Unimodular Roots of Special Littlewood Polynomials

Mercer, Idris David

Canadian mathematical bulletin, Tome 49 (2006) no. 3, pp. 438-447

Résumé

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.

Détail

DOI : 10.4153/CMB-2006-043-x

Mots-clés : 26C10, 30C15, 42A05