Differential Galois groups, specializations and Matzat's conjecture

TL;DR

This paper investigates how differential Galois groups behave under specialization in families of differential equations and proves Matzat's conjecture that the absolute differential Galois group of a one-variable function field is free proalgebraic.

Contribution

It establishes the Zariski density of specializations of differential Galois groups and proves Matzat's conjecture in full generality.

Findings

Zariski density of specialization points for differential Galois groups

Proof of Matzat's conjecture for characteristic zero fields

The absolute differential Galois group is a free proalgebraic group

Abstract

We study families of linear differential equations parametrized by an algebraic variety $\mathcal{X}$ and show that the set of all points $x\in \mathcal{X}$, such that the differential Galois group at the generic fibre specializes to the differential Galois group at the fibre over $x$, is Zariski dense in $\mathcal{X}$. As an application, we prove Matzat's conjecture in full generality: The absolute differential Galois group of a one-variable function field over an algebraically closed field of characteristic zero is a free proalgebraic group.