A density theorem for parameterized differential Galois theory

Abstract

We study parameterized linear differential equations with coefficients depending meromorphically upon the parameters. As a main result, analogously to the unparameterized density theorem of Ramis, we show that the parameterized monodromy, the parameterized exponential torus and the parameterized Stokes operators are topological generators in Kolchin topology, for the parameterized differential Galois group introduced by Cassidy and Singer. We prove an analogous result for the global parameterized 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 parameterized 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.

To appear in Pacific Journal of Mathematics

References in corpus (2)

• Parameterized Picard-Vessiot extensions and Atiyah extensions
• Isomonodromic differential equations and differential categories

Cited by in corpus (9)

• Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations
• Computing the differential Galois group of a parameterized second-order linear differential equation
• Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation
• Difference algebraic relations among solutions of linear differential equations
• Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation
• Parameterized Differential Equations over k((t))(x)
• Some applications of the parameterized Picard-Vessiot theory
• Calculating Galois groups of third order linear differential equations with parameters
• Differential Galois Theory and Isomonodromic Deformations