A density theorem for the difference Galois groups of regular singular Mahler equations

TL;DR

This paper develops the analytic theory of Mahler equations by linking connection matrices to the structure of their difference Galois groups, showing these groups are Zariski-dense.

Contribution

It introduces a method to associate connection matrices with regular singular Mahler equations and proves their Galois groups are Zariski-dense using these matrices.

Findings

Connection matrices can be attached to regular singular Mahler equations.

These matrices can generate Zariski-dense subgroups of the Galois group.

The approach advances understanding of the Galois groups' structure.

Abstract

The difference Galois theory of Mahler equations is an active research area. The present paper aims at developing the analytic aspects of this theory. We first attach a pair of connection matrices to any regular singular Mahler equation. We then show that these connection matrices can be used to produce a Zariski-dense subgroup of the difference Galois group of any regular singular Mahler equation.