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arXiv · 1802.08653

Becker's conjecture on Mahler functions

Jason Bell·Frederic Chyzak·Michael Coons·Philippe Dumas
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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.

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Jason Bell, Frederic Chyzak, Michael Coons, Philippe Dumas. 2018-02-23. Becker's conjecture on Mahler functions. https://arxiv.org/abs/1802.08653

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