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arXiv · 2511.18877

Computing basis of solutions of any Mahler equation

Colin Faverjon·Marina Poulet
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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.

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Colin Faverjon, Marina Poulet. 2025-11-24. Computing basis of solutions of any Mahler equation. https://arxiv.org/abs/2511.18877

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