An Asymptotic Bound for Non-covering Congruence Systems over Fq[x]

Abstract

Fix a prime power $q$. Let $D_q(n)$ be the largest possible least degree of a polynomial omitted by a non-covering family of $n$ congruence classes in ${\mathbb{F}}_q[x]$. Assuming the known theorem that every non-covering family of $n$ classes omits a polynomial of degree less than $n$, we prove \[ D_q(n)=\frac{n}{q-1}+O_q(1). \] The upper bound combines a minimal-counterexample reduction to irreducible moduli with a truncated inclusion--exclusion (Brun sieve) argument. A nested-modulus construction gives the matching lower bound. This is a follow-up to the author's 2025 work.