On covering systems of polynomial rings over finite fields

TL;DR

This paper extends the concept of covering systems from integers to polynomial rings over finite fields, proving bounds on the degrees of moduli and resolving the minimum modulus problem in this new setting.

Contribution

It introduces a method to bound the degrees of moduli in polynomial covering systems, adapting the distortion method to finite fields, and disproves a related conjecture.

Findings

Bound on the smallest degree of moduli depending only on s and q

Resolution of the minimum modulus problem for polynomial rings over finite fields

Disproof of Azlin's conjecture

Abstract

In 1950, Erd\H{o}s posed a question known as the minimum modulus problem on covering systems for $\mathbb{Z}$, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough in 2015, as he proved that the minimum modulus of any covering system with distinct moduli does not exceed $10^{16}$. Recently, Balister, Bollob\'as, Morris, Sahasrabudhe, and Tiba developed a versatile method called the distortion method and significantly reduced Hough's bound to $616,000$. In this paper, we apply this method to present a proof that the smallest degree of the moduli in any covering system for $\mathbb{F}_q[x]$ of multiplicity $s$ is bounded by a constant depending only on $s$ and $q$. Consequently, we successfully resolve the minimum modulus problem for $\mathbb{F}_q[x]$ and disprove a conjecture by Azlin.