GKZ systems with irregular singularities

具有不规则奇点的 GKZ 系统

• 批准号: 13640172
• 负责人: TAKAYAMA Nobuki
• 金额: $ 1.79万
• 依托单位: KOBE UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640172/
• 关键词: A-hypergeometric system; A-hypergeometric functions; irregular singularities; mathematical software; computer algebra; D-modules; numerical analysis; Kummer type identities; 自由分解

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.

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。

项目成果

M.Maekawa: "The Design and Implementation of OpenXM-RFC 100 and 101"Proceedings of the Fifth Asian Symposium. 102-111 (2001)

M.Maekawa：“OpenXM-RFC 100 和 101 的设计与实现”第五届亚洲研讨会论文集。

T. Oaku & Y. Shiraki & N. Takayama: "Algorithms for D-modules and Numerical Analysis"Proceedings of ASCM 2003. in press. (2003)

T·奥库

M. Joswig & N. Takayama (Editors): "Algebra, Geometry, and Software Systems"Springer. in press (2003)

M·乔斯维格

F. Castro-Jimenez & N. Takayama: "Slopes of a Hypergeometric System Associated to a Monomial Curve"Transaction of the American Mathematical Society. in press. (2003)

F·卡斯特罗-希门尼斯

H.Majima: "Quadratic relations for confluent hypergeometric functions"Tohoku Mathematical Journal. 52. 489-514 (2000)

H.Majima：“汇合超几何函数的二次关系”东北数学杂志。