On Ramisʹs solution of the local inverse problem of differential Galois theory Original Research Article

Recently, Ramis gave necessary and sufficient conditions for a linear algebraic group to be the Galois group of a Picard-Vessiot extension of the field image of germs of meromorphic functions at zero. In this paper, we give equivalent simple group theoretic conditions, and show how these generalize previous conditions of Kovacic in the solvable case.