Intrinsic Approach to Galois Theory of q-Difference Equations.

Format:

Book

Author/Creator:

Vizio, Lucia Di.

Contributor:

Hardouin, Charlotte.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.279

Language:

English

Subjects (All):

Galois theory.

Difference equations.

Physical Description:

1 online resource (88 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"The Galois theory of difference equations has witnessed a major evolution in the last two decades. In the particular case of q-difference equations, authors have introduced several different Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of q-difference equations and we use it to establish an arithmetical description of some of the Galois groups attached to q-difference systems"-- Provided by publisher.

Contents:

Cover

Title page

Introduction

Grothendieck conjecture for -difference equations

Intrinsic Galois groups

Comparison with Malgrange-Granier Galois theory for non-linear differential equations

Acknowledgments

Part 1. Introduction to -difference equations

Chapter 1. Generalities on -difference modules

1.1. Basic definitions

1.2. -difference modules, systems and equations

1.3. Some remarks on solutions

1.4. Trivial -difference modules

Chapter 2. Formal classification of singularities

2.1. Regularity

2.2. Irregularity

Part 2. Triviality of -difference equations with rational coefficients

Chapter 3. Rationality of solutions, when is an algebraic number

3.1. The case of algebraic, not a root of unity

3.2. Global nilpotence.

3.3. Proof of Theorem 3.8 (and of Theorem 3.6)

Chapter 4. Rationality of solutions when is transcendental

4.1. Statement of the main result

4.2. Regularity and triviality of the exponents

4.3. Proof of Theorem 4.2

4.4. Link with iterative -difference equations

Chapter 5. A unified statement

Part 3. Intrinsic Galois groups

Chapter 6. The intrinsic Galois group

6.1. Definition and first properties

6.2. Arithmetic characterization of the intrinsic Galois group

6.3. Finite intrinsic Galois groups

6.4. Intrinsic Galois group of a -difference module over \C( ), for ̸=0,1

Chapter 7. The parametrized intrinsic Galois group

7.1. Differential and difference algebra

7.2. Parametrized intrinsic Galois groups

7.3. Characterization of the parametrized intrinsic Galois group by curvatures

7.4. Parametrized intrinsic Galois group of a -difference module over \C( ), for ̸=0,1

7.5. The example of the Jacobi Theta function

Part 4. Comparison with the non-linear theory.

Chapter 8. Preface to Part 4. The Galois -groupoid of a -difference system, by Anne Granier

8.1. Definitions

8.2. A bound for the Galois -groupoid of a linear -difference system

8.3. Groups from the Galois -groupoid of a linear -difference system

Chapter 9. Comparison of the parametrized intrinsic Galois group with the Galois -groupoid

9.1. The Kolchin closure of the Dynamics and the Malgrange-Granier groupoid

9.2. The groupoid \Gal{ ( )}

9.3. The Galois -groupoid \Galan{ ( )} vs the intrinsic parametrized Galois group

9.4. Comparison with known results

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9781470472306

1470472309

OCLC:

1343250942