New classes of parameterized differential Galois groups

TL;DR

This paper investigates which groups can be realized as parameterized differential Galois groups, revealing limitations over certain fields and constructing new classes of groups that do occur as Galois groups.

Contribution

It demonstrates that many groups do not occur as Galois groups over K(x), and constructs new classes of groups that do occur over k((t))(x) using patching and Galois descent.

Findings

Many groups do not occur as Galois groups over K(x).

Certain groups, including generated unipotent and semisimple groups, do occur over k((t))(x).

New classes of parameterized differential Galois groups are identified.

Abstract

This paper is on the inverse parameterized differential Galois problem. We show that surprisingly many groups do not occur as parameterized differential Galois groups over K(x) even when K is algebraically closed. We then combine the method of patching over fields with a suitable version of Galois descent to prove that certain groups do occur as parameterized differential Galois groups over k((t))(x). This class includes linear differential algebraic groups that are generated by finitely many unipotent elements and also semisimple connected linear algebraic groups.