Comments on "Polytopes of polynomials with zeros in a prescribed set

In a recent paper by M. Fu and B.R. Barmish (ibid., vol.34, p.544-6, May 1989), the applicability of the edge theorem was significantly extended, mainly through relaxing a simple-connectedness assumption. The class of subsets of the complex plane to which the main result applies is referred to as those subsets whose complements are pathwise connected on the Riemann sphere. The authors give a definition of pathwise connected on the Riemann sphere that differs markedly from the literal meaning. The commenter shows by example that although the result given by the authors is correct, it does not apply to all subsets whose complements are pathwise connected on the Riemann sphere in the usual sense. The commenter introduces a result which strengthens the main result of the above paper. As a special case of this result, it is shown that for the case of real polynomials, the applicability of the edge theorem does extend to subsets symmetric with respect to the real axis whose complements are pathwise connected on the Riemann sphere in the usual sense.<>