The boundary of the numerical range of matrix polynomials (Q1373311)

For a polynomial \(L(\lambda) =A_m \lambda^m+ \cdots +A_1 \lambda +A_0\) with coefficients \(A_i\) being \(n\times n\) complex matrices, the numerical range \(NR [L (\lambda)] =\{\lambda; \exists 0\neq x\in \mathbb{C}^n\), \(x^* L(\lambda) x=0\}\) is well known [cf. \textit{C. K. Li} and \textit{L. Rodman}, SIAM J. Matrix. Anal. Appl. 15, No., 4, 1256-1265 (1994; Zbl 0814.15023), \textit{A. S. Markus}, Introduction to the spectral theory of polynomial operator pencils (1988; Zbl 0678.47005)]. If \(L(\lambda) =\lambda \cdot I-A\), then \(NR [L(\lambda)]\) reduces to the classical numerical range of \(A:NR(A) =\{\lambda; x^*Ax =\lambda\), \(x\in \mathbb{C}^n\), \(|x|=1\}\). The set \(NR [L(\lambda)]\) is not always connected, and, if it is connected, it is not always convex. In a previous paper [Comput. Math. Appl. 31, No. 4-5, 41-47 (1996; Zbl 0874.15021)], the authors introduced the notion of a sharp point as some \(\lambda_0\in NR [L(\lambda)]\) (belonging to the boundary) such that, if, for a connected component \(w_s\) of \(NR [L (\lambda)]\), there exists a disk \(S(\lambda_0,r)\), \(r>0\), and two angles \(\varphi_1\), \(\varphi_2\) with \(0\leq \varphi_1< \varphi_2 \leq 2\pi\), such that \(\text{Re} (e^{i \theta} \lambda_0) =\max \{\text{Re} z; e^{-i \theta} z\in w_s \cap S(\lambda_0,r)\}\) for all \(\theta \in [\varphi_1, \varphi_2]\). In this paper, some properties of the sharp points are obtained, such as, for example: if \(\lambda_0 \in \partial NR [L(\lambda)]\), then 0 is a boundary point of the numerical range of the matrix \(L (\lambda_0)\). The authors also investigate the degenerate case of sharp points -- the isolated points.