On Newman and Littlewood polynomials with prescribed number of zeros inside the unit disk

Abstract

We study {0, 1} and {−1, 1} polynomials f (z), called Newman and Littlewood polynomials, that have a prescribed number N (f ) of zeros in the open unit disk D = {z ∈ C : |z| < 1}. For every pair (k, n) ∈ N 2 , where n ≥ 7 and k ∈ [3, n − 3], we prove that it is possible to find a {0, 1}-polynomial f (z) of degree deg f = n with non-zero constant term f (0) = 0, such that N (f ) = k and f (z) = 0 on the unit circle ∂D. On the way to this goal, we answer a question of D. W. Boyd from 1986 on the smallest degree Newman polynomial that satisfies |f (z)| > 2 on the unit circle ∂D. This polynomial is of degree 38 and we use this special polynomial in our constructions. We also identify (without a proof) all exceptional (k, n) with k ∈ {1, 2, 3, n − 3, n − 2, n − 1}, for which no such {0, 1}-polynomial of degree n exists: such pairs are related to regular (real and complex) Pisot numbers.Similar, but less complete results for {−1, 1} polynomials are established. We also look at the products of spaced Newman polynomials and consider the rotated large Littlewood polynomials. Lastly, based on our data, we formulate a natural conjecture about the statistical distribution of N (f ) in the set of Newman and Littlewood polynomials.