Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.04106 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912335626502144 |
|---|---|
| author | Rissner, Roswitha Werner, Nicholas J. |
| author_facet | Rissner, Roswitha Werner, Nicholas J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04106 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting core sets in matrix rings over finite fields Rissner, Roswitha Werner, Nicholas J. Rings and Algebras 16S50, 15A15, 15B33, 13F20 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. |
| title | Counting core sets in matrix rings over finite fields |
| topic | Rings and Algebras 16S50, 15A15, 15B33, 13F20 |
| url | https://arxiv.org/abs/2405.04106 |