On covering systems of polynomial rings over finite fields

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.