Construction et classification de certaines solutions alg\'ebriques des syst\`emes de Garnier

TL;DR

This paper classifies all algebraic solutions of Garnier systems constructed via isomonodromic deformations, extending previous results and explicitly constructing one such solution.

Contribution

It provides a complete classification of algebraic solutions of Garnier systems obtained through Kitaev's method, generalizing prior work on Painleve VI.

Findings

Classified all algebraic solutions from isomonodromic deformations

Bounded exponents and degrees of pull-back maps

Explicit construction of one solution

Abstract

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.