On the geometry of numerical ranges over finite fields

Numerical ranges over a certain family of finite fields were classified in 2016 by a team including our fifth author [5]. Soon afterward Ballico generalized these results to all finite fields and published some new results about the cardinality of the finite field numerical range [1], [2]. In this paper we study the geometry of these finite fields using the boundary generating curve, first introduced by Kippenhahn in 1951 [8], [9]. We restrict our study to square matrices of dimension 2, with at least one eigenvalue in Fq2.