Comptes Rendus
Research article - Number theory
Distribution of matrices over 𝔽 q [x]
Yibo Ji1 
1 Cornell University, United States
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 883-893

In this paper, we count the number of matrices A=(A i,j )∈𝒪⊂Mat n×n (𝔽 q [x]) where deg(A i,j )≤k,1≤i,j≤n, deg(detA)=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 )·q (n-1)(nk-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].

Dans cet article, nous comptons le nombre de matrices A=(A i,j )∈𝒪⊂Mat n×n (𝔽 q [x]) où degA i,j ≤k,1≤i,j≤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 )·q (n-1)(nk-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].

Received:
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Accepted:
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DOI: 10.5802/crmath.616
Classification: 15B33
Keywords: Counting formula, Finite field, Polynomial ring
Mots-clés : Formule de comptage, Corps fini, Anneau polynomial

Yibo Ji 1

1 Cornell University, United States
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Distribution of matrices over $\mathbb{F}_q[x]$},
     journal = {Comptes Rendus. Math\'ematique},
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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

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