Differential Galois Groups over Laurent Series Fields

TL;DR

This paper demonstrates that any linear algebraic group over Laurent series fields of characteristic zero can be realized as a differential Galois group of a linear differential equation, solving the inverse problem in this setting.

Contribution

It provides a positive solution to the inverse differential Galois problem over Laurent series fields using patching methods, extending known results to a broader class of fields.

Findings

Any linear algebraic group over Laurent series fields occurs as a differential Galois group.

The method applies to function fields with one or multiple variables.

The approach confirms the realizability of all affine group schemes as differential Galois groups.

Abstract

In this manuscript, we apply patching methods to give a positive answer to the inverse differential Galois problem over function fields over Laurent series fields of characteristic zero. More precisely, we show that any linear algebraic group (i.e. affine group scheme of finite type) over such a Laurent series field does occur as the differential Galois group of a linear differential equation with coefficients in any such function field (of one or several variables).