Universality of rational canonical form for random matrices over a finite field
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915772113092608 |
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| author | Shen, Jiahe |
| author_facet | Shen, Jiahe |
| contents | In this note, we study the distribution of the rational canonical form of a random matrix over the finite field $\mathbb{F}_p$, whose entries are independent and $ε$-balanced with $ε\in(0,1-1/p]$. We show that, as the matrix size tends to infinity, the statistics converge to independent Cohen-Lenstra distributions, demonstrating the universality of this asymptotic behavior. In particular, we recover, as a special case, the uniform setting proved by Fulman in his thesis in 1997.
Our proof uses the fact that the rational canonical form data of $A_n$ and the $\mathbb{F}_p[t]$-module structure of the function field cokernel $\Cok(tI_n-A_n)$ determine each other uniquely. Consequently, our question can be reformulated, equivalently, as the asymptotic distribution problem for this cokernel, which has been established by Cheong-Yu (arXiv:2303.09125). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16225 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universality of rational canonical form for random matrices over a finite field Shen, Jiahe Probability Number Theory In this note, we study the distribution of the rational canonical form of a random matrix over the finite field $\mathbb{F}_p$, whose entries are independent and $ε$-balanced with $ε\in(0,1-1/p]$. We show that, as the matrix size tends to infinity, the statistics converge to independent Cohen-Lenstra distributions, demonstrating the universality of this asymptotic behavior. In particular, we recover, as a special case, the uniform setting proved by Fulman in his thesis in 1997. Our proof uses the fact that the rational canonical form data of $A_n$ and the $\mathbb{F}_p[t]$-module structure of the function field cokernel $\Cok(tI_n-A_n)$ determine each other uniquely. Consequently, our question can be reformulated, equivalently, as the asymptotic distribution problem for this cokernel, which has been established by Cheong-Yu (arXiv:2303.09125). |
| title | Universality of rational canonical form for random matrices over a finite field |
| topic | Probability Number Theory |
| url | https://arxiv.org/abs/2510.16225 |