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Gröbner Deformations of Hypergeometric Differential Equations 2000 ed. [Kõva köide]

  • Formaat: Hardback, 254 pages, kõrgus x laius: 235x155 mm, kaal: 576 g, 5 Illustrations, black and white; VIII, 254 p. 5 illus., 1 Hardback
  • Sari: Algorithms and Computation in Mathematics 6
  • Ilmumisaeg: 12-Nov-1999
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540660658
  • ISBN-13: 9783540660651
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  • Formaat: Hardback, 254 pages, kõrgus x laius: 235x155 mm, kaal: 576 g, 5 Illustrations, black and white; VIII, 254 p. 5 illus., 1 Hardback
  • Sari: Algorithms and Computation in Mathematics 6
  • Ilmumisaeg: 12-Nov-1999
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540660658
  • ISBN-13: 9783540660651
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.

Arvustused

".. 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)

Basic Notions
1(50)
Grobner Bases in the Weyl Algebra
2(9)
Weight Vectors and Non-term Orders
11(5)
The Gauss Hypergeometric Equation
16(12)
Holonomic Systems
28(12)
Integrals of Products of Linear Forms
40(11)
Solving Regular Holonomic Systems
51(52)
The Grobner Fan
51(11)
Semi-Continuity of the Holonomic Rank
62(4)
Torus Action and Frobenius Ideals
66(11)
Regular Holonomic Systems
77(11)
Canonical Series Solutions
88(8)
Construction of Series Solutions
96(7)
Hypergeometric Series
103(48)
GKZ-Hypergeometric Ideal for Generic Parameters
104(5)
Standard Pairs and Triangulations
109(11)
The Hypergeometric Fan
120(7)
Logarithm-free Hypergeometric Series
127(7)
Lower Bound for the Holonomic Rank
134(4)
Unimodular Triangulations
138(13)
Rank versus Volume
151(42)
The Fake Indical Ideal and an Upper Bound
154(7)
Hypergeometric Functions from Toric Curves
161(6)
Koszul Complexes and Cohen-Macaulay Property
167(7)
Integer Programming and Parametric b-functions
174(7)
The Exceptional Hyperplane Arrangement
181(5)
w-flatness
186(7)
Integration of D-modules
193(48)
b-functions for Holonomic D-ideals
193(6)
Computing Restrictions
199(12)
Powers of Polynomials
211(8)
Hypergeometric Intergrals
219(8)
Computing Integrals
227(8)
Asymptotic Expansions of Hypergeometric Integrals
235(6)
Appendix 241(4)
References 245(6)
Index 251