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Gröbner Deformations of Hypergeometric Differential Equations
Mutsumi Saito
Bernd Sturmfels
Nobuki Takayama
出版
Springer Science & Business Media
, 2013-03-09
主題
Mathematics / Mathematical Analysis
Mathematics / Counting & Numeration
Computers / Programming / Algorithms
Mathematics / Geometry / Algebraic
Science / Physics / Mathematical & Computational
Mathematics / Combinatorics
Mathematics / Calculus
Mathematics / Numerical Analysis
ISBN
366204112X
9783662041123
URL
http://books.google.com.hk/books?id=kijrCAAAQBAJ&hl=&source=gbs_api
EBook
SAMPLE
註釋
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.