Becker's conjecture on Mahler functions

TL;DR

This paper proves Becker's conjecture, demonstrating that for any $k$-regular power series, there exists a polynomial multiple making the series satisfy a Mahler-type functional equation with a regular inverse polynomial.

Contribution

The paper confirms Becker's conjecture by showing the rational function can be chosen as a polynomial times a $k$-regular inverse, in the strongest possible form.

Findings

The rational function $R(z)$ can be taken as $z^3 ext{Q}(z)$ for some explicit $3$.

The inverse of $Q(z)$ is $k$-regular.

The conjecture is proved in the best-possible form.

Abstract

In 1994, Becker conjectured that if $F(z)$ is a $k$-regular power series, then there exists a $k$-regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahler-type functional equation with polynomial coefficients where the initial coefficient satisfies $a_0(z)=1$. In this paper, we prove Becker's conjecture in the best-possible form; we show that the rational function $R(z)$ can be taken to be a polynomial $z^\gamma Q(z)$ for some explicit non-negative integer $\gamma$ and such that $1/Q(z)$ is $k$-regular.