Becker's conjecture on Mahler functions

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 $z^γ Q(z)$ for some explicit non-negative integer $γ$ and such that $1/Q(z)$ is k-regular.
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Submitted on : Tuesday, October 2, 2018 - 10:10:52 AM
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Jason Bell, Frédéric Chyzak, Michael Coons, Philippe Dumas. Becker's conjecture on Mahler functions. Transactions of the American Mathematical Society, American Mathematical Society, In press, 372, pp.3405--3423. ⟨10.1090/tran/7762 ⟩. ⟨hal-01885598⟩

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