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

In this paper, we count the number of matrices $A = (A_{i,j} )\in $\mathcal{O}$ \subset Mat_{n\times n}($\mathbb{F}$q[x])$where$deg(A{i,j})\leq k, 1\leq i,j\leq n$,$deg(\det A) = t$,and$$\mathcal{O}$$agivenorbitof$GL_n($\mathbb{F}$_q[x])$. By an elementary argument, we show that the above number is exactly $\mathrm{\#}G{L}_n({\mathbb{F}}_q)\cdot q^{(n-1)(nk-t)}$. 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.