AI 中文总结
该研究确定有限域上稀疏随机矩阵有理典范形式普适性的 sharp 阈值，通过满射矩量法证明阈值与有限域余核、$\mathbb{Z}_p$ 上随机矩阵模型的阈值一致，还构造了对应最小次数不可约多项式的临界稀疏障碍。
AI 中文摘要
我们研究有限域上稀疏随机矩阵的有理典范形式。设 $A_n\in \operatorname{Mat}_n(\mathbb{F}_p)$ 具有独立的 $\alpha_n$-平衡元素。我们证明：若 $\liminf_{n\to\infty}\frac{n\alpha_n}{\log n}>1$，则对 $\mathbb{F}_p$ 上任意固定的首一不可约多项式集合，$A_n$ 的对应主分块会联合收敛到 Fulman 在其论文中研究的均匀模型所对应的渐近独立 Cohen-Lenstra 分布。全有理典范形式的稀疏 sharp 阈值与 Lee 此前针对有限域上余核所得的阈值、以及 Jung-Lee-Yu 针对 $\mathbb{Z}_p$ 上随机矩阵模型所得的阈值一致。我们的证明基于函数域 $\mathbb{F}_p[t]$ 上的满射矩量法，应用于有限模 $\operatorname{Cok}_{\mathbb{F}_p[t]}(tI_n-A_n)$，其主分解记录了有理典范形式。我们还构造了 $d$ 次临界稀疏障碍，表明最小次数为 $d$ 的不可约多项式相关统计量的阈值为 $1/d$。
英文摘要
We study the rational canonical form of sparse random matrices over a finite field. Suppose $A_n\in \operatorname{Mat}_n(\mathbb{F}_p)$ has independent and $α_n$-balanced entries. We prove that if $$ \liminf_{n\to\infty}\frac{nα_n}{\log n}>1, $$ then, for every fixed collection of distinct monic irreducible polynomials over $\mathbb{F}_p$, the corresponding primary partitions of $A_n$ converge jointly to the same asymptotically independent Cohen-Lenstra distributions as in the uniform model studied by Fulman in his thesis. The sharp sparsity threshold for the full rational canonical form coincides with the threshold previously obtained by Lee for finite-field cokernels and by Jung-Lee-Yu for random matrix models over $\mathbb{Z}_p$. Our proof is based on the surjection moment method over the function field $\mathbb{F}_p[t]$, applied to the finite module $\operatorname{Cok}_{\mathbb{F}_p[t]}(tI_n-A_n)$, whose primary decomposition records the rational canonical form. We also construct degree-$d$ critical sparse obstructions, suggesting a polynomial-dependent threshold $1/d$ for statistics associated with irreducible polynomials of minimal degree $d$.
CommentsThis paper supersedes arXiv:2510.16225. 15 pages. Comments welcome!
