On a theorem by Kippenhahn

TL;DR

This paper examines Kippenhahn's theorem relating algebraic curves to matrix numerical ranges, proving a key converse and discussing challenges in higher-dimensional generalizations in real algebraic geometry.

Contribution

The paper proves the converse of Kippenhahn's theorem and discusses the difficulties in extending it to higher dimensions.

Findings

The converse of Kippenhahn's theorem is established.

Higher-dimensional generalizations remain challenging in real algebraic geometry.

The original theorem's assumptions are clarified and validated.

Abstract

Kippenhahn discovered a real algebraic plane curve whose convex hull is the numerical range of a matrix. The correctness of this theorem was called into question when Chien and Nakazato found an example where the spatial analogue fails. They showed that the mentioned plane curve indeed lies inside the numerical range. We prove the easier converse direction of the theorem. Finding higher-dimensional generalizations of Kippenhahn's theorem is a challenge in real algebraic geometry.