Linear differential equations with finite differential Galois group

Abstract

For a differential operator L of order n over C(z) with a finite (differential) Galois group G ⊂ GL(C n ), there is an algorithm, by M. van Hoeij and J.-A. Weil, which computes the associated evaluation of the invariants ev :The procedure proposed here does the opposite: it uses a theorem of E. Compoint and computes the operator L from a given evaluation h. Moreover it solves a part of the inverse problem of producing L for a given representation of a finite group G. Another part considered here, is finding irreducible G-invariant curves Z ⊂ P(C n ) with Z/G of genus zero and constructing evaluations from this. The theory developed here is illustrated by various examples, and relates to and continues classical work of H.