References & Citations
Mathematics > Number Theory
Title: Distributions of Matrices over $\mathbb{F}_q[x]$
(Submitted on 31 May 2022 (v1), last revised 13 Nov 2023 (this version, v5))
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.
Submission history
From: Yibo Ji [view email][v1] Tue, 31 May 2022 19:31:36 GMT (10kb)
[v2] Mon, 16 Jan 2023 22:43:18 GMT (10kb)
[v3] Mon, 8 May 2023 18:46:12 GMT (10kb)
[v4] Wed, 10 May 2023 01:51:09 GMT (10kb)
[v5] Mon, 13 Nov 2023 15:19:21 GMT (361kb)
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