Computing basis of solutions of any Mahler equation

TL;DR

This paper introduces a comprehensive algorithm to compute a complete basis of solutions for any Mahler equation, extending beyond previous limitations to include solutions over Puiseux series.

Contribution

It provides the first general algorithm capable of computing solutions for any Mahler equation and decomposing them over Puiseux series, advancing the understanding of Mahler functions.

Findings

Algorithm computes a basis of solutions for any Mahler equation.

Decomposition of solutions over Puiseux series is achieved.

Fundamental matrix of solutions for Mahler systems is constructed.

Abstract

Mahler equations arise in a wide range of contexts including the study of finite automata, regular sequences, algebraic series over Fp(z), and periods of Drinfeld modules. Introduced a century ago by K. Mahler to study the transcendence of certain complex numbers, they have recently been the subject of several works establishing a deep connection between such transcendence properties and the nature of their solutions. While numerous studies have investigated these solutions, existing algorithms can only compute them in specific rings: rational functions, power series, Puiseux series, or Hahn series. This paper solves the problem by providing an algorithm that computes a complete basis of solutions for any Mahler equation, along with a decomposition of each solution over the field of Puiseux series. Along the way, we describe an algorithm that computes a fundamental matrix of solutions for any Mahler system.