Regular singular Mahler equations and Newton polygons

TL;DR

This paper advances the understanding of Mahler equations by improving algorithms to determine regular singularities at zero, utilizing Newton polygons and Puiseux series for more efficient analysis.

Contribution

The authors enhance an existing algorithm for identifying regular singularities in Mahler equations, reducing complexity and extending applicability to equations with Puiseux coefficients.

Findings

Improved algorithm for regular singularity detection

Reduced computational complexity

Extended scope to Puiseux coefficient equations

Abstract

Though Mahler equations have been introduced nearly one century ago, the study of their solutions is still a fruitful topic for research. In particular, the Galois theory of Mahler equations has been the subject of many recent papers. Nevertheless, long is the way to a complete understanding of relations between solutions of Mahler equations. One step along this way is the study of singularities. Mahler equations with a regular singularity at 0 have rather "nice" solutions: they can be expressed with the help of Puiseux series and solutions of equations with constant coefficients. In a previous paper, the authors described an algorithm to determine whether an equation is regular singular at 0 or not. Exploiting information from the Frobenius method and Newton polygons, we improve this algorithm by significantly reducing its complexity, by providing some simple criterion for an equation to be regular singular at 0, and by extending its scope to equations with Puiseux coefficients.