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

Abstract

This is a largely expository paper, providing a self-contained account on the results of Schäfke and Singer (2017, 2019), in the cases denoted there 2Q and 2M. These papers supplied new proofs to the main theorems of Adamczewski and Bell (2017) and Bézivin and Boutabaa (1992), 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.