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Gröbner Deformations of Hypergeometric Differential Equations

  • Textbook
  • © 2000

Overview

Authors:
  1. Mutsumi Saito

    1. Department of Mathematics, Hokkaido University, Sapporo, Japan

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  2. Bernd Sturmfels

    1. Department of Mathematics, University of California, Berkeley, USA

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  3. Nobuki Takayama

    1. Department of Mathematics, Kobe University, Kobe, Japan

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  • This book is the first on this topic
  • It is a research monograph that can also be used as a course text for graduate courses in symbolic computation, algebraic geometry, and algebraic analysis
  • Includes supplementary material: sn.pub/extras

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

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Table of contents (5 chapters)

  1. Front Matter

    Pages I-VIII
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  2. Basic Notions

    • Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama
    Pages 1-50

  3. Solving Regular Holonomic Systems

    • Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama
    Pages 51-102

  4. Hypergeometric Series

    • Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama
    Pages 103-150

  5. Rank versus Volume

    • Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama
    Pages 151-191

  6. Integration of D-modules

    • Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama
    Pages 193-240

  7. Back Matter

    Pages 241-254
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Keywords

About this book

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.

Reviews

".. The book is very well written and, despite the deep results it contains, it is easy to read. Each chapter provides good and nice examples illustrating all main notions. In the reviewer's opinion this book can be addressed not only to researchers but also to beginners in D-module theory and expecially in algorithmic D-module theory."

Francisco Jesus Castro-Jimenez, Mathematical Reviews, Issue 2001i

"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 book is well written. (...) The monograph requires a consequent reading in order to discover all the beauties and the surprising connections between several different branches of mathematics, coming together in the text. This book contains a number of original research results on holonomic systems and hypergeometric functions. The reviewer is sure that it will be the standard reference for computational aspects and research on D-modules in the future. It raises many open problems for future work in this area.

(Zentralblatt für Mathematik und ihre Grenzgebiete 0946.13021)

"... The book is very well written and, despite the deep results it contains, it is easy to read. Each chapter provides good and nice examples illustrating all main notions. In the reviewer's opinion this book can be addressed not only to reearchers but also to beginners in D-module theory and especially in algorithmic D-module theory."

(F. J. Castro-Jimenez, Mathematical Reviews 2002)

Authors and Affiliations

  • Department of Mathematics, Hokkaido University, Sapporo, Japan

    Mutsumi Saito

  • Department of Mathematics, University of California, Berkeley, USA

    Bernd Sturmfels

  • Department of Mathematics, Kobe University, Kobe, Japan

    Nobuki Takayama

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