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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Académie des sciences
2024-10-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.616/ |
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| author | Ji, Yibo |
| author_facet | Ji, Yibo |
| author_sort | Ji, Yibo |
| collection | DOAJ |
| description | 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]. |
| format | Article |
| id | doaj-art-57b455bd61bf47bba1972902113dfbef |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-57b455bd61bf47bba1972902113dfbef2025-02-07T11:22:49ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-10-01362G888389310.5802/crmath.61610.5802/crmath.616Distribution of matrices over $\mathbb{F}_q[x]$Ji, Yibo0https://orcid.org/0000-0002-8873-6711Cornell University, United StatesIn 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].https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.616/Counting formulaFinite fieldPolynomial ring |
| spellingShingle | Ji, Yibo Distribution of matrices over $\mathbb{F}_q[x]$ Comptes Rendus. Mathématique Counting formula Finite field Polynomial ring |
| title | Distribution of matrices over $\mathbb{F}_q[x]$ |
| title_full | Distribution of matrices over $\mathbb{F}_q[x]$ |
| title_fullStr | Distribution of matrices over $\mathbb{F}_q[x]$ |
| title_full_unstemmed | Distribution of matrices over $\mathbb{F}_q[x]$ |
| title_short | Distribution of matrices over $\mathbb{F}_q[x]$ |
| title_sort | distribution of matrices over mathbb f q x |
| topic | Counting formula Finite field Polynomial ring |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.616/ |
| work_keys_str_mv | AT jiyibo distributionofmatricesovermathbbfqx |