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

Ji, Yibo

Comptes Rendus. Mathématique, Tome 362 (2024) no. G8, pp. 883-893

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

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

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

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