On a Conjecture of Erdős over Function Fields

Abstract

Using Katz's equidistribution framework, we show that for any squarefree polynomial $f\in {\mathbb{F}}_q[t]$ of degree $n\ge 2$, every residue class modulo $f$ can be represented as a product of two monic irreducible polynomials of degree at most $n$, provided $q$ is sufficiently large in terms of $n$. This gives the function-field analogue of a conjecture of Erdős in the large-$q$ regime. Sawin previously proved this representation with stronger square-root cancellation via a higher-dimensional sheaf-theoretic construction. This note presents a one-dimensional argument that yields a natural $q^{-1/2}$ saving.

Revised introduction