On Littlewood and Newman polynomial multiples of Borwein polynomials
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- by P. Drungilas, J. Jankauskas and J. Šiurys PDF
- Math. Comp. 87 (2018), 1523-1541 Request permission
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 $|z|=1$ 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.References
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Additional Information
- P. Drungilas
- Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
- MR Author ID: 724133
- Email: pdrungilas@gmail.com
- J. Jankauskas
- Affiliation: Mathematik und Statistik, Montanuniversität Leoben, Franz Josef Straße 18, A-8700 Leoben, Austria
- MR Author ID: 825362
- ORCID: 0000-0001-9770-7632
- 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
- 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) - © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 1523-1541
- MSC (2010): Primary 11R09, 11Y16, 12D05, 11R06
- DOI: https://doi.org/10.1090/mcom/3258
- MathSciNet review: 3766397