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

TL;DR

This paper derives an explicit formula for counting matrices over polynomial rings with degree constraints and determinant degree, revealing an equidistribution phenomenon analogous to classical integer matrix results.

Contribution

It provides a new exact counting formula for matrices over $F_q[x]$ with specified degree and determinant degree, extending classical distribution results.

Findings

Exact count of matrices with degree constraints and fixed determinant degree

Demonstrates equidistribution over $F_q[x]$ analogous to integer case

Provides a strong form of distribution result over polynomial rings

Abstract

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 $\mathcal{O}$ a given orbit of $GL_n(\mathbb{F}_q[x])$. By an elementary argument, we show that the above number is exactly $\# GL_n(\mathbb{F}_q)\cdot q^{(n-1)(nk-t)}$. 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.