Robust Hurwitzness of complex polynomials in convex and compact domain

A necessary and sufficient condition for a family of complex polynomials with coefficients varying in a convex and compact domain to be strictly Hurwitz is given. The result can imply several known results on the robust Hurwitzness of complex polynomials. In particular, it is shown that the result covers the `edge theorem´ for polytopes of polynomials in the case of strict Hurwitzness and requires weaker conditions than the edge theorem. It is also shown that the results on the robust strict Hurwitzness of diamond polynomials can be implied by the present result. Applying this result, the authors derive a new result on the robust positivity of a complex diamond rational function which is more advanced than that obtained by the authors in 1991