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

Rongyin Wang

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

Number Theory 2026-07-30 v1

Authors: Rongyin Wang

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.

Keywords

congruence systems non-covering families polynomial rings asymptotic bounds Brun sieve minimal counterexample nested moduli

Cite

@article{arxiv.2607.27538,
  title  = {An Asymptotic Bound for Non-covering Congruence Systems over Fq[x]},
  author = {Rongyin Wang},
  journal= {arXiv preprint arXiv:2607.27538},
  year   = {2026}
}