Stratifications of polynomial spaces

Abstract

In the paper we construct some stratifications of the space of monic polynomials in real and complex cases. These stratifications depend on properties of roots of the polynomials on some given semialgebraic subset of $\Bbb R$ or $\Bbb C$. We prove differential triviality of these stratifications. In the real case the proof is based on properties of the action of the group of interval exchange transformations on the set of all monic polynomials of some given degree. Finally we compare stratifications corresponding to different semialgebraic subsets