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

Hongyu Wang, Yizhi Zhang

TL;DR

The paper studies core sets in $M_n(\mathbb{F}_q)$ under a refined similarity-class partition, focusing on when the zero ideal $N(S)$ is a two-sided ideal. It introduces polynomial tools $\mathcal{N}_F(A,g)$ and $\mathcal{B}_F(A,g)$ to derive a practical core-set criterion, treating linear and higher-degree minimal-polynomial factors separately. For linear factors, core-set validity reduces to a linear-space condition on $\mathrm{Im}\,f_a(A)$, while for higher-degree factors it passes to a splitting field $K$ and uses Galois symmetry to reduce the problem to sums of $\mathrm{Im}\,f_\alpha(A)$ equalling $K^n$. The authors establish a universal bound $|S|\le 4|\mathcal{C}|/q$ for non-core subsets and prove that, as $q\to\infty$, almost all subsets of $M_n(\mathbb{F}_q)$ are core sets (indeed pure core sets), highlighting the asymptotic dominance of core structures over large finite fields.

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 $1$.

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

TL;DR

The paper studies core sets in under a refined similarity-class partition, focusing on when the zero ideal is a two-sided ideal. It introduces polynomial tools and to derive a practical core-set criterion, treating linear and higher-degree minimal-polynomial factors separately. For linear factors, core-set validity reduces to a linear-space condition on , while for higher-degree factors it passes to a splitting field and uses Galois symmetry to reduce the problem to sums of equalling . The authors establish a universal bound for non-core subsets and prove that, as , almost all subsets of are core sets (indeed pure core sets), highlighting the asymptotic dominance of core structures over large finite fields.

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 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 . We prove that as increases, the proportion of pure core sets among subsets of tends to .
Paper Structure (7 sections, 26 theorems, 17 equations)

This paper contains 7 sections, 26 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminary Propositions
  3. Equivalent Polynomial Characterization of Core Sets
  4. Analysis of Linear Factors
  5. Splitting of Higher-Degree Factors
  6. Equivalence Class Partition in the Splitting Field
  7. Asymptotic Proportion of Pure Core Sets

Key Result

Proposition 1.3

([4, Proposition 3.1,3.4]) Let $S\subseteq M_n(F)$, then (1) $N(S)$ is a left ideal of $M_n(F)[x]$. (2) Let $I$ be an index set, $\{S_i\}_{i\in I}$ a family of core sets, then $\bigcup\limits_{i\in I}S_i$ is a core set. (3) Let $a\in F$, then $S$ is a core set $\iff S+a$ is a core set. (4) Let $0\ne

Theorems & Definitions (51)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Definition 1.4
  • Proposition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 41 more