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Article Dans Une Revue Pacific Journal of Mathematics Année : 2014

A DENSITY THEOREM IN PARAMETERIZED DIFFERENTIAL GALOIS THEORY

Thomas Dreyfus

Résumé

We study parameterized linear differential equations with coefficients depending meromorphically upon the parameters. As a main result, analogously to the un-parameterized density theorem of Ramis, we show that the parameterized monodromy, the parameterized exponential torus and the parameterized Stokes operators are topo-logical generators in Kolchin topology, for the parameterized differential Galois group introduced by Cassidy and Singer. We prove an analogous result for the global pa-rameterized 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 parameter-ized 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 parameterized Hukuhara-Turrittin theorem, we show that the Galois group descends to a smaller field, whose field of constants is not differentially closed.
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Dates et versions

hal-01897295 , version 1 (17-10-2018)

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Thomas Dreyfus. A DENSITY THEOREM IN PARAMETERIZED DIFFERENTIAL GALOIS THEORY. Pacific Journal of Mathematics, 2014, 271 (1), pp.87 - 141. ⟨10.2140/pjm.2014.271.87⟩. ⟨hal-01897295⟩
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