Real Mahler Series

TL;DR

This paper proves that under certain conditions, a Hahn series that satisfies both alpha- and beta-Mahler equations must be a rational function, extending understanding of Mahler functions in the context of Hahn series.

Contribution

It establishes a new rigidity result for Hahn series satisfying multiple Mahler equations, showing they are necessarily rational functions under specific conditions.

Findings

Hahn series satisfying both alpha- and beta-Mahler are rational functions.

The result applies under non-degeneracy conditions on alpha and beta.

Extends classical Mahler function theory to Hahn series.

Abstract

Let $\alpha$ and $\beta$ be positive real numbers. Let $F(x) \in K[[x^\Gamma]]$ be a Hahn series. We prove that if $F(x)$ is both $\alpha$-Mahler and $\beta$-Mahler then it must be a rational function, $F(x) \in K(x)$, assuming some non-degeneracy conditions on $\alpha$ and $\beta$.