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Can oval descriptions be simplified by choosing alternative or additional poles?

Determine whether the polynomial oval description of the Kippenhahn curve (the boundary of the numerical range) can be simplified by selecting poles other than just the eigenvalues or by augmenting the set of poles, for example in the 3×3 case where the eigenvalue-only oval relation is algebraically unwieldy.

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Background

After showing that, for 3×3 matrices, an oval description using only the eigenvalues as poles exists but is often extremely complicated, the author raises the possibility that choosing different poles or adding more poles might yield a simpler relation. In particular, Example A0 demonstrates that while a three-pole (eigenvalue-only) oval description exists, it can be dramatically more complex than the corresponding Kippenhahn equation.

This remark highlights a conceptual gap: although eigenvalues are natural poles, they may not be optimal for expressing a succinct oval relation. The author explicitly flags this as an unresolved question and declines to pursue it further in the paper.

References

The question remains whether the above formula can be simplified by choosing other poles than just the eigenvalues or by adding more poles to the mix. As we will see, it is often advantageous to choose more poles than the absolute minimum. However, we will not attempt to answer this question here.

— Coval description of the boundary of a numerical range and the secondary values of a matrix (2410.03744 - Blaschke, 1 Oct 2024) in Remark following Proposition 2 (P2), end of Introduction