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 $\mathcal{D}=\{z\in \mathbb{C}:|z|<1\}$. For every pair $(k,n)\in {\mathbb{N}}^2$, where $n\ge 7$ and $k\in [3,n-3]$, we prove that it is possible to find a $\{0,1\}$--polynomial $f(z)$ of degree $\text{deg }f=n$ with non--zero constant term $f(0)\ne 0$, such that $N(f)=k$ and $f(z)\ne 0$ on the unit circle $\partial \mathcal{D}$. 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 $\partial \mathcal{D}$. 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 $k\in \{1,2,3,n-3,n-2,n-1\}$, 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.

Submitted for publiation,v1.0, 9 pages, 7 figures, 8 tables