A Density Theorem for the Difference Galois Groups of Regular Singular Mahler Equations

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.

Hahn series and Mahler equations: Algorithmic aspects

“…This is well illustrated by the work of Shäfke and Singer in [37] which gives a new proof of a conjecture of Loxton and van der Poorten — previously established by Adamczewski and Bell in [1] — and therefore, a new proof of Cobham's theorem in automata theory by using tools coming from the theory of functional equations. Here are some references [1–8, 10, 11, 13–19, 21, 22, 26, 28–38].…”