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Construction et classification de certaines solutions algébriques des systèmes de Garnier

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Résumé

Les déformations isomonodromiques d’équations fuchsiennes d’ordre 2 sur la sphère de Riemann sont paramétrées par les solutions des systèmes de Garnier. Le but de cet article est de construire des solutions algébriques exotiques, i.e. correspondant à des déformations d’équation fuchsienne à monodromie Zariski dense. Précisément, nous classifions toutes les solutions algébriques (complètes) exotiques construites par la méthode de pull-back de Doran-Kitaev: elles se déduisent des déformations isomonodromiques données en tirant en arrière une équation fuchsienne donnée E par une famille de revêtements ramifiés ϕ t . Nous introduisons tout d’abord les structures orbifoldes associées et sous-jacentes à une équation fuchsienne. Ceci nous permet d’avoir une version raffinée de la formule de Riemann Hurwitz qui nous permet rapidement de montrer que E doit être hypergéométrique. Ensuite, on arrive à borner le degré de ϕ et les exposants, puis enfin à lister tous les cas possibles. Ceci généralise un résultat d’ à C. Doran dans le cas de l’équation de Painlevé VI. Nous construisons explicitement une de ces solutions.

Abstract

IsomonodromicdeformationsofFuchsianequationsoforder2onRiemann sphere are parameterized by the solutions of Garnier system. The purpose of this paper is to construct algebraic solutions exotic, i.e. corresponding to deformations of Fuchsian equation with Zariski dense monodromy. Specifically, we classify all the algebraic solutions (complete) exotic constructed by the method of pull-back of Doran-Kitaev: they are deduced from the data isomonodromic deformations pulling back a Fuchsian equation E given by a family of branched coverings ϕ t . We first introduce the structures and associated orbifoldes underlying Fuchsian equation. This allows us to have are fined version of the Riemann Hurwitz formula that allows us quickly to show that E must be hypergeometric. Then we come to limit the degree of ϕ and exponents, and finally to Painlevé VI. We explicitly construct one of these solutions.

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Authors and Affiliations

  1. IRMAR, Unité Mixte de Recherche 6625 du CNRS, Université de Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France

    Karamoko Diarra

  2. DER de Mathématiques et d’informatique, FAST, Université de Bamako, BP: E 3206, Mali, France

    Karamoko Diarra

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  1. Karamoko Diarra
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Diarra, K. Construction et classification de certaines solutions algébriques des systèmes de Garnier. Bull Braz Math Soc, New Series 44, 129–154 (2013). https://doi.org/10.1007/s00574-013-0006-x

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  • DOI: https://doi.org/10.1007/s00574-013-0006-x

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