Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON CERTAIN MULTIPLES OF LITTLEWOOD AND NEWMAN POLYNOMIALS

  • Received : 2017.10.03
  • Accepted : 2018.05.02
  • Published : 2018.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Polynomials with all the coefficients in {0, 1} and constant term 1 are called Newman polynomials, whereas polynomials with all the coefficients in {-1, 1} are called Littlewood polynomials. By exploiting an algorithm developed earlier, we determine the set of Littlewood polynomials of degree at most 12 which divide Newman polynomials. Moreover, we show that every Newman quadrinomial $X^a+X^b+X^c+1$, 15 > a > b > c > 0, has a Littlewood multiple of smallest possible degree which can be as large as 32765.

Keywords

Acknowledgement

Grant : Number Systems, Spectra and Rational Fractal Tiles

Supported by : Research Council of Lithuania, FWF

References

  1. S. Akiyama, J. M. Thuswaldner, and T. Zaimi, Comments on the height reducing property II, Indag. Math. (N.S.) 26 (2015), no. 1, 28-39. https://doi.org/10.1016/j.indag.2014.07.002
  2. P. Borwein and K. G. Hare, Some computations on the spectra of Pisot and Salem numbers, Math. Comp. 71 (2002), no. 238, 767-780. https://doi.org/10.1090/S0025-5718-01-01336-9
  3. P. Drungilas, J. Jankauskas, and J. Siurys, On Littlewood and Newman polynomial multiples of Borwein polynomials, Math. Comp. 87 (2018), no. 311, 1523-1541.
  4. A. Dubickas and J. Jankauskas, On Newman polynomials which divide no Littlewood polynomial, Math. Comp. 78 (2009), no. 265, 327-344. https://doi.org/10.1090/S0025-5718-08-02138-8
  5. C. Frougny, Representations of numbers and finite automata, Math. Systems Theory 25 (1992), no. 1, 37-60. https://doi.org/10.1007/BF01368783
  6. K. G. Hare and M. J. Mossinghoff, Negative Pisot and Salem numbers as roots of Newman polynomials, Rocky Mountain J. Math. 44 (2014), no. 1, 113-138. https://doi.org/10.1216/RMJ-2014-44-1-113
  7. F. Johansson, Arb: effcient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Comput. 66 (2017), no. 8, 1281-1292. https://doi.org/10.1109/TC.2017.2690633
  8. K.-S. Lau, Dimension of a family of singular Bernoulli convolutions, J. Funct. Anal. 116 (1993), no. 2, 335-358. https://doi.org/10.1006/jfan.1993.1116
  9. OpenMP Architecture Review Board, OpenMP, Application Program Interface Version 4.0, 2013, http://www.openmp.org/mp-documents/OpenMP4.0.0.pdf.
  10. R. Salem, A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan, Duke Math. J. 11 (1944), 103-108. https://doi.org/10.1215/S0012-7094-44-01111-7
  11. R. Salem, Power series with integral coefficients, Duke Math. J. 12 (1945), 153-172. https://doi.org/10.1215/S0012-7094-45-01213-0
  12. R. Salem, Algebraic Numbers and Fourier Analysis, D. C. Heath and Co., Boston, MA, 1963.
  13. D. Stankov, On spectra of neither Pisot nor Salem algebraic integers, Monatsh. Math. 159 (2010), no. 1-2, 115-131. https://doi.org/10.1007/s00605-008-0048-0
  14. The Sage Developers, SageMath, the Sage Mathematics Software System (Version 7.6), 2017; http://www.sagemath.org.
  15. T. Zaimi, On real parts of powers of complex Pisot numbers, Bull. Aust. Math. Soc. 94 (2016), no. 2, 245-253. https://doi.org/10.1017/S0004972716000125