Becker’s conjecture on Mahler functions

Bell, Jason; Chyzak, Frédéric; Coons, Michael; Dumas, Philippe

doi:10.1090/tran/7762

Becker’s conjecture on Mahler functions
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by Jason P. Bell, Frédéric Chyzak, Michael Coons and Philippe Dumas PDF

Trans. Amer. Math. Soc. 372 (2019), 3405-3423

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 nonnegative integer $\gamma$ and such that $1/Q(z)$ is $k$-regular.

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