Grbner Deformations of Hypergeometric Differential Equations [Hardcover]

The theory of Gr?bner bases is a main tool for dealing with rings of differential operators. This book reexamines the concept of Gr?bner 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.In recent years, new algorithms for dealing with rings of differential operators have been discovered and implemented. A main tool is the theory of Gr?bner bases, which is reexamined here from the point of view of geometric deformations. Perturbation techniques have a long tradition in analysis; Gr?bner 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 Gr?bner 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.1. Basic Notions.- 2. Solving Regular Holonomic Systems.- 3. Hypergeometric Series.- 4. Rank versus Volume.- 5. Integration of D-modules.- References.

.. 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 l£)