Differential Embedding Problems over Laurent series fields

TL;DR

This paper addresses the inverse differential Galois problem over Laurent series fields, providing solutions to split differential embedding problems using patching techniques and prior results.

Contribution

It introduces methods to solve inverse differential Galois problems over specific Laurent series fields and applies these to split embedding problems induced from rational function fields.

Findings

Solved the inverse differential Galois problem over the fraction field of $k[[t,x]]$.

Provided solutions to split differential embedding problems over $k((t))(x)$.

Utilized patching techniques and previous results to achieve these solutions.

Abstract

We solve the inverse differential Galois problem over the fraction field of $k[[t,x]]$ and use this to solve split differential embedding problems over $k((t))(x)$ that are induced from $k(x)$. The proofs use patching as well as prior results on inverse problems and embedding problems.