Smooth trivial vector bundle structure of the space of Hurwitz polynomials (Q976227)

The authors consider a Hurwitz real polynomial, that is, a real polynomial with all its roots with negative real part. It is known that the set of Hurwitz real polynomials is an open set and that it is not a convex set. Recently it was proved that the space of Hurwitz real polynomials of degree \(n\) with positive coefficients, denoted by \(\mathcal{H}^{+}_{n}\), is contractible. In this paper the authors prove that \(\mathcal{H}^{+}_{n}\) is a smooth trivial line bundle over \(\mathcal{H}^{+}_{n-1}\) and as a consequence they conclude that \(\mathcal{H}^{+}_{n}\) is a smooth trivial vector bundle over \(\mathcal{H}^{+}_{n-k}\) of rank \(k\).