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Algebra, Geometry, Analysis in non-linear equations

代数、几何、非线性方程分析

基本信息

批准号：
15340004
负责人：
UMEMURA Hiroshi
金额：
$ 8.58万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

One of the central theme of this research is the differential Galois theory of infinite dimension. The head investigator presented in 1996 one such theory. Malgrange interested in this theory himself proposed a general differential Galois theory. Umemura's differential Galois theory is a Galois theory of differential field extension. Namely when a differential field extension L/K is given, we construct a kind of Galois closure of the extension, The Galois group p is th infinitesimal automorphism group of this Galois closure. On the other hand, Malgrange's Galois roupois is attached to a foliation on a variety. Namely when a foliation F on a variety is given, the Galois groupoid is the smallest algebraic Lie groupoid whose Lie algebra contains the tangent vectors to the foliation F.These two definitions are seemingly different but specialists observed that they coincide in Examples. In recent three years the development in this direction was remarkable. On can show in the absolute case L/K, by which we mean the base field K is a subfield in the constant field of L, these two definitions are equivalent. The proof is done through the universalOn the other hand one of the investigator of this project, M. Noumi at Kobe University studied the most general Painleve equation or the Master equation called the elliptic Painleve equation. He showed among other things that as Riccati solutions to the elliptic Painleve equation, there appear hyperelliptic geometric functions. This result is one of the most remarkable results in this field of research.

这项研究的中心主题之一是无限维微分伽罗瓦理论。首席研究员于 1996 年提出了这样一种理论。对该理论感兴趣的马尔格朗日本人提出了广义微分伽罗瓦理论。梅村的微分伽罗瓦理论是微分场扩展的伽罗瓦理论。即当给定一个微分域外延L/K时，我们构造该外延的一种伽罗瓦闭包，伽罗瓦群p就是这个伽罗瓦闭包的第一个无穷小自同构群。另一方面，马尔格朗日的 Galois roupois 附着在品种的叶子上。也就是说，当给定簇上的叶状 F 时，伽罗瓦群群是最小的代数李群群，其李代数包含叶状 F 的切向量。这两个定义看似不同，但专家观察到它们在示例中是一致的。近三年来，这个方向的发展是显着的。 On可以表示在绝对情况下L/K，即基域K是L的常域中的子域，这两个定义是等价的。证明是通过通用性完成的。另一方面，该项目的研究者之一，神户大学的M. Noumi研究了最通用的Painleve方程或称为椭圆Painleve方程的Master方程。除其他外，他还表明，作为椭圆 Painleve 方程的 Riccati 解，出现了超椭圆几何函数。这一结果是该研究领域最引人注目的成果之一。