A density theorem for parameterized differential Galois theory

Details A density theorem for parameterized differential Galois theory A density theorem for parameterized differential Galois theory

2012-03-13

arxiv.org/abs/1203.2904v1

Mathematics

Classical Analysis and ODEs

Scientific paper

We study parameterized linear differential equations with coefficients depending meromorphically upon the parameters. As a main result, analogous to Ramis's theorem in the unparameterized case, we show that the parameterized monodromy, the parameterized exponential torus and the parameterized Stokes operators are topological generators of the parameterized differential Galois group in Kolchin's topology. We prove an analogous result for the global parameterized differential Galois group and we give a sufficient condition on a group for being a global parameterized differential Galois group. As an application, we give a characterization of the completely integrable equations and we give a partial answer to a question of Sibuya. Moreover, we prove a parameterized Turrittin theorem, which allow to show directly that the Galois group descends to a smaller field whose field of constant is only algebraically closed rather than differentially closed.

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