Differential algebra explores properties of solutions of systems of (ordinary or partial, linear or non-linear) differential equations from an algebraic point of view. It includes as special cases algebraic systems as well as differential systems with algebraic constraints. This algebraic theory of Joseph F Ritt and Ellis R Kolchin is further enriched by its interactions with algebraic geometry, Diophantine geometry, differential geometry, model theory, control theory, automatic theorem proving, combinatorics, and difference equations.Differential algebra now plays an important role in computational methods such as symbolic integration and symmetry analysis of differential equations. These proceedings consist of tutorial and survey papers presented at the Second International Workshop on Differential Algebra and Related Topics at Rutgers University, Newark in April 2007. As a sequel to the proceedings of the First International Workshop, this volume covers more related subjects, and provides a modern and introductory treatment to many facets of differential algebra, including surveys of known results, open problems, and new, emerging, directions of research. It is therefore an excellent companion and reference text for graduate students and researchers.
Author(s): P. Cassidy, Li Guo, William F. Keigher, Phyllis J. Cassidy, William Y. Sit
Edition: 1st
Publisher: World Scientific Publishing Company
Year: 2002
Language: English
Commentary: 31748
Pages: 320
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
