Counting core sets in matrix rings over finite fields
Roswitha Rissner, Nicholas J. Werner
TL;DR
This work analyzes core sets in the matrix ring M_2(F_q) through the lens of null ideals. By partitioning matrices into similarity classes categorized as LIN, IRR, SQR, and SQD, the authors derive exact counts of core versus noncore subsets within each class, exploiting structural criteria for core-ness (notably when S is quadratic, a subset is core iff there exist A,B∈S with A−B invertible). They show that, as q grows, almost all subsets are core and, in fact, almost all core subsets are purely core, with explicit asymptotic product formulas over the four class types. The paper also provides detailed constructions (L-modules and B-sets) that illuminate noncore behavior in SQD classes and presents explicit small-q examples to illustrate the combinatorial complexity of core sets not purely core. Overall, the results reveal a strong asymptotic predominance of core behavior in M_2(F_q) and give a precise combinatorial framework for counting core subsets by similarity class.
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.