Coefficients of squares of Newman polynomials

TL;DR

This paper constructs sparse Newman polynomials with large degree that defy a conjecture by Yu, showing their squared coefficients can be significantly smaller relative to their maximum, using randomization techniques.

Contribution

It introduces a method to produce sparse Newman polynomials with specific coefficient properties, disproving a previous conjecture.

Findings

Existence of sparse Newman polynomials with large degree and small maximum coefficient ratio

Disproof of Yu's conjecture on Newman polynomial coefficients

Application of randomization to achieve desired polynomial sparsity

Abstract

We show that there are polynomials $p_N$ of arbitrarily large degree $N$, with coefficients equal to 0 or 1 (Newman polynomials), such that $$ \liminf_{N \to \infty} N \Linf{p_N^2} \bigl / p_N^2(1) < 1, $$ where $\Linf{q}$ denotes the maximum coefficient of the polynomial $q$ and which, at the same time, are sparse: $p_N(1)/N \to 0$. This disproves a conjecture of Yu \cite{yu}. We build on some previous results of Berenhaut and Saidak \cite{berenhaut-saidak} and Dubickas \cite{dubickas} whose examples lacked the sparsity. This sparsity we create from these examples by randomization.