Topology of spaces of hyperbolic polynomials and combinatorics of resonances

Abstract

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 $m\ge 2$. 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 $\lambda$.