Computational progress on the unfair 0-1 polynomial Conjecture

TL;DR

This paper develops an algorithm to test the unfair 0-1 polynomial conjecture for polynomial factors up to degree 15, significantly narrowing the search for counterexamples by computationally ruling out most cases.

Contribution

It introduces a novel algorithm to verify the conjecture for candidate factors, providing extensive computational evidence supporting the conjecture's validity for degrees up to 15.

Findings

Out of over 7 million candidate factors, only 975 remain unresolved.

The algorithm successfully rules out the existence of certain factorizations in most cases.

Results strongly support the conjecture for polynomials of degree up to 15.

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 $a(x) b(x) = c(x)$.