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Gröbner Deformations of Hypergeometric Differential Equations

  • Mutsumi Saito
  • Bernd Sturmfels
  • Nobuki Takayama

Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 6)

Table of contents

  1. Front Matter
    Pages I-VIII
  2. Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama
    Pages 1-50
  3. Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama
    Pages 51-102
  4. Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama
    Pages 103-150
  5. Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama
    Pages 151-191
  6. Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama
    Pages 193-240
  7. Back Matter
    Pages 241-254

About this book

Introduction

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 in this book are particularly useful for studying the systems of multidimensional hypergeometric partial differentiel equations introduced by Gel'fand, Kapranov and Zelevinsky. The Gröbner deformation of these GKZ hypergeometric systems reduces problems concerning hypergeometric functions to questions about commutative monomial ideals, and thus 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 it raises many open problems for future research in this rapidly growing area of computational mathematics '

Keywords

Gröbner Basen Gröbner bases Hypergeometric function Hypergeometrische Funktionen Weyl algebra combinatorial commutative algebra differential equation holonome Systeme holonomic systems hypergeometric functions kombinatorische kommutative Algebra

Authors and affiliations

  • Mutsumi Saito
    • 1
  • Bernd Sturmfels
    • 2
  • Nobuki Takayama
    • 3
  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of MathematicsKobe UniversityKobeJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-04112-3
  • Copyright Information Springer-Verlag Berlin Heidelberg 2000
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-08534-5
  • Online ISBN 978-3-662-04112-3
  • Series Print ISSN 1431-1550
  • Buy this book on publisher's site