Algorithmic aspects of Newman polynomials and their divisors

TL;DR

This paper investigates which integer polynomials divide Newman polynomials, providing new bounds, explicit examples, and non-divisibility results to advance understanding of polynomial divisibility properties.

Contribution

It offers new bounds on Mahler measures for divisibility, constructs explicit Newman polynomials divisible by Lehmer's polynomial, and identifies polynomials that do not divide any Newman polynomial.

Findings

A degree-10 polynomial with Mahler measure ~1.4194 divides no Newman polynomial.

Constructed Newman polynomials divisible by Lehmer's polynomial squared up to degree 150.

No Newman polynomial is divisible by Lehmer's polynomial cubed up to degree 160.

Abstract

We study the problem of determining which integer polynomials divide Newman polynomials. In this vein, we first give results concerning the $8438$ known polynomials with Mahler measure less than $1.3$. We then exhibit a list of polynomials that divide no Newman polynomial. In particular, we show that a degree-10 polynomial of Mahler measure \text{approximately} 1.419404632 divides no Newman polynomial, thereby improving the best known upper bound for any universal constant $\sigma$, if it exists, such that every integer polynomial of Mahler measure less than $\sigma$ divides a Newman polynomial. Finally, letting $l(x)$ denote Lehmer's polynomial, we explicitly construct Newman polynomials divisible by $l(x)^2$ with degrees up to $150$, and show that no Newman polynomial is divisible by $l(x)^3$ up to degree $160$.