Simplifying oval descriptions by choosing alternative or additional poles

Determine whether the oval relation that describes the boundary curve ∂W(A) of the numerical range of a complex matrix A can be simplified by selecting poles other than the eigenvalues or by augmenting the set of poles with additional points, and identify choices of poles that yield a simpler algebraic relation among the distance coordinates r(·) than the description using only eigenvalues as poles.

Background

The paper shows that while every algebraic plane curve (including Kippenhahn curves of numerical ranges) admits an oval description in terms of distances r(·) to finitely many poles, using only the eigenvalues as poles generally yields very complicated relations, even for 3×3 matrices.

The author then develops a coval (tangent-based) framework that often leads to remarkably simple descriptions involving eigenvalues and additional points called secondary values. This remark raises the unresolved issue of whether an oval (distance-based) description can likewise be simplified by choosing poles other than just the eigenvalues or by adding more poles.