The regular singular inverse problem in differential Galois theory

TL;DR

This paper proves that any linear algebraic group over an algebraically closed field of characteristic zero can be realized as the differential Galois group of a regular singular linear differential equation with rational coefficients.

Contribution

It establishes a universal realization result for linear algebraic groups as differential Galois groups in the regular singular case.

Findings

Every linear algebraic group over an algebraically closed field of characteristic zero is realizable as a differential Galois group.

Constructs regular singular linear differential equations with prescribed Galois groups.

Advances the understanding of the inverse problem in differential Galois theory.

Abstract

We show that every linear algebraic group over an algebraically closed field of characteristic zero is the differential Galois group of a regular singular linear differential equation with rational function coefficients.