On Ramis's solution of the local inverse problem of differential Galois theory (Q1916426)

\textit{J. P. Ramis} [About the inverse problem in differential Galois theory: The differential Abhyankar conjecture. In: Braaksma, B. L. J. (ed.) et al., The Stokes phenomenon and Hilbert's 16th problem. Proceedings of the workshop, Groningen, 261-278 (1996; Zbl 0860.12003)] gave necessary and sufficient conditions for a linear algebraic group to be the Galois group of a Picard-Vessiot extension of the field \(C\{ x\}[ x^{-1}]\) of germs of meromorphic functions at zero. In this paper, the authors give equivalent simple group theoretic conditions. The main result is the following: Theorem. Let \(G\) be a linear algebraic group. The following statements are equivalent: (1) \(G\) is the Galois group of some Picard-Vessiot extension of \(C\{ x\}[ x^{-1} ]\). (2) The following three conditions hold: (a) \(G/ G^0\) is cyclic, (b) the dimension of \(R_u/[ R_u, G^0]\) is at most 1, (3) \(G/ G^0\) acts trivially on \(R_u/[ R_u, G^0]\).