Classification of algebraic solutions of irregular Garnier systems

TL;DR

This paper classifies algebraic solutions of irregular Garnier systems into two types, providing complete lists for rank 2 cases based on their Galois groups and deformation origins.

Contribution

It offers a comprehensive classification of algebraic solutions for irregular Garnier systems, including explicit lists for rank 2 cases, distinguishing classical and pull-back solutions.

Findings

Classified algebraic solutions into two types.

Provided complete lists for rank N=2 solutions.

Connected solutions to Galois groups and deformation origins.

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.