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

The numerical range of an n x n matrix polynomial P(lambda) = A(m)lambda(m) + A(m-1)lambda(m-1) + .... + A(1)lambda + A(0) is defined by W(P) = {lambda is an element of C : x * P (lambda)x = 0, x is an element of C-n, x not equal 0}. For the linear pencil P(lambda) = Ilambda - A, the range W(P) coincides with the numerical range of matrix A, F(A) = {x*Ax: x is an element of C-n, 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. (C) 2002 Elsevier Science Inc. All rights reserved.