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arXiv · 2510.17612

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

Likun Xie
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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.

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Likun Xie. 2025-10-20. On a Conjecture of Erd\H{o}s over Function Fields. https://doi.org/10.1007/s00209-026-04077-6

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