Classification of algebraic solutions of irregular Garnier systems

Abstract

We prove that algebraic solutions of Garnier systems in the irregular case are of two types. The classical ones come from isomonodromic deformations of linear equations with diagonal or dihedral differential Galois group; we give a complete list in the rank N = 2 case (two indeterminates).The pull-back ones come from deformations of coverings over a fixed degenerate hypergeometric equation; we provide a complete list when the differential Galois group is SL 2 (C). By the way, we have a complete list of algebraic solutions for the rank N = 2 irregular Garnier systems.

A nonclassical algebraic solution of a 3-variable irregular Garnier system

“…On the other hand, for irregular Garnier systems, there is a complete classification of algebraic solutions due to Diarra-Loray [14]. They have shown that any algebraic solution of an irregular Garnier system is one of the following types:…”

“…They have shown that N-variable irregular Garnier systems with N > 3 admit only classical algebraic solutions. Moreover, up to canonical transformations (see [17]), there are exactly three nonclassical algebraic solutions for N-variable irregular Garnier systems with N > 1 (see [14,Theorem 2]). Remark that these algebraic solutions are pull-back type.…”

“…When N = 2, the list of explicit irregular Garnier systems is provided in [26,25]. By finding fixed systems on P 1 and algebraic families of ramified covers P 1 → P 1 , Diarra-Loray [14] have given explicit forms of those two algebraic solutions under Kimura's notation [26]. On the other hand, for the last of the three algebraic solutions, N = 3.…”

“…In particular, we are interested in nonclassical algebraic solutions for irregular Garnier systems of rank N > 1. By [14,Theorem 2], up to canonical transformations, there are exactly three nonclassical algebraic solutions for N-variable irregular Garnier systems with N > 1. The corresponding algebraic isomonodromic families are constructed by the pull-back method.…”

“…The corresponding algebraic isomonodromic families are constructed by the pull-back method. In fact, the two of the three nonclassical algebraic solutions were already described in [14,Section 9] by this method. Remark that the irregular Garnier systems corresponding to these two solutions are 2-variable.…”