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On Littlewood and Newman polynomial multiples of Borwein polynomials


Authors: P. Drungilas, J. Jankauskas and J. Šiurys
Journal: Math. Comp. 87 (2018), 1523-1541
MSC (2010): Primary 11R09, 11Y16, 12D05, 11R06
DOI: https://doi.org/10.1090/mcom/3258
Published electronically: September 19, 2017
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Abstract: A Newman polynomial has all the coefficients in $ \{0,1\}$ and constant term 1, whereas a Littlewood polynomial has all coefficients in $ \{-1,1\}$. We call $ P(X)\in \mathbb{Z}[X]$ a Borwein polynomial if all its coefficients belong to $ \{-1,0,1\}$ and $ P(0)\neq 0$. By exploiting an algorithm which decides whether a given monic integer polynomial with no roots on the unit circle $ \vert z\vert=1$ has a non-zero multiple in $ \mathbb{Z}[X]$ with coefficients in a finite set $ \mathcal {D}\subset \mathbb{Z}$, for every Borwein polynomial of degree at most 9 we determine whether it divides any Littlewood or Newman polynomial. In particular, we show that every Borwein polynomial of degree at most 8 which divides some Newman polynomial divides some Littlewood polynomial as well. In addition to this, for every Newman polynomial of degree at most 11, we check whether it has a Littlewood multiple, extending the previous results of Borwein, Hare, Mossinghoff, Dubickas and Jankauskas.


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Additional Information

P. Drungilas
Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
Email: pdrungilas@gmail.com

J. Jankauskas
Affiliation: Mathematik und Statistik, Montanuniversität Leoben, Franz Josef Straße 18, A-8700 Leoben, Austria
Email: jonas.jankauskas@gmail.com

J. Šiurys
Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
Email: jonas.siurys@mif.vu.lt

DOI: https://doi.org/10.1090/mcom/3258
Keywords: Borwein polynomial, Littlewood polynomial, Newman polynomial, Pisot number, Salem number, Mahler measure, polynomials of small height
Received by editor(s): September 23, 2016
Received by editor(s) in revised form: December 16, 2016
Published electronically: September 19, 2017
Additional Notes: The first author was supported by the Research Council of Lithuania grant MIP-049/2014
The second author was supported by project P27050 Fractals and Words: Topological, Dynamical, and Combinatorial Aspects funded by the Austrian Science Fund (FWF)
Article copyright: © Copyright 2017 American Mathematical Society

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