Pure Core Sets of $n \times n$ Matrices over Finite Fields

TL;DR

This paper explores the structure and prevalence of pure core sets in matrix spaces over finite fields, revealing that as the field size grows, almost all subsets tend to be pure core sets.

Contribution

It provides new bounds on non-core set sizes and analyzes how the structure of core sets depends on polynomial factors and field size.

Findings

Proves the proportion of pure core sets approaches 1 as field size increases.

Establishes upper bounds on non-core set sizes in matrix similarity classes.

Analyzes the impact of minimal polynomial factors on core set structure.

Abstract

This paper studies the structure of core sets under different similarity classes. We investigate the influence of factors of the minimal polynomial with different degrees on the structure of core sets. When $F$ is a finite field of prime order, we study the upper bound on the size of a non-core set in a similarity class in $M_n(F)$. We prove that as $|F|$ increases, the proportion of pure core sets among subsets of $M_n(F)$ tends to $1$.