Least omitted degrees in non-covering polynomial congruence systems
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For fixed prime power q, take n polynomial congruence classes in F_q[x] that do not cover the ring. How large can the least degree of an omitted polynomial be? Call the extremal value D_q(n).
Why it matters and what counts as progress
Why it matters: The growth rate measures how far a non-covering congruence system can postpone its first uncovered polynomial.
What counts as progress: A sharp leading term n/(q−1)+O_q(1), improved additive bounds, or counterexamples to the stronger proposed exact floor formula.
How the problem developed
- 30 Jul 2026 · Asymptotic bound [1]
Rongyin Wang Proved D_q(n)=n/(q−1)+O_q(1) for fixed prime-power q, matching the leading term for the largest least omitted degree in a non-covering family of n polynomial congruence classes.
AI contribution
An Asymptotic Bound for Non-covering Congruence Systems over Fq[x]
For each fixed prime power q, the greatest possible least degree D_q(n) omitted by a non-covering family of n congruence classes in F_q[x] satisfies D_q(n)=n/(q−1)+O_q(1). ChatGPT supplied a truncated CRT/Brun-sieve approach used in the upper bound.
Sources
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