Overview

In recent years, new algorithms for dealing with rings of differential operators have been discovered and implemented. A main tool is the theory of Grobner bases, which is reexamined here from the point of view of geometric deformations. Perturbation techniques have a long tradition in analysis; Grobner 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 Grobner 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.

Full Product Details

Author:   Mutsumi Saito ,  Bernd Sturmfels ,  Nobuki Takayama
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   2000 ed.
Volume:   6
Dimensions:   Width: 15.50cm , Height: 1.50cm , Length: 23.50cm
Weight:   0.576kg
ISBN:  

9783540660651

ISBN 10:   3540660658
Pages:   254
Publication Date:   12 November 1999
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1. Basic Notions.- 2. Solving Regular Holonomic Systems.- 3. Hypergeometric Series.- 4. Rank versus Volume.- 5. Integration of D-modules.- References.

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. <p>Francisco Jesus Castro-Jimenez, Mathematical Reviews, Issue 2001i <p> In recent years the theory of GrAbner 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. <p>(Zentralblatt fA1/4r Mathematik und ihre Grenzgebiete 0946.13021) <p>.,. 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. <p>(F. J. Castro-Jimenez, Mathematical Reviews 2002)

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