Gröbner Deformations of Hypergeometric Differential Equations (Algorithms and Computation in Mathematics) - Softcover

Saito, Mutsumi; Sturmfels, Bernd; Takayama, Nobuki

Softcover

ISBN 10: 3642085342 ISBN 13: 9783642085345

Publisher: Springer, 2011

Synopsis

The theory of Gröbner bases is a main tool for dealing with rings of differential operators. This book reexamines the concept of Gröbner bases from the point of view of geometric deformations. The algorithmic methods introduced in this book are particularly useful for studying the systems of multidimensional hypergeometric PDE's introduced by Gelfand, Kapranov, and Zelevinsky. A number of original research results are contained in the book, and many open problems are raised for future research in this rapidly growing area of computational mathematics.

About this title

From the Back Cover

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 '

Review

".. 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)

Edition details

• Publisher: Springer
• Publication date: 2011
• ISBN 10: 3642085342
• ISBN 13: 9783642085345
• Binding: Paperback
• Language: English
• Number of pages: 262