A general theory of André’s solution algebras
[Une théorie générale des algèbres de solutions d’André]
Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 2103-2129.

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.

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.

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DOI : https://doi.org/10.5802/aif.3383
Classification : 18M25,  12H20,  12H10,  11J91
Mots clés : catégories tannakiennes, modules différentiel itérés, modules aux differences, algèbres de solutions, fonctions de Mahler
@article{AIF_2020__70_5_2103_0,
     author = {Nagy, Levente and Szamuely, Tam\'as},
     title = {A general theory of Andr\'e{\textquoteright}s solution algebras},
     journal = {Annales de l'Institut Fourier},
     pages = {2103--2129},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {5},
     year = {2020},
     doi = {10.5802/aif.3383},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3383/}
}
Nagy, Levente; Szamuely, Tamás. A general theory of André’s solution algebras. Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 2103-2129. doi : 10.5802/aif.3383. https://aif.centre-mersenne.org/articles/10.5802/aif.3383/

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