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E-raamat: Galois Theories of Linear Difference Equations: An Introduction

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Galois Theories of Linear Difference Equations: An IntroductionSuurem pilt
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This book is a collection of three introductory tutorials coming out of three courses given at the CIMPA Research School ``Galois Theory of Difference Equations'' in Santa Marta, Columbia, July 23-August 1, 2012. The aim of these tutorials is to introduce the reader to three Galois theories of linear difference equations and their interrelations. Each of the three articles addresses a different galoisian aspect of linear difference equations. The authors motivate and give elementary examples of the basic ideas and techniques, providing the reader with an entry to current research. In addition each article contains an extensive bibliography that includes recent papers; the authors have provided pointers to these articles allowing the interested reader to explore further.
Preface ix
Algebraic and Algorithmic Aspects of Linear Difference Equations
1(42)
Michael F. Singer
1 Introduction
1(1)
2 Algebraic varieties
2(6)
2.1 Varieties and ideals
3(1)
2.2 Irreducible varieties
4(2)
2.3 Morphisms and coordinate rings
6(2)
2.4 Problems
8(1)
3 Linear algebraic groups
8(5)
3.1 Linear algebraic groups
8(3)
3.2 Lie-Kolchin theorem
11(1)
3.3 Torsors
11(1)
3.4 Problems
12(1)
4 Picard-Vessiot extensions
13(7)
4.1 Difference rings and fields
13(1)
4.2 Linear difference equations and Picard-Vessiot extensions
14(5)
4.3 Applications
19(1)
4.4 Problems
20(1)
5 Picard-Vessiot groups
20(7)
5.1 Galois groups of PV extensions
20(1)
5.2 PV extensions and torsors
21(2)
5.3 Applications
23(2)
5.4 Galois correspondence
25(2)
5.5 Problems
27(1)
6 Computational questions
27(9)
6.1 Calculating PV groups and algebraic relations among solutions of linear difference equations
27(4)
6.2 Liouvillian sequences
31(5)
7 Hints and answers to problems
36(7)
7.1 Problems for Section 2
36(1)
7.2 Problems for Section 3
37(1)
7.3 Problems for Section 4
38(1)
7.4 Problems for Section 5
39(1)
References
40(3)
Galoisian Approach to Differential Transcendence
43(60)
Charlotte Hardouin
1 Introduction
43(2)
2 Differential algebraic equations from an algebraic point of view
45(18)
2.1 Differential rings
45(3)
2.2 Differential ideals
48(2)
2.3 Differential polynomial rings and differential algebras
50(7)
2.4 Differential fields
57(6)
3 Differential algebraic geometry
63(15)
3.1 Differential algebraic sets
64(2)
3.2 κ-δ-coordinate rings and κ-δ-morphisms
66(4)
3.3 Dimension of a κ-δ-closed set
70(1)
3.4 From algebraic geometry to differential algebraic geometry and vice versa
71(1)
3.5 Linear κ-δ-groups
71(7)
4 Parametrized Picard-Vessiot theory
78(25)
4.1 The framework
79(1)
4.2 Parametrized Picard-Vessiot rings
80(8)
4.3 The parametrized Galois group
88(4)
4.4 Applications to differential transcendence and isomonodromic problems
92(7)
References
99(4)
Analytic Study of q-Difference Equations
103(61)
Jacques Sauloy
1 Introduction and motivation: q-analogies
106(9)
1.1 Euler, the master of the series
106(4)
1.2 q-analogues and q-degeneracies
110(4)
1.3 Exercises on Section 1
114(1)
2 Some difference and q-difference algebra
115(11)
2.1 Difference fields
115(2)
2.2 Difference operators
117(1)
2.3 Difference equations
118(2)
2.4 Difference systems
120(2)
2.5 Difference modules
122(2)
2.6 Exercises on Section 2
124(2)
3 Fuchsian q-difference equations and systems
126(13)
3.1 Definitions and criteria
126(1)
3.2 Local reduction and classification
127(2)
3.3 Solving fuchsian systems
129(3)
3.4 Local and global classification of fuchsian systems
132(2)
3.5 Two examples
134(2)
3.6 Exercises on Section 3
136(3)
4 Galois and q-monodromy groups of fuchsian systems
139(11)
4.1 Local fuchsian differential Galois group and monodromy group
139(4)
4.2 Local fuchsian q-difference Galois group and q-monodromy group
143(5)
4.3 Global fuchsian q-difference Galois group and q-monodromy group
148(1)
4.4 Exercises on Section 4
149(1)
5 Slopes and filtration
150(14)
5.1 The Newton polygon of an operator
150(1)
5.2 Factorisation of operators
151(5)
5.3 The Adams filtration
156(3)
5.4 Birkhoff-Guenther normal form
159(4)
5.5 Growth of solutions
163(1)
5.6 Exercises on Section 5
164(1)
A Confluence
164(1)
A.1 What lies behind q-degeneracies
164(1)
A.2 Degeneracy of basic functions when q → 1
165(1)
A.3 Degeneracy of canonical solutions when q → 1
166(1)
A.4 Degeneracy of Birkhoff's matrix when q → 1
167(1)
A.5 Basic hypergeometric series again
168(1)
A.6 Exercises on Appendix A
169(1)
References
169
Charlotte Hardouin and Jacques Sauloy, Institut de Mathematiques de Toulouse, France.

Michael F. Singer, North Carolina State University, Raleigh, NC, USA.
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