Computational progress on the unfair 0-1 polynomial Conjecture

Abstract

Let $c(x)$ be a monic integer polynomial with coefficients $0$ or $1$. Write $c(x) = a(x) b(x)$ where $a(x)$ and $b(x)$ are monic polynomials with non-negative real (not necessarily integer) coefficients. The unfair 0--1 polynomial conjecture states that $a(x)$ and $b(x)$ are necessarily integer polynomials with coefficients $0$ or $1$. Let $a(x)$ be a candidate factor of a (currently unknown) 0--1 polynomial. We will assume that we know if a coefficient is $0$, $1$ or strictly between $0$ and $1$, but that we do not know the precise value of non-integer coefficients. Given this candidate $a(x)$, this paper gives an algorithm to either find a $b(x)$ and $c(x)$ with $a(x) b(x) = c(x)$ such that $b(x)$ has non-negative real coefficients and $c(x)$ has coefficients $0$ or $1$, or (often) shows that no such $c(x)$ and $b(x)$ exist. Using this algorithm, we consider all candidate factors with degree less than or equal to 15. With the exception of 975 candidate factors (out of a possible 7141686 cases), this algorithm shows that there do not exist $b(x)$ with non-negative real coefficients and $c(x)$ with coefficients $0$ or $1$ such that $a(x) b(x) = c(x)$.