整个域上的分析函数| xy d21e d21e d2 | <ri d21e d2，| xy d22e d22e d2 | ri d2qe d2qe d2qe d2qe d2qe d2qe d2qe d2ri d2 | d2ri d2其中q = |π| <1（π是k的主要元素）。通过对称性，我们稍微扩展了功能的分析域。我们还看到定理是否符合字段K是否是完整但不是局部紧凑的一般非架构的字段。

项目成果

遠藤幹彦: "Hartogs-Stawski's theorem in diserete valued fields" Lecture notes in pure and applied mathematics. (1998)

Mikihiko Endo：“离散值域中的 Hartogs-Stawski 定理”纯粹数学和应用数学讲义（1998）。

Endo M.: "Hartogs-Stawski's theorem in discrete valued fields."Lecture note in pure and applied mathematics.. 209. 77-96 (1999)

Endo M.：“离散值域中的 Hartogs-Stawski 定理。”纯粹数学和应用数学讲义.. 209. 77-96 (1999)

森本,木田,山崎: "円分数の素因数分解(その4)"上智大学数学講究録. 42. (1999)

森本、木田、山崎：《圆分数的素因数分解（第4部分）》上智大学数学讲座记录42。（1999）

SATO, Fumihiro & HIRONAKA, Yumiko: "Local densities of representations of quadratic forms over p-adic integers : the non-dyadic case"Journal of Number Theory.. 19.

佐藤文宏

森本、木田、山崎: "円分数の素因数分解(その4)"上智大学数学講究録. 42. (1999)

森本、木田、山崎：《日元分数的质因数分解（第4部分）》上智大学数学讲座记录42。（1999）