Algebra, Geometry, Analysis in non-linear equations

本研究的中心主题之一是无限维的微分伽罗瓦理论。首席研究员在1996年提出了一个这样的理论。对这一理论感兴趣的马格朗日自己提出了一个一般微分伽罗瓦理论。Umemura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扩展L/K时，构造该类扩展的一类伽罗瓦闭包，该伽罗瓦群p是该伽罗瓦闭包的无限小自同构群。另一方面，Malgrange's Galois roupois附着在一个品种的叶子上。也就是说，当给定一个品种上的叶形F时，伽罗瓦群是最小的代数李群，其李代数包含到叶形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著的。On可以表示在绝对情况下L/K，即基域K是L的常数域的子域，这两个定义是等价的。另一方面，该项目的研究者之一，神户大学的M. Noumi研究了最一般的painlevel方程或称为椭圆painlevel方程的主方程。他证明了椭圆型Painleve方程的Riccati解会出现超椭圆型几何函数。这个结果是这个研究领域中最引人注目的结果之一。

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.