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.

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

Inhalt

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