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A Precondition for the Edge Theorem

The stability version of the Edge Theorem states that a polytope of polynomials is D-stable if and only if its edge polynomials are all D-stable. Unfortunately, this statement is only true for restricted classes of polytopes and restricted classes of stability regions. If either of these restrictions is removed, the theorem will not be valid. In order to remove these constraints, this paper will present a simple precondition for the Edge Theorem. If this condition is not satisfied then the polytope is not D-stable. If this condition is satisfied then the stability version of the Edge Theorem is valid for all stability regions D and all polytopes of polynomials. The Generalized Edge Theorem of Soh and Berger is stated in a similar form. They consider stability regions D whose complements are simply connected in the complex plane. A special case of this theorem states that a polytope of polynomials is D-stable if and only if the edge polynomials are D-stable. It will be shown by counter example that this statement is not true. The precondition presented in this paper suggests ways of correcting this theorem.