AI 中文总结
本文针对函数域上的覆盖族最小基数问题，否定了Chow与Rimani'c的猜想，推导了相关下界并对低次多项式合冲三元组分类，得到$C_k(q)$的新估计。
AI 中文摘要
设$C_k(q)$为$\text{GF}(q)$上非零多项式族的最小基数，该多项式族对应的余维-$k$部分循环核覆盖整个系数空间。Chow与Rimani'c猜想$C_k(q)=1+q+\boldsymbol{\text{...}}+q^k$。我们通过在$\text{GF}(2)$上构造13个首一多项式（其$k=3$的核覆盖$\text{GF}(2)^7$），否定了无限制猜想；特别地，$C_3(2)\boldsymbol{\text{≤}}13\boldsymbol{<}15$。对于大小为$N=q^k+S$且$\text{GF}(q)$线性秩为$d$的一般覆盖族，我们证明$S\boldsymbol{\text{≫}}d^{2/3}\biggl(\frac{\boldsymbol{\text{log}}(2q)}{\boldsymbol{\text{log}}(\boldsymbol{\text{e}}Nq^k/S)}\biggr)^{2/3}$。因此，对每个固定的$k\boldsymbol{\text{≥}}2$及所有足够大的$q$，$C_k(q)\boldsymbol{\text{≥}}q^k+c_kq^{2/3}$。当$k=2$时，二阶矩覆盖论证的整数重数改进给出$\boldsymbol{\text{lim inf}}_{q\to\boldsymbol{\text{∞}}}(C_2(q)-q^2)/q\boldsymbol{\text{≥}}\boldsymbol{\text{ṽ}}c_2$，其中$\boldsymbol{\text{ṽ}}c_2$是显式单变量变分常数，数值为$\boldsymbol{\text{ṽ}}c_2=0.5829944375\boldsymbol{\text{...}}$。我们还对允许两个独立低次多项式合冲的三元组进行分类，并证明条件无分组下界为$q^k+(1/2-o(1))q^{k-1}$。
英文摘要
Let $C_k(q)$ be the least cardinality of a family of nonzero polynomials over $\mathbb F_q$ whose associated codimension-$k$ partial-circulant kernels cover the full coefficient space. Chow and Rimani'c conjectured that $C_k(q)=1+q+\cdots+q^k$. We disprove the unrestricted conjecture by constructing thirteen monic polynomials over $\mathbb F_2$ whose $k=3$ kernels cover $\mathbb F_2^7$; in particular, $C_3(2)\le 13<15$. For a general covering family of size $N=q^k+S$ and $\mathbb F_q$-linear rank $d$, we prove $S\gg d^{2/3}\left(\frac{\log(2q)}{\log(eNq^k/S)}\right)^{2/3}$. Consequently, for every fixed $k\ge 2$ and all sufficiently large $q$, $C_k(q)\ge q^k+c_kq^{2/3}$. When $k=2$, an integer-multiplicity refinement of the second-moment covering argument yields $\liminf_{q\to\infty}(C_2(q)-q^2)/q\ge \widetilde c_2$, where $\widetilde c_2$ is an explicit one-variable variational constant with numerical value $\widetilde c_2=0.5829944375\ldots$. We also classify triples admitting two independent low-degree polynomial syzygies and prove a conditional packet-free lower bound of size $q^k+(1/2-o(1))q^{k-1}$.
Comments25 pages, no figures. Work-in-progress formalization and related code will be developed and maintained at https://github.com/hxypqr/lonely-runners-function-fields. Corrected the historical attribution in the Introduction. No mathematical changes
