Product representations of polynomials over finite fields

TL;DR

This paper investigates the structure and size of subsets in finite fields that avoid certain polynomial value configurations, extending classical number theory problems into the finite field setting.

Contribution

It introduces a finite field analogue of Verstra"ete's conjecture, providing new insights into polynomial value sets over finite fields and their combinatorial properties.

Findings

Established bounds for subset sizes avoiding polynomial value configurations

Extended classical number theory problems to finite fields

Provided new conjectures and directions for future research

Abstract

Erd\H{o}s, S\'ark\"ozy, and S\'os studied the asymptotics of the maximum size of a subset of $\{1,2,\ldots, N\}$ such that it does not contain $k$ distinct elements whose product is a perfect square. More generally, Verstra\"ete proposed a conjecture regarding the asymptotic behavior of the same quantity with the set of perfect squares replaced by the value set of a polynomial in $\mathbb{Z}[x]$. In this paper, we study a finite field analogue of Verstra\"ete's conjecture.