Construction et classification de certaines solutions algébriques des systèmes de Garnier

Karamoko Diarra (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

In this paper, we classify all (complete) non elementary algebraic solutions of Garnier systems that can be constructed by Kitaev's method: they are deduced from isomonodromic deformations defined by pulling back a given fuchsian equation E by a family of ramified covers. We first introduce orbifold structures associated to a fuchsian equation. This allow to get a refined version of Riemann-Hurwitz formula and then to promtly deduce that E is hypergeometric. Then, we can bound exponents and degree of the pull-back maps and further list all possible ramification cases. This generalizes a result due to C. Doran for the Painleve VI case. We explicitely construct one of these solutions.

Mots clés: Équations différentielles ordinaires; Déformations isomonodromiques; Familles de Hurwitz

Domaines: Géométrie algébrique [math.AG]

Karamoko Diarra. Construction et classification de certaines solutions algébriques des systèmes de Garnier. Boletim da Sociedade Brasileira de Matemática / Bulletin of the Brazilian Mathematical Society, 2013, 44 (1), pp.129-154. ⟨10.1007/s00574-013-0006-x⟩. ⟨hal-00656992⟩