On Ramis's solution of the local inverse problem of differential Galois theory.
Résumé
Recently, J.P. Ramis gave necessary and sufficient conditions for a linear algebraic group to be the Galois group of a Picard-Vessiot extension of the field ${\bf C}\{x\}[x^{-1}]$ of germs of meromorphic functions at zero. The conditions of Ramis are stated in terms of the Lie algebra of the group. In this paper, we give equivalent simple group theoretic conditions, and show how these generalize previous conditions of Kovacic in the solvable case.
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