Necessary and sufficient conditions for a polytope of real polynomials to contain a Hurwitz polynomial

Necessary and sufficient conditions for a polytope of real polynomials to contain a Hurwitz polynomial are established. More specifically, it is proved that a polytope of real polynomials contains a stable polynomial if and only if a certain semipolytope of the polytope contains a stable polynomial. The existence theorem is followed by an algorithm to determine a stable polynomial in the polytope, if one exists. The polytope problem for discrete polynomials is also solved. Two illustrative examples are included.