M/E23

Kippenhahn discovered a real algebraic plane curve whose convex hull is the numerical range of a matrix. The correctness of this theorem was questioned when Chien and Nakazato found an example where the spatial analogue fails. They also showed that the mentioned plane curve indeed lies inside the numerical range. We prove the easier converse direction of Kippenhahn’s theorem. Further, we point out that finding higher-dimensional generalizations of Kippenhahn’s theorem contributes to the geometry of quantum mechanics and is a challenge in real algebraic geometry.

Stephan Weis

Université libre Bruxelles