Hahn series and Mahler equations: Algorithmic aspects

TL;DR

This paper explores the algorithmic problem of computing Hahn series solutions to linear Mahler equations, addressing the complexity of their supports and providing a constructive approach.

Contribution

It introduces an algorithmic method to compute Hahn series solutions for Mahler equations, including a way to handle their complex supports.

Findings

Developed an algorithm for Hahn series solutions

Constructed a computable well-ordered support structure

Addressed supports with infinitely many accumulation points

Abstract

Many articles have recently been devoted to Mahler equations, partly because of their links with other branches of mathematics such as automata theory. Hahn series (a generalization of the Puiseux series allowing arbitrary exponents of the indeterminate as long as the set that supports them is well-ordered) play a central role in the theory of Mahler equations. In this paper, we address the following fundamental question: is there an algorithm to calculate the Hahn series solutions of a given linear Mahler equation? What makes this question interesting is the fact that the Hahn series appearing in this context can have complicated supports with infinitely many accumulation points. Our (positive) answer to the above question involves among other things the construction of a computable well-ordered receptacle for the supports of the potential Hahn series solutions.