Point equation of the boundary of the numerical range of a matrix polynomial

AbstractThe numerical range of an n×n matrix polynomialP(λ)=Amλm+Am−1λm−1+⋯+A1λ+A0is defined byW(P)=λ∈C:x*P(λ)x=0,x∈Cn,x≠0.For the linear pencil P(λ)=Iλ−A, the range W(P) coincides with the numerical range of matrix A, F(A)={x*Ax:x∈Cn,x*x=1}. In this paper, we obtain necessary conditions for the origin to be a boundary point of F(A). As a consequence, an algebraic curve of degree at most 2n(n−1)m, which contains the boundary of W(P), is constructed