UNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS

Title & Authors
UNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS
Drungilas, Paulius;

Abstract
The main result of this paper shows that every reciprocal Littlewood polynomial, one with {-1, 1} coefficients, of odd degree at least 7 has at least five unimodular roots, and every reciprocal Little-wood polynomial of even degree at least 14 has at least four unimodular roots, thus improving the result of Mukunda. We also give a sketch of alternative proof of the well-known theorem characterizing Pisot numbers whose minimal polynomials are in $\small{A_N=\{[{X^d+ \sum\limits^{d-1}_{k=0} a_k\;X^k{\in} \mathbb{Z}[X]\;:\;a_k={\pm}N,\;0{\leqslant}k{\leqslant}d-1}\}}$ for positive integer $\small{N{\geqslant}2}$.
Keywords
Pisot numbers;Littlewood polynomials;unimodular roots;reciprocal polynomiab;
Language
English
Cited by
1.
A family of self-inversive polynomials with concyclic zeros, Journal of Mathematical Analysis and Applications, 2013, 401, 2, 695
References
1.
P. Borwein, T. Erdelyi, R. Ferguson, and R. Lockhart, On the zeros of cosine polynomials: solution of an old problem of Littlewood (submitted)

2.
P. Borwein, T. Erdelyi, and F. Littmann, Zeros of polynomials with finitely many different coefficients, Trans. Amer. Math. Soc. (to appear)

3.
P. Borwein, K. G. Hare, and M. J. Mossinghoff, The Mahler measure of polynomials with odd coefficients, Bull. Lond. Math. Soc. 36 (2004), no. 3, 332-338

4.
D. W. Boyd, Pisot and Salem numbers in intervals of the real line, Math. Comp. 32 (1978), no. 144, 1244-1260

5.
T. Erdelyi, On the zeros of polynomials with Littlewood-type coefficient constraints, Michigan Math. J. 49 (2001), no. 1, 97-111

6.
J. Konvalina and V. Matache, Palindrome-polynomials with roots on the unit circle, C. R. Math. Acad. Sci. Soc. R. Can. 26 (2004), no. 2, 39-44

7.
P. Lakatos, On a number theoretical application of Coxeter transformations, Riv. Mat. Univ. Parma (6) 3 (2000), 293-301

8.
P. Lakatos, On polynomials having zeros on the unit circle, C. R. Math. Acad. Sci. Soc. R. Can. 24 (2002), no. 2, 91-96

9.
P. Lakatos, On zeros of reciprocal polynomials, Publ. Math. Debrecen 61 (2002), no. 3-4, 645-661

10.
P. Lakatos and L. Losonczy, On zeros of reciprocal polynomials of odd degree, J. Inequal. Pure Appl. Math. 4 (2003), no. 3, Article 60, 8 pp

11.
P. Lakatos, Self-inversive polynomials whose zeros are on the unit circle, Publ. Math. Debrecen 65 (2004), no. 3-4, 409-420

12.
M. Marden, Geometry of Polynomials, Second edition. Mathematical Surveys, No. 3 American Mathematical Society, Providence, R.I. 1966

13.
I. D. Mercer, Unimodular roots of special Littlewood polynomials, Canad. Math. Bull. 49 (2006), no. 3, 438-447

14.
K. Mukunda, Littlewood Pisot numbers, J. Number Theory 117 (2006), no. 1, 106-121

15.
R. Salem, Algebraic Numbers and Fourier Analysis, D. C. Heath and Co., Boston, Mass. 1963

16.
A. Schinzel, Self-inversive polynomials with all zeros on the unit circle, Ramanujan J. 9 (2005), no. 1-2, 19-23

17.
C. L. Siegel, Algebraic integers whose conjugates lie in the unit circle, Duke Math. J. 11 (1944), 597-602

18.
C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. Lond. Math. Soc. 3 (1971), 169-175

19.
J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall Series in Automatic Computation Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964