Distribution of matrices over $\mathbb{F}_q[x]$

Yibo Ji

doi:10.5802/crmath.616

Article de recherche - Théorie des nombres

Distribution of matrices over

𝔽_{q} [x]

[Distribution des matrices sur

𝔽_{q} [x]

]

Yibo Ji ¹

¹ Cornell University, United States

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 883-893.

Résumé
Abstract

Dans cet article, nous comptons le nombre de matrices $A = (A_{i, j}) \in 𝒪 \subset {Mat}_{n \times n} (𝔽_{q} [x])$ où $deg A_{i, j} \leq k, 1 \leq i, j \leq n, deg (det (A)) = t$ , et $𝒪$ est une orbite donnée de ${GL}_{n} (𝔽_{q} [x])$ . Par un argument élémentaire, nous montrons que le nombre ci-dessus est exactement $# {GL}_{n} (𝔽_{q}) \cdot q^{(n - 1) (n k - t)}$ . Cette formule donne un résultat d’équidistribution sur $𝔽_{q} [x]$ , qui est un analogue, sous forme forte, d’un résultat sur $ℤ$ prouvé dans [2] et [3].

In this paper, we count the number of matrices $A = (A_{i, j}) \in 𝒪 \subset {Mat}_{n \times n} (𝔽_{q} [x])$ where $deg (A_{i, j}) \leq k, 1 \leq i, j \leq n$ , $deg (det A) = t$ , and $𝒪$ is a given orbit of ${GL}_{n} (𝔽_{q} [x])$ . By an elementary argument, we show that the above number is exactly $# {GL}_{n} (𝔽_{q}) \cdot q^{(n - 1) (n k - t)}$ . This formula gives an equidistribution result over $𝔽_{q} [x]$ , which is an analogue, in strong form, of a result over $ℤ$ proved in [2] and [3].

Reçu le : 2023-05-23
Révisé le : 2023-11-07
Accepté le : 2024-01-19
Publié le : 2024-10-07

DOI : 10.5802/crmath.616

Classification : 15B33
Keywords: Counting formula, Finite field, Polynomial ring
Mot clés : Formule de comptage, Corps fini, Anneau polynomial

Affiliations des auteurs :

Yibo Ji ¹

¹ Cornell University, United States

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2024__362_G8_883_0,
     author = {Yibo Ji},
     title = {Distribution of matrices over $\mathbb{F}_q[x]$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {883--893},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.616},
     language = {en},
}

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Yibo Ji. Distribution of matrices over $\mathbb{F}_q[x]$. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 883-893. doi : 10.5802/crmath.616. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.616/

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