A DENSITY THEOREM IN PARAMETRIZED DIFFERENTIAL GALOIS THEORY

We study parametrized linear differential equations with coefficients depending meromorphically upon the parameters. As a main result, analogously to the unparametrized density theorem of Ramis, we show that the parametrized monodromy, the parametrized exponential torus and the parametrized Stokes operators are topological generators in the Kolchin topology for the parametrized differential Galois group introduced by Cassidy and Singer. We prove an analogous result for the global parametrized differential Galois group, which generalizes a result by Mitschi and Singer. These authors give also a necessary condition on a group for being a global parametrized differential Galois group; as a corollary of the density theorem, we prove that their condition is also sufficient. As an application, we give a characterization of completely integrable equations, and we give a partial answer to a question of Sibuya about the transcendence properties of a given Stokes matrix. Moreover, using a parametrized Hukuhara-Turrittin theorem, we show that the Galois group descends to a smaller field, whose field of constants is not differentially closed.