A density theorem for parameterized differential Galois theory

TL;DR

This paper establishes a density theorem for parameterized differential Galois groups, showing that certain monodromy and Stokes operators generate the group topologically, with applications to integrability and transcendence questions.

Contribution

It generalizes the unparameterized density theorem to the parameterized setting and characterizes the Galois group via topological generators, extending previous results by Cassidy, Singer, Mitschi, and Sibuya.

Findings

Parameterized monodromy and Stokes operators generate the Galois group topologically.

A necessary and sufficient condition for a group to be a global parameterized Galois group.

Application to characterizing integrable equations and analyzing Stokes matrix transcendence.

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.