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UNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS

  • Drungilas, Paulius
  • Published : 2008.05.31

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 $$A_N=\{[{X^d+ \sum\limits^{d-1}_{k=0} a_k\;X^k{\in} \mathbb{Z}[X]\;:\;a_k={\pm}N,\;0{\leqslant}k{\leqslant}d-1}\}$$ for positive integer $N{\geqslant}2$.

Keywords

Pisot numbers;Littlewood polynomials;unimodular roots;reciprocal polynomiab

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  1. A family of self-inversive polynomials with concyclic zeros vol.401, pp.2, 2013, https://doi.org/10.1016/j.jmaa.2012.12.048
  2. On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk vol.67, pp.03, 2015, https://doi.org/10.4153/CJM-2014-007-1