On covering systems of polynomial rings over finite fields

Abstract

In 1950, Erdős posed a question known as the minimum modulus problem on covering systems for Z \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 [Ann. of Math. (2) 181 (2015), no. 1, pp. 361–382] in 2015, as he proved that the minimum modulus of any covering system with distinct moduli does not exceed 10 16 10^{16} . Recently, Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [Invent. Math. 228 (2022), pp. 377–414] developed a versatile method called the distortion method and significantly reduced Hough’s bound to 616 , 000 616,000 . In this paper, we apply this method to present a proof that the smallest degree of the moduli in any covering system for F q [ x ] \mathbb {F}_q[x] of multiplicity s s is bounded by a constant depending only on s s and q q . Consequently, we successfully resolve the minimum modulus problem for F q [ x ] \mathbb {F}_q[x] and disprove a conjecture by Azlin [Covering Systems of Polynomial Rings Over Finite Fields, University of Mississippi, Electronic Theses and Dissertations. 39, 2011].