Linear Algebraic Groups as Parameterized Picard-Vessiot Galois Groups

TL;DR

This paper characterizes which linear algebraic groups can serve as Galois groups of parameterized Picard-Vessiot extensions over certain differential fields, linking group structure to differential Galois theory.

Contribution

It provides a necessary and sufficient condition based on the group's identity component for being a Galois group in this setting.

Findings

Characterization of Galois groups via group structure

Condition on the identity component's quotients

Extension of differential Galois theory results

Abstract

We show that a linear algebraic group is the Galois group of a parameterized Picard-Vessiot extension of k(x), x' = 1, for certain differential fields k, if and only if its identity component has no one dimensional quotient as a linear algebraic group.