1999 Fiscal Year Final Research Report Summary
P-adic integration and Hartogs-Stawski's theorem
| Project/Area Number |
10640045
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | RIKKYO UNIVERSITY |
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Principal Investigator |
ENDOU Mikihiko RIKKYO UNIV. COLLEGE OF SCIENCE, PROFESSOR, 理学部, 教授 (40062616)
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| Co-Investigator(Kenkyū-buntansha) |
SATOU Fumihiro RIKKYO UNIV. COLLEGE OF SCIENCE, PROFESSOR, 理学部, 教授 (20120884)
KIDA Yuuji RIKKYO UNIV. COLLEGE OF SCIENCE, PROFESSOR, 理学部, 教授 (30113939)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Hartogs / Stawski / p-adic analytic function / analyticity / p-adic integration / p-adic complex number / analytic domain |
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| Research Abstract |
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D2
then 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D2
where 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.
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