Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

IMPACT FACTOR 2018: 0.490

CiteScore 2018: 0.47

SCImago Journal Rank (SJR) 2018: 0.279
Source Normalized Impact per Paper (SNIP) 2018: 0.627

Mathematical Citation Quotient (MCQ) 2017: 0.26

See all formats and pricing
More options …
Volume 64, Issue 6


Polynomials with coefficients from a finite set

Javad Baradaran / Mohsen Taghavi
Published Online: 2015-01-10 | DOI: https://doi.org/10.2478/s12175-014-0282-y


This paper focuses on the problem concerning the location and the number of zeros of those polynomials when their coefficients are restricted with special conditions. The problem of the number of the zeros of reciprocal Littlewood polynomials on the unit circle $\mathbb{T}$ is discussed, the interest on bounds for the number of the zeros of reciprocal polynomials on the unit circle arose after 1950 when Erdös began introducing problems on zeros of various types of polynomials. Our main result is the problem of finding the number of zeros of complex polynomials in an open disk.

MSC: Primary 26D05, 41A10

Keywords: Littlewood polynomial; Pisot number; reciprocal polynomial

  • [1] BIDKHAM, M.— DEWAN, K. K.: Bounds for the zeros of polynomials, Indian J. Pure Appl. Math. 20 (1989), 768–772. Google Scholar

  • [2] BORWEIN, P.— ERDÉLYI, T.: Polynomials and Polynomial Inequalities. Grad. Texts in Math., Springer-Verlag, New York, 1995. http://dx.doi.org/10.1007/978-1-4612-0793-1CrossrefGoogle Scholar

  • [3] BORWEIN, P.— ERDÉLYI, T.— KÓS, G.: Littlewood-type problems on [0, 1], Proc. Lond. Math. Soc. (3) 79 (1999), 22–46. http://dx.doi.org/10.1112/S0024611599011831Google Scholar

  • [4] BORWEIN, P.— ERDÉLYI, T.— LITTMANN, F.: Zeros of polynomials with finitely many different coefficients, Trans. Amer. Math. Soc. (To appear). Google Scholar

  • [5] CAUCHY, A. L.: Exercises de mathématique, IV Annee de Bure Freres, Paris, 1829. Google Scholar

  • [6] DRUNGILAS, P.: Unimodular roots of reciprocal Littlewood polynomials, J. KoreanMath. Soc. 45 (2008), 835–840. Google Scholar

  • [7] ERDÉLYI, T.: On the zeros of polynomials with Littlewood-type coefficient constraints, Michigan Math. J. 49 (2001), 97–111. http://dx.doi.org/10.1307/mmj/1008719037CrossrefGoogle Scholar

  • [8] ERDÖS, P.— TURÀN, P.: On the distribution of roots of polynomials, Ann. of Math. 324 (1997), 105–119. Google Scholar

  • [9] LAKATOS, P.: On a number theoretical application of Coxeter transformations, Riv. Mat. Univ. Parma (8) 3 (2000), 293–301. Google Scholar

  • [10] LAKATOS, P.— LOSONCZY, L.: On zeros of reciprocal polynomials of odd degree, J. Inequal. Pure Appl. Math. 4 (2003), Article 60, 8 pp. Google Scholar

  • [11] MARDEN, M.: Geometry of Polynomials (2nd ed.). Math. Surveys Monogr. 3, Amer. Math. Soc., Providence, RI, 1966. Google Scholar

  • [12] MITRINOVIC, D. S.: Analytic Inequalities, Springer-Verlag, Berlin, 1970. http://dx.doi.org/10.1007/978-3-642-99970-3CrossrefGoogle Scholar

  • [13] MUKUNDA, K.: Littlewood Pisot numbers, J. Number Theory 117 (2006), 106–121. http://dx.doi.org/10.1016/j.jnt.2005.05.009CrossrefGoogle Scholar

  • [14] POLYA, G.— SZEGO, G.: Problems and Theorems in Analysis, Vol. I, Springer Verlag, Berlin, 1972. http://dx.doi.org/10.1007/978-1-4757-1640-5CrossrefGoogle Scholar

  • [15] RAHMAN, Q. I.— SCHMEISSER, G.: Analytic Theorey of Polynomials, Clarendon Press, Oxford, 2002. Google Scholar

  • [16] SALEM, R.: Algebraic Numbers and Fourier Analysis, D. C. Heath and Co., Boston, MA, 1963. Google Scholar

  • [17] SHAH, W. M.— LIMAN, A.: Bounds for the zeros of polynomials, Anal. Theory Appl. 20 (2004), 16–27. http://dx.doi.org/10.1007/BF02835255CrossrefGoogle Scholar

About the article

Published Online: 2015-01-10

Published in Print: 2014-12-01

Citation Information: Mathematica Slovaca, Volume 64, Issue 6, Pages 1397–1408, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.2478/s12175-014-0282-y.

Export Citation

© 2014 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in