The q-analogue of the wild fundamental group and the inverse problem of the Galois theory of q-difference equations
[Le q-analogue du groupe fondamental sauvage et le problème inverse de la théorie de Galois aux q-différences]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 1, pp. 171-226.

Nous avons défini dans [23,24] des q-analogues des dérivations étrangères pour les équations aux q-différences linéaires analytiques à pentes entières, et prouvé un théorème de densité (dans le groupe de Galois) et un théorème de liberté. Dans cet article, nous décrivons complètement le groupe fondamental sauvage et appliquons ce résultat au problème inverse en théorie de Galois des équations aux q-différences.

In [23,24], we defined q-analogues of alien derivations for linear analytic q-difference equations with integral slopes and proved a density theorem (in the Galois group) and a freeness theorem. In this paper, we completely describe the wild fundamental group and apply this result to the inverse problem in q-difference Galois theory.

Publié le :
DOI : 10.24033/asens.2241
Classification : 39A13; 34M50.
Keywords: $q$-difference equations, Stokes phenomenon, alien derivations, Galois theory, inverse problem.
Mot clés : Équations aux $q$-différences, phénomène de Stokes, dérivations étrangères, théorie de Galois, problème inverse. 
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     author = {Ramis, Jean-Pierre and Sauloy, Jacques},
     title = {The $q$-analogue of the wild fundamental group and the inverse problem of the {Galois} theory of~$q$-difference equations},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {171--226},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
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Ramis, Jean-Pierre; Sauloy, Jacques. The $q$-analogue of the wild fundamental group and the inverse problem of the Galois theory of $q$-difference equations. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 1, pp. 171-226. doi : 10.24033/asens.2241. http://www.numdam.org/articles/10.24033/asens.2241/

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