# 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.
Keywords :
Document type :
Journal articles
Domain :

Cited literature [15 references]

https://hal.inria.fr/hal-01885598
Contributor : Frédéric Chyzak <>
Submitted on : Tuesday, October 2, 2018 - 10:10:52 AM
Last modification on : Monday, December 9, 2019 - 4:34:34 PM
Long-term archiving on: Thursday, January 3, 2019 - 1:33:19 PM

### File

BellChyzakCoonsDumas-2018-BCM....
Files produced by the author(s)

### Citation

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⟩

Record views