Gröbner deformations of hypergeometric differential equations
type: Item
instance of: scholarly article
title: Gröbner deformations of hypergeometric differential equations
zbMATH Open document ID: 0946.13021
author: Mutsumi Saito
author: Bernd Sturmfels
author: Nobuki Takayama
published in: Algorithms and Computation in Mathematics
publication date: 2000-01-19T00:00:00Z
review text: In recent years the theory of Gröbner bases has found several applications in various fields of symbolic computations, in particular in applications related to combinatorics. The second author of the present book has opened the view for further applications in combinatorics [see \textit{B. Sturmfels}, ``Gröbner bases and convex polytopes'', Univ. Lect. Ser. 8 (Providence 1996; Zbl 0856.13020)]. The present book offers another application of Gröbner bases related to new algorithms for dealing with rings of differential operators. Here the Gröbner bases are reexamined from the point of geometric deformations. More precisely it provides symbolic algorithms for constructing holomorphic solutions to systems of linear partial differential equations with polynomial coefficients. Such a system is represented by a left ideal \(I\) in the Weyl algebra \(D = {\mathbb C} \langle x_1, \ldots, x_n, \partial_1, \ldots, \partial_n \rangle.\) A Gröbner deformation of the left ideal \(I\) is the initial ideal \(\text{in}_{(-w,w)}(I) \subset D\) with respect to some generic weight vector \(w = (w_1, \ldots, w_n)\) with real coordinates \(w_i.\) By methods from computational commutative algebra there is an explicit solution basis for the Gröbner deformation \(\text{in}_{(-w,w)}(I).\) The problem is to extend it to a solution basis for \(I.\) This is solved under the natural hypothesis that the given \(D\)-ideal \(I\) is regular holonomic. This is valid, for instance, for the \(D\)-ideals representing hypergeometeric integrals. The first chapter of the book is devoted to the basic notations. The classical Gauss' hypergeometric function is expressed in the Gel'fand-Kapranov-Zelevinsky (GKZ) scheme. It contains also an introduction to holonomic systems of differential equations from the Gröbner basis point of view. The second chapter is concerned with solving regular holonomic systems. The holomorphic solutions of \(I\) around a generic point in \({\mathbb C}^n\) form a vector space. If \…
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 13-02
Mathematics Subject Classification ID: 68-02
Mathematics Subject Classification ID: 33D60
Mathematics Subject Classification ID: 13N10
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 16S32
Mathematics Subject Classification ID: 14F10
zbMATH DE Number: 1391025
zbMATH Keywords: Gröbner deformation
zbMATH Keywords: hypergeometric differential equation
zbMATH Keywords: Gröbner bases
zbMATH Keywords: Gröbner fan
zbMATH Keywords: \(D\)-modules
zbMATH Keywords: holonomic systems
zbMATH Keywords: toric variety
describes a project that uses: Kan
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1391025