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

TL;DR

This paper investigates the zeros of Newman and Littlewood polynomials with prescribed counts inside the unit disk, providing constructions, answering longstanding questions, and exploring distributions of zeros.

Contribution

It constructs Newman polynomials with specific zero counts, answers a 1986 question about minimal degree polynomials exceeding magnitude 2 on the unit circle, and discusses exceptional cases related to Pisot numbers.

Findings

Constructed Newman polynomials with prescribed zeros inside the unit disk.

Identified degree 38 as the smallest for Newman polynomials with |f(z)| > 2 on the unit circle.

Proposed a conjecture on the distribution of zeros in Newman and Littlewood polynomials.

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 $n \geq 7$ 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.