Differential Embedding Problems over Laurent series fields

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.

13 pages; simplified Sections 1-3 (resulting in a renumbering of the statements in Section 1-2), added Proposition 1.4, added Remark 3.4, streamlined proof of Theorem 4.2, added Remark 4.4