A class of quadratic matrix equations over finite fields

TL;DR

This paper provides explicit formulas for counting solutions to quadratic matrix equations over finite fields, analyzes their orbit structure under conjugation, and identifies invariants and generating sets for their algebraic properties.

Contribution

It introduces a formula for solution counts, characterizes orbit separation via invariants, and describes the algebraic structure of these solution sets.

Findings

Explicit solution count formula for quadratic matrix equations.

Orbit separation achieved through characteristic polynomial invariants.

Generated the vanishing ideal for solution orbits.

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.