A general theory of André’s solution algebras
Annales de l'Institut Fourier, Volume 70 (2020) no. 5, pp. 2103-2129.

We extend Yves André’s theory of solution algebras in differential Galois theory to a general Tannakian context. As applications, we establish analogues of his correspondence between solution fields and observable subgroups of the Galois group for iterated differential equations in positive characteristic and for difference equations. The use of solution algebras in the difference algebraic context also allows a new approach to recent results of Philippon and Adamczewski–Faverjon in transcendence theory.

Nous étendons la théorie des algèbres de solutions d’Yves André en théorie de Galois différentielle à un contexte général tannakien. Comme application nous obtenons des analogues de sa correspondance entre corps de solutions et sous-groupes observables du groupe de Galois différentiel pour les équations différentielles itérées en caractéristique positive et pour les équations aux différences. L’utilisation des algèbres de solutions dans le cadre de l’algèbre aux différences permet également un nouveau point de vue sur des résultats récents de Philippon et d’Adamczewski–Faverjon en théorie de la transcendance.

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DOI: 10.5802/aif.3383
Classification: 18M25, 12H20, 12H10, 11J91
Keywords: Tannakian categories, iterative differential modules, difference modules, solution algebras, Mahler functions
Mot clés : catégories tannakiennes, modules différentiel itérés, modules aux differences, algèbres de solutions, fonctions de Mahler
Nagy, Levente 1; Szamuely, Tamás 2

1 Central European University, Nádor utca 9, H-1051 Budapest, (Hungary)
2 Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, (Italy) and Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13–15, H-1053 Budapest, (Hungary)
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Nagy, Levente; Szamuely, Tamás. A general theory of André’s solution algebras. Annales de l'Institut Fourier, Volume 70 (2020) no. 5, pp. 2103-2129. doi : 10.5802/aif.3383. http://archive.numdam.org/articles/10.5802/aif.3383/

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