Algebra, Geometry, Analysis in non-linear equations

这项研究的中心主题之一是无限维微分伽罗瓦理论。首席调查员在1996年提出了一个这样的理论。马尔格兰奇对这一理论很感兴趣，他自己提出了一个广义的伽罗瓦微分理论。内村的微分伽罗瓦理论是微分场扩张的伽罗瓦理论。即当给定一个微分域扩张L/K时，我们构造了这种扩张的一种Galois闭包，Galois群p是这个Galois闭包的第n个无穷小自同构群。另一方面，Malgrange的Galois roupois附着在一个品种的叶上。也就是说，当给定一个簇上的叶F时，Galois群胚是最小的代数李群胚，它的李代数包含叶F的切向量。这两个定义似乎不同，但专家们观察到它们在例子中是重合的。近三年来，这一方向的发展令人瞩目。在绝对情形下，可以表示L/K，即基场K是L常数域中的一个子域，这两个定义是等价的。另一方面，这个项目的研究人员之一神户大学的M.Noumi研究了最一般的Painleve方程或称为椭圆型Painleve方程的Master方程。他还证明了，作为椭圆型Painleve方程的Riccati解，出现了超椭圆几何函数。这一结果是这一研究领域最显著的成果之一。

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.