Differential Algebra & Related Topics

Contents......Page 14
Foreword......Page 6
Workshop Participants......Page 10
Workshop Program......Page 12
Preface......Page 15
1 Basic Definitions......Page 17
2 Triangular Sets and Pseudo-Division......Page 20
3 Invertibility of Initials......Page 23
4 Ranking and Reduction Concepts......Page 32
5 Characteristic Sets......Page 38
6 Reduction Algorithms......Page 41
7 Rosenfeld Properties of an Autoreduced Set......Page 45
8 Coherence and Rosenfeld\'s Lemma......Page 49
9 Ritt-Raudenbush Basis Theorem......Page 56
10 Decomposition Problems......Page 58
11 Component Theorems......Page 64
12 The Low Power Theorem......Page 69
Appendix: Solutions and hints to selected exercises......Page 77
References......Page 82
1 Introduction......Page 85
2 Differential rings......Page 86
3 Differential spectrum......Page 89
4 Structure sheaf......Page 92
5 Morphisms......Page 94
7 A-Zeros......Page 95
8 Differential spectrum of R......Page 97
9 AAD modules......Page 98
10 Global sections of AAD rings......Page 100
12 AAD reduction......Page 103
13 Based schemes......Page 104
14 Products......Page 105
References......Page 107
Introduction......Page 109
1 Differential Rings......Page 110
2 Kolchin\'s Irreducibility Theorem......Page 125
3 Descent for Projective Varieties......Page 129
4 Complements and Questions......Page 134
References......Page 136
2 Notation and conventions in differential algebra......Page 139
3 What is model theory?......Page 140
4 Differentially closed fields......Page 142
5 O-minimal theories......Page 156
6 Valued differential fields......Page 158
7 Model theory of difference fields......Page 160
References......Page 161
1 Introduction......Page 165
2 The derivation approach to the inverse problem......Page 169
3 The inverse problem for a 2 x 2 upper triangular matrix group......Page 172
4 Solvable groups......Page 178
References......Page 183
1 The basic concepts......Page 185
2 Universal Picard-Vessiot rings......Page 187
3 Regular singular equations......Page 192
4 Formal differential equations......Page 194
5 Multisummation and Stokes maps......Page 195
6 Meromorphic differential equations......Page 198
References......Page 202
1 Introduction......Page 205
2 Linear Differential Equations......Page 208
3 The Algorithm......Page 210
4 Remarks on the Algorithm......Page 214
5 Remarks on the Hypotheses......Page 217
6 Counterexamples......Page 218
7 An Alternate Approach......Page 220
Appendix - A MAPLE implementation......Page 223
References......Page 230
1 Integrals of Ordinary Differential Equations......Page 233
2 Linearized Equations......Page 237
3 Hamiltonian Systems - The Classical Formulation......Page 238
4 Normal Variational Equations......Page 248
5 Differential Galois Theory and Non-Integrability......Page 250
6 Preliminaries to the Applications......Page 252
7 Applications......Page 258
References......Page 267
Introduction......Page 271
1 Moving frames a tutorial......Page 272
2 Comparison of {uaK||K|>=0} with {IaK||K|>=0}......Page 283
3 Calculations with invariants......Page 287
References......Page 292
0 Introduction......Page 295
1 Definitions examples and basic properties......Page 297
2 Free Baxter algebras......Page 301
3 Further applications of free Baxter algebras......Page 312
References......Page 317