Elliptic numerical ranges of \(3 \times 3\) companion matrices (Q924345)

The companion matrix associated with a polynomial \[ p(z)=(z-z_1)(z-z_2)(z-z_3)=z^3+az^2+bz+c, \] where \(z_1, z_2, z_3\) are given complex numbers, is \[ A(z_1,z_2,z_3)=\left( \begin{matrix} 0 & 1 & 0\\ 0 & 0 & 1\\ -c & -b & -a \end{matrix}\right). \] It is shown that for any two complex numbers \(z_1\) and \(z_2\) there exists at least one complex number \(z_3\) (and at most five) such that the companion matrix \(A(z_1,z_2,z_3)\) has an elliptic numerical range with foci at \(z_1\) and \(z_2\).