GKZ systems with irregular singularities

我们得到了以下结果。(1)超几何系统E_<k，n>定义在Grassmanian簇上，具有许多优良的性质。给出了一种算法方法，导出了系统E_<k，n>解的库默型公式. (2)给出了包括GKZ超几何函数在内的多元超几何函数数值计算的一种新方法。我们的方法将一个给定的超几何方程转化为一个线性偏微分方程系统，可以很容易地应用标准的数值分析技术。这种转换是通过使用Grobner基方法完成的。(3)斜率是在不规则奇点处的基本不变量。我们确定了与单项曲线相关的GKZ系统的斜率。

We have obtained the following results.(1) The hypergeometric system E_<k, n> is defined on the Grassmanian variety and it has a lot of nice properties. We gave an algorithmic method to derive Kummer type formulas for solutions of the system E_<k, n>. (2) We gave a new method for numerical evaluation of hypergeometric functions in several variables including GKZ hypergeometric functions. Our method translates a given hypergeometric equations into a systems of linear partial differential equations for which standard techniques of numerical analysis can be easily applied. This translation is done by using the Grobner basis method. (3) Slope is a fundamental invariant at irregular singular points. We determined the slopes for GKZ systems associated to monomial curves.