# Free differential Galois groups

## TL;DR

This paper proves Matzat's freeness conjecture for the absolute differential Galois group of rational function fields over algebraically closed fields with countably infinite transcendence degree, revealing its free proalgebraic structure.

## Contribution

It establishes the first proof of Matzat's freeness conjecture in a significant case, linking differential embedding problems to the group's freeness.

## Key findings

- Proved Matzat's freeness conjecture for algebraically closed fields with countably infinite transcendence degree.
- Connected differential embedding problems to the freeness of the differential Galois group.
- Demonstrated the absolute differential Galois group is free as a proalgebraic group in the studied case.

## Abstract

We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of differential embedding problems to the condition that the absolute differential Galois group is free as a proalgebraic group. Building on this, we prove Matzat's freeness conjecture in the case that the field of constants is algebraically closed of countably infinite transcendence degree over the rationals. This is the first known case of the twenty year old conjecture.