Higher-rank numerical ranges and Kippenhahn polynomials

We prove that two n-by-n matrices A and B have their rank-k numerical ranges Λk(A) and Λk(B) equal to each other for all , if and only if their Kippenhahn polynomials and coincide. The main tools for the proof are the Li-Sze characterization of higher-rank numerical ranges, Weyl’s perturbation theorem for eigenvalues of Hermitian matrices and Bézout’s theorem for the number of common zeros for two homogeneous polynomials.