Counting core sets in matrix rings over 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.04106
- arXiv:
- arXiv:2405.04106
- Bibcode:
- 2024arXiv240504106R
- Keywords:
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- Mathematics - Rings and Algebras;
- 16S50;
- 15A15;
- 15B33;
- 13F20