Algorithmic aspects of Newman polynomials and their divisors
Musbahu Idris, Jean-Marc Sac-Épée
TL;DR
The paper investigates which integer polynomials divide Newman polynomials by combining two complementary computational frameworks. A MILP-based method tests divisibility for polynomials with Mahler measure below $1.3$, yielding widespread divisibility certificates and isolating three unresolved cases; a disk-based obstruction approach identifies polynomials with no Newman multiples, tightening lower bounds on the universal constant $\sigma$ and producing several new examples with Mahler measures below $1.5560$. The work also advances the study of repeated zeros by constructing Newman polynomials divisible by $l(x)^2$ up to degree $150$ and proving $l(x)^3$ does not divide any Newman polynomial up to degree $160$, significantly extending prior results. Overall, the paper provides new algorithmic tools and extensive data that sharpen our understanding of divisibility by Newman polynomials and suggest a flexible framework for related coefficient-restricted polynomial problems.
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 $σ$, if it exists, such that every integer polynomial of Mahler measure less than $σ$ 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 $160$.