Distributions of Matrices over ${\mathbb{F}}_q[x]$

Abstract

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

Comments are welcome. Some typos were corrected. To appear in Comptes Rendus Mathématique