ISBN: 3-540-66065-8
TITLE: Grbner Deformations of Hypergeometric Differential Equations
AUTHOR: Saito, Mutsumi; Sturmfels, Bernd; Takayama, Nobuki
TOC:

1. Basic Notions 1
1.1 Grbner Bases in the Weyl Algebra 2
1.2 Weight Vectors and Non-term Orders 11
1.3 The Gauss Hypergeometric Equation 16
1.4 Holonomic Systems 28
1.5 Integrals of Products of Linear Forms 40
2. Solving Regular Holonomic Systems 51
2.1 The Grbner Fan 51
2.2 Semi-Continuity of the Holonomic Rank 62
2.3 Torus Action and Frobenius Ideals 66
2.4 Regular Holonomic Systems 77
2.5 Canonical Series Solutions 88
2.6 Construction of Series Solutions 96
3. Hypergeometric Series 103
3.1 GKZ-Hypergeometric Ideal for Generic Parameters 104
3.2 Standard Pairs and Triangulations 109
3.3 The Hypergeometric Fan 120
3.4 Logarithm-free Hypergeometric Series 127
3.5 Lower Bound for the Holonomic Rank 134
3.6 Unimodular Triangulations 138
4. Rank versus Volume 151
4.1 The Fake Indicial Ideal and an Upper Bound 154
4.2 Hypergeometric Functions from Toric Curves 161
4.3 Koszul Complexes and Cohen-Macaulay Property 167
4.4 Integer Programming and Parametric b-functions 174
4.5 The Exceptional Hyperplane Arrangement 181
4.6 w-flatness 186
5. Integration of D-modules 193
5.1 b-functions for Holonomic D-ideals 193
5.2 Computing Restrictions 199
5.3 Powers of Polynomials 211
5.4 Hypergeometric Integrals 219
5.5 Computing Integrals 227
5.6 Asymptotic Expansions of Hypergeometric Integrals 235
Appendix 241
References 245
Index 251
END
