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Unimodular Roots of Special Littlewood Polynomials

Published online by Cambridge University Press:  20 November 2018

Idris David Mercer
Idris David Mercer*
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6 e-mail: idmercer@math.sfu.ca
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Abstract

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We call $\alpha \left( z \right)={{a}_{0}}+{{a}_{1}}z+\cdot \cdot \cdot +{{a}_{n-1}}{{z}^{n-1}}$ a Littlewood polynomial if ${{a}_{j}}=\pm 1$ for all $j$. We call $\alpha \left( z \right)$ self-reciprocal if $\alpha \left( z \right)={{z}^{n-1}}\alpha \left( 1/z \right)$, and call $\alpha \left( z \right)$ skewsymmetric if $n=2m+1$ and ${{a}_{m+j}}={{\left( -1 \right)}^{j}}{{a}_{m-j}}$ for all $j$. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in $\mathbb{C}$ tend to be skewsymmetric. In this paper, we prove that a skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle, as well as providing a new proof of the known result that a self-reciprocal Littlewood polynomial must have a zero on the unit circle.

Keywords

26C1030C1542A05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 3 , 01 September 2006 , pp. 438 - 447
Copyright
Copyright © Canadian Mathematical Society 2006

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