Solvable-by-finite groups as differential Galois groups
Let \(C\) be an algebraically closed field of characteristic zero and let \(K=C(x)\) be the field of rational functions over \(C\) with the derivation \(d/dx\). Let \(G\) be an affine algebraic group defined over \(C\). The inverse problem of differential Galois theory asks whether there is a Picard-Vessiot extension \(E \supset K\) whose differential Galois group is isomorphic to \(G\). The authors consider the (stronger, in case \(G\) is not connected) problem of finding a principal homogeneous space \(V\) for \(G\) and a derivation on the function field \(K(V)\) such that \(E=K(V)\), with the given action of \(G\), is such a Picard-Vessiot extension. They prove this when the identity component of \(G\) is solvable, and their proof is constructive up to finding finite Galois extensions of \(C(x)\). In fact their proof, which is explained with careful attention to exposition, potentially covers a much wider class of groups (and has been extended to all groups by \textit{Julia Hartmann} [On the inverse problem in differential Galois theory. Thesis Heidelberg (2002; Zbl 1063.12005)]). Because of this attention to background, this paper is an excellent introduction to this approach to the inverse problem.
inverse problem of differential Galois theory
scientific article; zbMATH DE number 2074273
scientific article