Algorithmic aspects of Newman polynomials and their divisors

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 $160$.