The lowest-degree polynomials with non-negative coefficients

TL;DR

This paper investigates the minimal degree of polynomials with non-negative coefficients that are divisible by a given polynomial, extending known results from quadratics to cubics and improving bounds for general polynomials.

Contribution

It determines the lowest degree for cubic polynomials and refines bounds for the minimal degree for arbitrary polynomials with non-negative coefficients.

Findings

Lowest degree for cubic polynomials established

Bounds for general polynomials improved

Characterization of divisibility conditions with non-negative coefficients

Abstract

A polynomial $p\in\mathbb{R}[x]$ is a divisor of some polynomial $0\neq f\in\mathbb{R}[x]$ with non-negative coefficients if and only if $p$ does not have a positive real root. The lowest possible degree of such $f$ for a given $p$ is known for quadratic polynomials. We provide it for cubic polynomials and improve known bounds of this value for a general polynomial.