Containment regions for zeros of polynomials from numerical ranges of companion matrices (Q1611879)

The results on containment regions for the numerical range of certain companion matrices (bounded diagonal matrices) of a given polynomial \(P_n(x)=x^n-a_1x^{n-1}-\dots-a_n(a_1,\dots, a_n\in {\mathbb C}\), \(a_n\neq 0\), \(\deg P_n\geq 3\)) are applied to obtain containment regions for the zeros of~\(P_n\). In some cases the obtained regions are better than the known ones, as several examples show.