Smooth trivial vector bundle structure of the space of Hurwitz polynomials

Due to the importance of the robustness of stable systems, the properties of the set of Hurwitz real polynomials are interesting to study. Among these properties is that the set is an open and not a convex set, and recently it was proved that the set of Hurwitz polynomials of degree nn with positive coefficients, Hn+, is contractible. Now, we will prove that the set Hn+ is a smooth trivial vector bundle over Hn−k+ of rank kk.