Polytopes of Polynomials with Zeros in a Prescribed Region

Abstract

In Bartlett, Hollot and Lin [2], 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 n-th order real polynomials is contained in a simply connected region D if and only if the zeros of all polynomial along the exposed edges of P are contained in D. This paper is motivated by the fact that the requirement of simple connectedness of D may be too restrictive in applications such as dominant pole assignment and filter design where the separation of zeros is required. In this paper, we extend the "edge criterion" in [2] to handle any region D whose complement Dc has the following property: Every point d Dc lies on some continuous path which remains within Dc and is unbounded. This requirement is typically verified by inspection and allows for a large class of disconnected regions. We also allow for polynomials with complex coefficients.