Hyperbolic polynomials are those polynomials all of whose roots are real. The set of all monic hyperbolic polynomials of degree \(n\) is identified with a subspace \(\text{Hyp}^n\) of \({\mathbb R}^n\) (via the ordered roots with multiplicities). There is a natural stratification of \(\text{Hyp}^n\) whose closed strata \(\text{Hyp}^n_\lambda\) are indexed by number partitions \(\lambda\) of \(n\) (coming from the root multiplicities). This paper is concerned primarily with the homology of the strata of \(\text{Hyp}^n\) and lies in the area of the study of the topology of subsets of spaces of smooth maps as given by V. A. Vassiliev [Complements of discriminants of smooth maps: topology and applications, Trans. Math. Mono., Am. Math. Soc., Providence (1994; Zbl 0826.55001)]. B. Shapiro and V. Welker [Result. Math. 33, No. 3-4, 338-355 (1998; Zbl 0919.57022)] have given a combinatorial construction of a finite simplicial complex \(\delta_\lambda\) whose double suspension is homeomorphic to the one point compactification of \(\text{Hyp}^n_\lambda\). The paper under review gives conditions on \(\lambda\) that imply that the one point compactification of \(\text{Hyp}^n_\lambda\) is contractible. Additional results are given concerning the homology of these compactified strata and homomorphisms of their homology groups induced by the author’s notion of resonances. An ultimate goal for this line of research is to find an algorithm to compute the homology of the compactified strata. The methods are mainly combinatorial; in particular, use is made of discrete Morse theory as developed by R. Forman [ Adv. Math. 134, No. 1, 90-145 (1998; Zbl 0896.57023)].