Algebraic Approach to Differential Equations

CONTENTS......Page 8
Preface......Page 6
Acknowledgments......Page 7
Introduction......Page 9
1. Cauchy Theorem......Page 10
2. Sheaves of Holomorphic Functions......Page 12
3. Sheaf Version of Cauchy Theorem......Page 19
4. Local Monodromy......Page 21
5. Fuchs Theory......Page 29
6. Index of Differential Operators at Singular Points......Page 30
7. Division Tools......Page 32
8. Generalized Solutions......Page 41
9. Holonomic D-Modules......Page 47
10. Regular D-Modules......Page 48
11. A Local Version of the Riemann-Hilbert Correspondence in One Variable (in Collaboration with F. Gudiel Rodriguez)......Page 51
12. D-Modules on a Riemann Surface......Page 56
References......Page 58
Introduction......Page 60
1.1. Linear differential operators......Page 61
1.2. Order and total order......Page 65
1.4. The graded rings grB(An) and grF (An)......Page 69
1.6. F–filtrations on An-modules......Page 73
1.7. The –order and the –symbol map......Page 74
1.8. Graded associated module......Page 75
1.9. Induced filtrations......Page 76
1.10. Good filtrations......Page 78
1.11. Rational functions......Page 81
2.1. Classical characteristic vectors......Page 82
2.2. Characteristic variety......Page 83
2.3. Dimension of an An–module......Page 87
2.4. Hilbert polynomial......Page 89
2.5. Bernstein's inequality......Page 93
2.6. Holonomic An–modules......Page 94
2.7. C[x]f is holonomic......Page 97
2.8. The Bernstein polynomial......Page 100
3.1. Logarithmic derivations......Page 103
3.2. The ideal Ann......Page 106
3.3. Logarithmic differential forms......Page 111
More rings of linear differential operators......Page 115
Division theorem in A......Page 117
Groebner bases in An......Page 120
Buchberger's algorithm in An......Page 122
References......Page 123
1. Whitney Conditions......Page 127
2. Stratifications......Page 129
3. Constructible Sheaves......Page 131
4. Whitney Stratifications......Page 132
5. Milnor Fibrations......Page 135
7. Neighbouring Cycles......Page 136
8. Constructible Complexes......Page 137
9. Vanishing Cycles......Page 138
10. Holonomic D-Modules......Page 141
References......Page 142
1. Introduction......Page 144
2.1.2. Second method......Page 147
2.1.3. Third method : the stationary phase method......Page 148
2.2. Airy and the steepest-descent method in dimension 1......Page 153
2.2.2. Integrability and space of allowed paths of integration......Page 154
2.2.3. The steepest-descent method......Page 156
2.2.4. Localisation......Page 158
2.2.5. First method......Page 159
2.2.6. Second method : reduction to an incomplete Laplace transform (1)......Page 160
2.2.7. Third method : reduction to an incomplete Laplace transform (2)......Page 162
2.2.8. The asymptotics......Page 163
2.2.9. Remark 1 : geometric monodromy......Page 164
2.2.10. Remark 2 : an example of local system......Page 165
2.2.11. Exercise......Page 166
2.3.1. Two division lemmas......Page 167
2.3.2. An application......Page 169
3. Integrals of Holomorphic Differential Forms along Cycles......Page 170
3.1.2. The Ehresmann fibration theorem......Page 171
3.1.3. Applications......Page 173
3.2. The polynomial case......Page 176
3.2.1. A fibration theorem and consequences......Page 177
3.2.2. The finite rank case......Page 178
3.2.3. Examples......Page 180
3.3. Localisation near an isolated singularity......Page 183
4.1. Allowed chains of integration......Page 185
4.2. The steepest-descent method......Page 188
4.3. Localisation......Page 190
4.4.1. Reduction to an incomplete Laplace transform......Page 193
4.4.2. The asymptotics......Page 196
4.5. An example......Page 197
4.6. To go further......Page 199
Appendix A.1. Simplex and chain......Page 200
Appendix A.2. Boundary operator......Page 201
Appendix A.3. Homomorphism induced by a continuous map......Page 203
Appendix A.4. Homology of a pair......Page 204
Appendix A.5. Homology with support in a family......Page 207
Appendix A.6. Homology and fibre bundle......Page 208
Appendix A.7. Integration......Page 209
Appendix B.1. Definition and main properties......Page 210
Appendix B.2. Some applications......Page 212
Appendix B.2.2. Second application......Page 213
References......Page 215
1.1. Classical hypergeometric series (Gauss)......Page 218
1.3. Hypergeometric integral (Euler)......Page 219
2. Modern Approach......Page 220
3. Local Systems......Page 221
4. Hypergeometric Pairing......Page 222
5. Cohomology Groups Hp(M, )......Page 223
6.1. Generalities......Page 224
6.3. Brieskorn algebra......Page 225
6.5. The invariant......Page 226
7. Resonance......Page 227
8. The NBC Complex......Page 229
References......Page 231
Bernstein-Sato Polynomials and Functional Equations M. Granger......Page 233
Contents......Page 235
1.1 Definitions......Page 236
1.2 A review of a number of elementary facts about b-functions......Page 238
1.3 A first list of examples......Page 239
1.4 Remarks on variants of the definition......Page 240
2.1 Holonomic modules......Page 241
2.2 Bernstein equation......Page 243
2.3 Generalisations......Page 244
2.4 Semi-invariants of prehomogeneous spaces......Page 246
3.1 Roots of b and analytic continuation of Y (f)fs......Page 249
3.2 Asymptotic expansion and Mellin transforms......Page 251
3.3 Application to analytic continuation......Page 253
3.4 Application to the division of distribution......Page 255
4.1 Quasi-homogeneous polynomials......Page 256
4.2 Semi-quasi-homogeneous germs......Page 259
4.3 Calculation of the Bernstein-Sato polynomial of a quasi-homogeneous polynomial......Page 261
5.1 The proof in the analytic case......Page 264
5.2 An application: Holonomicity of the module......Page 267
5.3 Links between various Bernstein-Sato polynomials......Page 268
6.1 The rationality of the zeros......Page 269
6.2 Derived results......Page 270
7.1.1 Definition of D(U) on an open subset of Cn......Page 271
7.1.2 Behaviour under change of coordinates......Page 272
7.1.4 Principal symbols, and graded associated sheaves......Page 274
7.1.5 Coherence......Page 276
7.2.1 What is a differential system?......Page 278
7.2.2 Regular connections......Page 279
7.3 Good filtrations and coherence conditions......Page 281
7.4.1 Case of a monogeneous module M= D......Page 284
7.4.2 General case......Page 285
7.4.3 A finiteness property......Page 286
8.1.2 Analyticity of Mellin transforms......Page 290
8.2 Appendix B: Regular sequences, and application to the annihilator AnnDfs, in the isolated case......Page 293
References......Page 296
2. Systems of partial differential equations......Page 300
3. Differential Groups......Page 302
4. Differential Lie Algebras......Page 304
4.5. Lie groups and Lie algebras......Page 305
5.1. Additive groups......Page 307
6. Family of Elliptic Curves......Page 308
7.1. Connections in the sense of Ehresmann (or \foliated bundles")......Page 311
7.2. Connections and differential equations......Page 312
7.3. From now on, the connections will be supposed flat......Page 313
7.5. Connections on principal bundles......Page 314
7.7. Connections with respect to a foliation......Page 315
8. Simple Groups......Page 316
8.2. Generalizations......Page 318
References......Page 319