Splitting sums of binary polynomials

TL;DR

This paper investigates a polynomial analogue of a classical arithmetic problem over [x], establishing that five is the minimal number of polynomials needed to prevent all pairwise sums from having a specific form.

Contribution

It proves that five is the smallest size of a polynomial set over [x] where pairwise sums avoid a particular polynomial form, extending classical additive number theory.

Findings

The minimal number m is 5 for the given property.

Sets of fewer than 5 polynomials can have all pairwise sums of the specified form.

The result characterizes the structure of polynomial sums over [x].

Abstract

We study an analogue of a classical arithmetic problem over the ring of polynomials. We prove that $m = 5$ is the minimal number such that the sums of any two distinct polynomials in a set of $m$ polynomials over $\F_2[x]$ cannot all be of the form $x^k(x+1)^{\ell}$.