Counting core sets in matrix rings over finite fields

TL;DR

This paper investigates the frequency of core subsets within the matrix ring over finite fields, providing exact counts for small matrices and showing that almost all subsets are core as the field size grows.

Contribution

It offers the first explicit enumeration of core subsets in $M_2(F_q)$ and proves their prevalence asymptotically as the field size increases.

Findings

Exact counts for core subsets in each similarity class of $M_2(F_q)$

Not all subsets are core, but prevalence approaches 1 as $q o

Almost all subsets are core in large finite fields.

Abstract

Let $R$ be a commutative ring and $M_n(R)$ be the ring of $n \times n$ matrices with entries from $R$. For each $S \subseteq M_n(R)$, we consider its (generalized) null ideal $N(S)$, which is the set of all polynomials $f$ with coefficients from $M_n(R)$ with the property that $f(A) = 0$ for all $A \in S$. The set $S$ is said to be core if $N(S)$ is a two-sided ideal of $M_n(R)[x]$. It is not known how common core sets are among all subsets of $M_n(R)$. We study this problem for $2 \times 2$ matrices over $\mathbb{F}_q$, where $\mathbb{F}_q$ is the finite field with $q$ elements. We provide exact counts for the number of core subsets of each similarity class of $M_2(\mathbb{F}_q)$. While not every subset of $M_2(\mathbb{F}_q)$ is core, we prove that as $q \to \infty$, the probability that a subset of $M_2(\mathbb{F}_q)$ is core approaches 1. Thus, asymptotically in~$q$, almost all subsets of $M_2(\mathbb{F}_q)$ are core.