Inhaltsverzeichnis
Contents v - vii
ALGEBRAIC THEORY 1
1 Picard-Vessiot rings 4
1.1 Existence and uniqueness of Picard-Vessiot rings 7
1.2 The Galois group 8
1.3 Galois correspondence for difference equations 16
1.4 Difference modules and fibre functors 23
2 Algorithms for difference equations 28
2.1 Difference equations of order one 28
2.2 Difference equations in diagonal form 31
2.3 Difference equations of order two 33
3 The inverse problem for difference equations 35
4 The ring S of sequences 45
5 An excursion in positive characteristic 52
5.1 Generalities 52
5.2 Modules over K[T,T-1] 56
5.3 Difference Galois groups 57
5.4 Comparing characteristic 0 and p 58
6 Difference modules over P 60
6.1 Classification of difference modules over P 60
6.2 The universal Picard-Vessiot ring of P 63
6.3 Fields of constants which are not algebraically closed 65
6.4 Automorphisms of the universal Picard-Vessiot ring of P 65
6.5 Difference equations over C((z-1)) and the formal Galois group 66
ANALYTIC THEORY 68
7 Classification and canonical forms 71
7.1 A classification of singularities 71
7.2 Canonical forms 75
8 Semi-regular difference equations 77
8.1 Introduction 77
8.2 Some easy asymptotics 78
8.3 The connection matrix of a semi-regular equation 80
8.4 The theorem of Malgrange and Sibuya 84
8.5 Regular difference equations 86
8.6 Inverse problems for semi-regular equations 88
9 Mild difference equations 95
9.1 Asymptotics for mild equations 95
9.2 Connection matrices of mild equations 96
9.3 Tame differential modules 105
9.4 Inverse problems for mild equations 106
10 Examples of equations and Galois groups 111
10.1 Calculating connection matrices 111
10.2 Classification of order one equations 116
10.3 More on difference Galois groups 119
10.4 Mild difference and differential equations 122
10.5 Very mild difference modules and multisummability 124
10.6 Very mild differential modules 125
11 Wild difference equations 127
11.1 Introduction 127
11.2 Multisummability of formal solutions 128
11.3 The Quadrant Theorem 129
11.4 0n the Gamma function 130
11.5 An example 131
11.6 Solutions on a right half plane 133
11.7 Solutions on an upper half plane 137
11.8 Analytic equivalence classes of difference equations 140
11.9 An example 144
12 q-difference equations 149
12.1 Formal aspects 149
12.2 Analytic properties 153
12.2.1 Regular singular equations over k0 154
12.2.2. Equations over C(z) 156
12.3 Construction of the connection map 157
12.3.1 Meromorphic vector bundles 157
12.3.2 The connection map of a regular equation 159
12.3.3 The connection map of a regular equation 162
12.3.4 Inverse problems 166
Bibliography 175
Index 179
Notations 180