Polytopes of polynomials with zeros in a prescribed set

In the publication by A.C. Bartlett, C.V. Holot, and H. Lin (Proc. Amer. Contr. Conf., Minneapolis, MN, 1987) a fundamental result is established on the zero locations of a family of polynomials. It is shown that the zeros of a polytope P of nth-order real polynomials are contained in a simply connected set D if and only if the zeros of all polynomials along the edges of P are contained in D. The present authors are motivated by the fact that the requirement of simple connectedness of D may be too restrictive and applications such as dominant pole assignment and filter design where the separation of zeros is required. They extend the edge criterion of Bartlett et al. to handle any set D whose complement D^c has the following property: every point D ∈ D^c lies on some continuous path which remains within D^c and is unbounded. This requirement is typically verified by inspection and allows for a large class of disconnected sets