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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

Becker's conjecture on Mahler functions


Authors: Jason P. Bell, Frédéric Chyzak, Michael Coons and Philippe Dumas
Journal: Trans. Amer. Math. Soc. 372 (2019), 3405-3423
MSC (2010): Primary 11B85, 30B10; Secondary 68R15
DOI: https://doi.org/10.1090/tran/7762
Published electronically: May 31, 2019
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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 nonnegative integer $ \gamma $ and such that $ 1/Q(z)$ is $ k$-regular.


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Additional Information

Jason P. Bell
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada
Email: jpbell@uwaterloo.ca

Frédéric Chyzak
Affiliation: INRIA, Université Paris–Saclay, Paris, France
Email: Frederic.Chyzak@inria.fr

Michael Coons
Affiliation: School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, New South Wales, Australia
Email: Michael.Coons@newcastle.edu.au

Philippe Dumas
Affiliation: INRIA, Université Paris–Saclay, Paris, France
Email: Philippe.Dumas@inria.fr

DOI: https://doi.org/10.1090/tran/7762
Keywords: Automatic sequences, regular sequences, Mahler functions
Received by editor(s): April 20, 2018
Received by editor(s) in revised form: November 23, 2018
Published electronically: May 31, 2019
Additional Notes: The research of the first author was partially supported by an NSERC Discovery Grant.
The third author was visiting the Alfréd Rényi Institute of the Hungarian Academy of Sciences during the time this research was undertaken; he thanks the Institute and its members for their kindness and support.
Article copyright: © Copyright 2019 by the authors