On a Conjecture of Erd\H{o}s over Function Fields

TL;DR

This paper proves a function-field analogue of Erdős's conjecture using Katz's equidistribution, showing that for large finite fields, every residue class modulo a squarefree polynomial can be expressed as a product of two monic irreducible polynomials of bounded degree.

Contribution

It provides a new one-dimensional argument that achieves a natural $q^{-1/2}$ saving, simplifying previous higher-dimensional sheaf-theoretic approaches.

Findings

Every residue class modulo a squarefree polynomial can be represented as a product of two monic irreducibles for large q.

The method yields a $q^{-1/2}$ saving, improving the understanding of polynomial factorizations over finite fields.

The approach offers a simpler proof of a function-field analogue of Erdős's conjecture.

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 Erd\H{o}s 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.