On the Geometry of Numerical Ranges Over Finite Fields

TL;DR

This paper investigates the geometric structure of numerical ranges over finite fields, focusing on boundary curves for 2x2 matrices with eigenvalues in an extension field, building on prior classifications and generalizations.

Contribution

It introduces a geometric approach using boundary generating curves to analyze finite field numerical ranges, extending previous classifications to a geometric framework.

Findings

Characterization of boundary generating curves for 2x2 matrices over finite fields

Extension of numerical range classification to geometric analysis

Insights into the shape and properties of finite field numerical ranges

Abstract

Numerical ranges over a certain family of finite fields were classified in 2016 by a team including our fifth author. Soon afterward, in 2017 Ballico generalized these results to all finite fields and published some new results about the cardinality of the finite field numerical range. In this paper we study the geometry of these finite fields using the boundary generating curve, first introduced by Kippenhahn in 1951. We restrict our study to square matrices of dimension 2, with at least one eigenvalue in $\mathbb F_{q^2}$.