On Newman and Littlewood polynomials with a 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 $\mathbb {D} = \{z \in \mathbb {C}: \lvert z \rvert < 1\}$. For every pair $(k, n) \in \mathbb {N}^2$, where $n \geq 7$ and $k \in [3, n-3]$, we prove that it is possible to find a $\{0, 1\}$–polynomial $f(z)$ of degree $\deg {f}=n$ with non–zero constant term $f(0) \ne 0$, such that $N(f)=k$ and $f(z) \ne 0$ on the unit circle $\partial \mathbb {D}$. On the way to this goal, we answer a question of D. W. Boyd from 1986 on the smallest degree Newman polynomial that satisfies $\lvert f(z) \rvert > 2$ on the unit circle $\partial \mathbb {D}$. 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.