Grobner asymptotic expansion for regular holononic systems

建立了一种分析正则全息系统无穷远点渐近行为的方法。一阶近似由初始系统决定。对于GKZ超几何系统，其相应的系统本质上是单项理想，因此可以用组合方法对它们进行分析。（1）Bayer和Sturmfels最近证明了单项理想可以通过图论和阶梯来研究。该方法可用于研究GKZ超几何系统。（2）我们确定渐近性态的方法将为研究有理解和整体解奠定基础。某些超几何系统是Painleve系统的特解。Painleve系统和超几何系统在有理解、同构问题和整体解等方面的研究将会产生令人兴奋的互动。(3)It是确定非正则奇点的渐近性态的一个重要问题。然而，它仍然是开放的。

We establish a method to analyze asymptotic behavior of regular holonopic systems at infinity. The first order approximation is governed by the initial system. In case of GKZ hypergeometric systems, the correspondiny systems are essentially monomial ideals and hence can be analyzed by nsins combinatorial methods for them.Our research project develops to the following new directions.(1)Bayer and Sturmfels showed recently that monomial ideals can be studied through graph theory an stairs. Their method can be applied to study GKZ hypergeometric systems.(2)Our method to determine asymptotic behavior will be a foundation to study the rational solutions and the global solutions. Some hypergeometric systems are special solutions of Painleve systems. There will be an exciting interaction between studies on Painleve systems and hypergeometric systems on the rational solutions, isomorphism problem and the global solutions.(3)It is an important problem to determine the asymptotic behaviors around an irregular singular point. It, however, is still open.