  In this paper we study the topology of the strata, indexed by number
partitions $\lambda$, in the natural stratification of the space of monic
hyperbolic polynomials of degree $n$. We prove stabilization theorems for
removing an independent block or an independent relation in $\lambda$. We also
prove contractibility of the one-point compactifications of the strata indexed
by a large class of number partitions, including $\lambda=(k^m,1^r)$, for
$m\geq 2$. Furthermore, we study the maps between the homology groups of the
strata, induced by imposing additional relations (resonances) on the number
partition $\lambda$, or by merging some of the blocks of $\lambda$.
