Distribution of matrices over $\mathbb{F}_q[x]$
In this paper, we count the number of matrices $A = (A_{i,j} )\in \mathcal{O} \subset \mathrm{Mat}_{n\times n}(\mathbb{F}_q[x])$ where $\deg (A_{i,j})\le k, 1\le i,j\le n$, $\deg (\det A) = t$, and $\mathcal{O}$ is a given orbit of $\mathrm{GL}_n(\mathbb{F}_q[x])$. By an elementary argument, we show...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2024-10-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.616/ |
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| Summary: | In this paper, we count the number of matrices $A = (A_{i,j} )\in \mathcal{O} \subset \mathrm{Mat}_{n\times n}(\mathbb{F}_q[x])$ where $\deg (A_{i,j})\le k, 1\le i,j\le n$, $\deg (\det A) = t$, and $\mathcal{O}$ is a given orbit of $\mathrm{GL}_n(\mathbb{F}_q[x])$. By an elementary argument, we show that the above number is exactly $\# \mathrm{GL}_n(\mathbb{F}_q)\cdot q^{(n-1)(nk-t)}$. This formula gives an equidistribution result over $\mathbb{F}_q[x]$, which is an analogue, in strong form, of a result over $\mathbb{Z}$ proved in [2] and [3]. |
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| ISSN: | 1778-3569 |