An algorithm to recognize regular singular Mahler systems

TL;DR

This paper develops an algorithm to determine whether Mahler systems are regular singular at 0, filling a gap in the theory by providing an effective characterization specific to Mahler systems, unlike existing methods for differential and q-difference systems.

Contribution

It introduces the first effective algorithm to decide regular singularity at 0 for Mahler systems, extending the analytic theory to this class.

Findings

Provides an algorithm to recognize regular singular Mahler systems at 0.

Characterizes Mahler systems equivalent to constant systems.

Enables application of Schlesinger's density theorem analogs.

Abstract

This paper is devoted to the study of the analytic properties of Mahler systems at 0. We give an effective characterisation of Mahler systems that are regular singular at 0, that is, systems which are equivalent to constant ones. Similar characterisations already exist for differential and (q-)difference systems but they do not apply in the Mahler case. This work fills in the gap by giving an algorithm which decides whether or not a Mahler system is regular singular at 0. In particular, it gives an effective characterisation of Mahler systems to which an analog of Schlesinger's density theorem applies.