On divisors of sums of polynomials

TL;DR

This paper demonstrates that for large enough sets of degree n polynomials over finite fields, the sum of two such polynomials often contains an irreducible divisor of large degree, revealing structural properties of polynomial sums.

Contribution

It establishes conditions under which sums of polynomials from large sets have irreducible divisors of significant degree, advancing understanding of polynomial sum structures over finite fields.

Findings

Large sets of polynomials ensure sums have large irreducible divisors

Existence of irreducible divisors depends on set size

Results apply to polynomials over finite fields

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be sets of polynomials of degree $n$ over a finite field. We show, that if $\mathcal{A}$ and $\mathcal{B}$ are large enough, then $A+B$ has an irreducible divisor of large degree for some $A\in\mathcal{A}$ and $B\in \mathcal{B}$.