On the structure of certain $\Gamma$-difference modules

TL;DR

This paper provides an expository overview of $ ext{Gamma}$-difference modules, revisiting key results on rationality of power series satisfying q-difference or Mahler equations, and introduces a new case with generalized dihedral groups.

Contribution

It offers a self-contained account of previous results using $ ext{Gamma}$-difference modules and extends the theory to a new case with non-abelian groups and finite characteristic.

Findings

Revisited and simplified proofs of rationality results for q-difference and Mahler equations.

Introduced a new case with generalized dihedral $ ext{Gamma}$-groups.

Extended main theorems to finite characteristic settings.

Abstract

This is a largely expository paper, providing a self-contained account on the results of [Sch-Si1, Sch-Si2], in the cases denoted there 2Q and 2M. These papers of Sch\"afke and Singer supplied new proofs to the main theorems of [Bez-Bou, Ad-Be], on the rationality of power series satisfying a pair of independent q-difference, or Mahler, equations. We emphasize the language of $\Gamma$-difference modules, instead of difference equations or systems. Although in the two cases mentioned above this is only a semantic change, we also treat a new case, which may be labeled 1M1Q. Here the group $\Gamma$ is generalized dihedral rather than abelian, and the language of equations is inadequate. In the last section we explain how to generalize the main theorems in case 2Q to finite characteristic.