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Record Nr. |
UNINA9910955333303321 |
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Autore |
Vizio Lucia Di |
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Titolo |
Intrinsic Approach to Galois Theory of |
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Pubbl/distr/stampa |
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Providence : , : American Mathematical Society, , 2022 |
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©2022 |
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ISBN |
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Edizione |
[1st ed.] |
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Descrizione fisica |
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1 online resource (88 pages) |
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Collana |
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Memoirs of the American Mathematical Society ; ; v.279 |
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Classificazione |
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Altri autori (Persone) |
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Disciplina |
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Soggetti |
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Galois theory |
Difference equations |
Difference and functional equations -- Difference equations -- Difference equations, scaling ($q$-differences) |
Field theory and polynomials -- Differential and difference algebra -- Difference algebra |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Nota di contenuto |
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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. |
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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. |
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Sommario/riassunto |
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"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"-- |
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