Unimodular roots of special littlewood polynomials

Unimodular roots of special littlewood polynomials Article

Mercer, ID. (2006). Unimodular roots of special littlewood polynomials . Canadian Mathematical Bulletin, 49(3), 438-447. 10.4153/CMB-2006-043-x

We call α(z) = a0 + a1z + ⋯ + a n-1z^n-1 a Littlewood polynomial if aj = ±1 for all j. We call α(z) self-reciprocal if α(z) = z ^n-1α(1/z), and call α(z) skewsymmetric if n = 2m + 1 and a^m+j = (-1)^jam-j for all j. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in 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. © Canadian Mathematical Society 2006.