On Newman and Littlewood multiples of Borwein polynomials

TL;DR

This paper investigates divisibility properties of Borwein polynomials by Newman and Littlewood polynomials, using an algorithm to determine multiples with coefficients in finite sets, and extends known results for degrees up to 11.

Contribution

It introduces an algorithm to decide if certain polynomials have multiples with coefficients in finite sets and applies it to classify Borwein polynomials up to degree 9.

Findings

Borwein polynomials of degree ≤8 dividing Newman polynomials also divide Littlewood polynomials.

Every Newman polynomial of degree ≤11 has a Littlewood multiple.

Extended previous results on polynomial divisibility for degrees up to 11.

Abstract

A Newman polynomial has all the coefficients in $\{ 0,1\}$ and constant term 1, whereas a Littlewood polynomial has all coefficients in $\{-1,1\}$. We call $P(X)\in\mathbb{Z}[X]$ a Borwein polynomial if all its coefficients belong to $\{ -1,0,1\}$ and $P(0)\neq 0$. By exploiting an algorithm which decides whether a given monic integer polynomial with no roots on the unit circle $|z|=1$ has a non-zero multiple in $\mathbb{Z}[X]$ with coefficients in a finite set $\mathcal{D} \subset \mathbb{Z}$, for every Borwein polynomial of degree at most 9 we determine whether it divides any Littlewood or Newman polynomial. In particular, we show that every Borwein polynomial of degree at most 8 which divides some Newman polynomial divides some Littlewood polynomial as well. In addition to this, for every Newman polynomial of degree at most 11, we check whether it has a Littlewood multiple, extending the previous results of Borwein, Hare, Mossinghoff, Dubickas and Jankauskas.