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A class of quadratic matrix equations over finite fields

Yin Chen, Xinxin Zhang

Abstract

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can be separated by classical conjugation invariants defined by characteristic polynomials. We also find a generating set for the vanishing ideal of these orbits.

A class of quadratic matrix equations over finite fields

Abstract

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can be separated by classical conjugation invariants defined by characteristic polynomials. We also find a generating set for the vanishing ideal of these orbits.

Paper Structure

This paper contains 4 sections, 13 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Conjugation Actions and Rational Canonical Forms
  3. $|\mathcal{N}(n,q)|(n\geqslant 3)$
  4. Separating Invariants

Key Result

Proposition 2.1

If an $n\times n$ matrix $X\in\mathcal{N}(n,q)$, then $Y\in\mathcal{N}(n,q)$ for all $Y\in[X]$.

Theorems & Definitions (28)

  • Proposition 2.1
  • proof
  • Example 2.2: $n=2$
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 3.4
  • ...and 18 more