On the Geometry of Numerical Ranges Over Finite Fields

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 $\mathbb F_{q^2}$.