Linear differential equations with finite differential Galois group

TL;DR

This paper develops an algorithmic approach to reconstruct linear differential operators with finite Galois groups from evaluations, advancing inverse Galois problems and invariant theory in differential equations.

Contribution

It introduces a method to compute differential operators from evaluations using a theorem by Compoint, complementing existing algorithms and addressing inverse problems for finite Galois groups.

Findings

Algorithm for reconstructing operators from evaluations

Construction of G-invariant curves with genus zero quotients

Illustrative examples connecting to classical work

Abstract

For a differential operator $L$ of order $n$ over $C(z)$ with a finite (differential) Galois group $G\subset {\rm GL}(C^n)$, there is an algorithm, by M. van Hoeij and J.-A.~Weil, which computes the associated evaluation of the invariants $ev:C[X_1,\dots ,X_n]^G\rightarrow C(z)$. 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 $Z\subset \mathbb{P}(C^n)$ 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.A.~Schwarz, G.~Fano, F.~Klein and A.~Hurwitz.