Grobner Deformations of Hypergeometric Differential Equations

by Saito, Mutsumi; Sturmfels, Bernd; Takayama, Nobuki

• ISBN13: 9783540660651
• ISBN10: 3540660658
• Format: Hardcover
• Copyright: 1999-12-01
• Publisher: Springer Verlag

Summary

In recent years, new algorithms for dealing with rings of differential operators have been discovered and implemented. A main tool is the theory of Gröbner bases, which is reexamined here from the point of view of geometric deformations. Perturbation techniques have a long tradition in analysis; Gröbner deformations of left ideals in the Weyl algebra are the algebraic analogue to classical perturbation techniques. The algorithmic methods introduced here are particularly useful for studying the systems of multidimensional hypergeometric PDEs introduced by Gelfand, Kapranov and Zelevinsky. The Gröbner deformation of these GKZ hypergeometric systems reduces problems concerning hypergeometric functions to questions about commutative monomial ideals, and leads to an unexpected interplay between analysis and combinatorics. This book contains a number of original research results on holonomic systems and hypergeometric functions, and raises many open problems for future research in this area.

Table of Contents

Basic Notions	1	(50)
Grobner Bases in the Weyl Algebra	2	(9)
Weight Vectors and Non-term Orders	11	(5)
The Gauss Hypergeometric Equation	16	(12)
Holonomic Systems	28	(12)
Integrals of Products of Linear Forms	40	(11)
Solving Regular Holonomic Systems	51	(52)
The Grobner Fan	51	(11)
Semi-Continuity of the Holonomic Rank	62	(4)
Torus Action and Frobenius Ideals	66	(11)
Regular Holonomic Systems	77	(11)
Canonical Series Solutions	88	(8)
Construction of Series Solutions	96	(7)
Hypergeometric Series	103	(48)
GKZ-Hypergeometric Ideal for Generic Parameters	104	(5)
Standard Pairs and Triangulations	109	(11)
The Hypergeometric Fan	120	(7)
Logarithm-free Hypergeometric Series	127	(7)
Lower Bound for the Holonomic Rank	134	(4)
Unimodular Triangulations	138	(13)
Rank versus Volume	151	(42)
The Fake Indical Ideal and an Upper Bound	154	(7)
Hypergeometric Functions from Toric Curves	161	(6)
Koszul Complexes and Cohen-Macaulay Property	167	(7)
Integer Programming and Parametric b-functions	174	(7)
The Exceptional Hyperplane Arrangement	181	(5)
w-flatness	186	(7)
Integration of D-modules	193	(48)
b-functions for Holonomic D-ideals	193	(6)
Computing Restrictions	199	(12)
Powers of Polynomials	211	(8)
Hypergeometric Intergrals	219	(8)
Computing Integrals	227	(8)
Asymptotic Expansions of Hypergeometric Integrals	235	(6)
Appendix	241	(4)
References	245	(6)
Index	251