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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.


References [Enhancements On Off] (What's this?)

  • [1] A. C. Aitken, Determinants and Matrices, 9th ed. Interscience Pub., New York, 1956.
  • [2] Shigeki Akiyama, Jörg M. Thuswaldner, and Toufik Zaïmi, Comments on the height reducing property II, Indag. Math. (N.S.) 26 (2015), no. 1, 28-39. MR 3281687, https://doi.org/10.1016/j.indag.2014.07.002
  • [3] Mohamed Amara, Ensembles fermés de nombres algébriques, Ann. Sci. École Norm. Sup. (3) 83 (1966), 215-270 (1967) (French). MR 0237459
  • [4] Peter Borwein and Kevin G. Hare, Some computations on the spectra of Pisot and Salem numbers, Math. Comp. 71 (2002), no. 238, 767-780. MR 1885627, https://doi.org/10.1090/S0025-5718-01-01336-9
  • [5] David W. Boyd, Pisot and Salem numbers in intervals of the real line, Math. Comp. 32 (1978), no. 144, 1244-1260. MR 0491587, https://doi.org/10.2307/2006349
  • [6] David W. Boyd, Pisot numbers in the neighbourhood of a limit point. I, J. Number Theory 21 (1985), no. 1, 17-43. MR 804914, https://doi.org/10.1016/0022-314X(85)90010-1
  • [7] Artūras Dubickas and Jonas Jankauskas, On Newman polynomials which divide no Littlewood polynomial, Math. Comp. 78 (2009), no. 265, 327-344. MR 2448710, https://doi.org/10.1090/S0025-5718-08-02138-8
  • [8] J. Dufresnoy and Ch. Pisot, Etude de certaines fonctions méromorphes bornées sur le cercle unité. Application à un ensemble fermé d'entiers algébriques, Ann. Sci. Ecole Norm. Sup. (3) 72 (1955), 69-92 (French). MR 0072902
  • [9] Kevin G. Hare and Michael J. Mossinghoff, Negative Pisot and Salem numbers as roots of Newman polynomials, Rocky Mountain J. Math. 44 (2014), no. 1, 113-138. MR 3216012, https://doi.org/10.1216/RMJ-2014-44-1-113
  • [10] Roger A. Horn and Charles R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1994. Corrected reprint of the 1991 original. MR 1288752
  • [11] F. Johansson, Arb: a C library for ball arithmetic, ACM Communications in Computer Algebra 47 (4) (2013), 166-169, http://fredrikj.net/arb/.
  • [12] Dan Kalman, The generalized Vandermonde matrix, Math. Mag. 57 (1984), no. 1, 15-21. MR 729034, https://doi.org/10.2307/2690290
  • [13] Ka-Sing Lau, Dimension of a family of singular Bernoulli convolutions, J. Funct. Anal. 116 (1993), no. 2, 335-358. MR 1239075, https://doi.org/10.1006/jfan.1993.1116
  • [14] C. Méray, Sur un déterminant dont celui de Vandermonde n'est qu'un cas particulier, Rev. math. spéc. 9 (1899), 217-219.
  • [15] Michael J. Mossinghoff, Polynomials with restricted coefficients and prescribed noncyclotomic factors, LMS J. Comput. Math. 6 (2003), 314-325. MR 2051588, https://doi.org/10.1112/S1461157000000474
  • [16] A. M. Odlyzko and B. Poonen, Zeros of polynomials with $ 0,1$ coefficients, Enseign. Math. (2) 39 (1993), no. 3-4, 317-348. MR 1252071
  • [17] OpenMP Architecture Review Board, OpenMP Application Program Interface Version 4.0, 2013, http://www.openmp.org/mp-documents/OpenMP4.0.0.pdf
  • [18] Charles Pisot, La répartition modulo 1 et les nombres algébriques, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 7 (1938), no. 3-4, 205-248 (French). MR 1556807
  • [19] R. Salem, A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan, Duke Math. J. 11 (1944), 103-108. MR 0010149
  • [20] R. Salem, Power series with integral coefficients, Duke Math. J. 12 (1945), 153-172. MR 0011720
  • [21] Raphaël Salem, Algebraic Numbers and Fourier Analysis, D. C. Heath and Co., Boston, Mass., 1963. MR 0157941
  • [22] Dragan Stankov, On spectra of neither Pisot nor Salem algebraic integers, Monatsh. Math. 159 (2010), no. 1-2, 115-131. MR 2564390, https://doi.org/10.1007/s00605-008-0048-0
  • [23] W. A. Stein et al., Sage Mathematics Software (Version 7.2), The Sage Development Team, 2016, http://www.sagemath.org.

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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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