Large Fields in Differential Galois Theory

TL;DR

This paper addresses the inverse differential Galois problem over differential fields with large constant fields, providing solutions for all split differential embedding problems, including over ${ m Q}_p(x)$, advancing the understanding of differential Galois theory.

Contribution

It extends the inverse differential Galois problem solutions to fields with large constant fields and solves all split differential embedding problems over such fields.

Findings

Solved the inverse differential Galois problem over fields with infinite transcendence degree constants.

Established that all split differential embedding problems can be solved over these fields.

Provided solutions specifically over ${ m Q}_p(x)$, a significant case in the theory.

Abstract

We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over ${\mathbb Q}$. More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over ${\mathbb Q}_p(x)$.