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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2206.00072 |
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| _version_ | 1866929488536797184 |
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| author | Ji, Yibo |
| author_facet | Ji, Yibo |
| contents | In this paper, we count the number of matrices $A = (A_{i,j} )\in \mathcal{O} \subset Mat_{n\times n}(\mathbb{F}_q[x])$ where $deg(A_{i,j})\leq k, 1\leq i,j\leq n$, $deg(\det A) = t$, and $\mathcal{O}$ a given orbit of $GL_n(\mathbb{F}_q[x])$. By an elementary argument, we show that the above number is exactly $\# GL_n(\mathbb{F}_q)\cdot q^{(n-1)(nk-t)}$. This formula gives an equidistribution result over $\mathbb{F}_q[x]$ which is an analogue, in strong form, of a result over $\mathbb{Z}$ before. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_00072 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Distributions of Matrices over $\mathbb{F}_q[x]$ Ji, Yibo Number Theory 11C20, 15B33, 11T06 In this paper, we count the number of matrices $A = (A_{i,j} )\in \mathcal{O} \subset Mat_{n\times n}(\mathbb{F}_q[x])$ where $deg(A_{i,j})\leq k, 1\leq i,j\leq n$, $deg(\det A) = t$, and $\mathcal{O}$ a given orbit of $GL_n(\mathbb{F}_q[x])$. By an elementary argument, we show that the above number is exactly $\# GL_n(\mathbb{F}_q)\cdot q^{(n-1)(nk-t)}$. This formula gives an equidistribution result over $\mathbb{F}_q[x]$ which is an analogue, in strong form, of a result over $\mathbb{Z}$ before. |
| title | Distributions of Matrices over $\mathbb{F}_q[x]$ |
| topic | Number Theory 11C20, 15B33, 11T06 |
| url | https://arxiv.org/abs/2206.00072 |