On arrangements of the roots of a hyperbolic polynomial and of one of its derivatives

Abstract

We consider real monic {\em hyperbolic} polynomials in one real variable, i.e. polynomials having only real roots. Call {\em hyperbolicity domain} $\Pi$ of the family of polynomials $P(x,a)=x^n+a_1{x}^{n-1}+...+a_n$, $a_i,x\in \mathbf{R}$, the set $\{a\in {\mathbf{R}}^n|P$is hyperbolic $\}$. The paper studies a stratification of $\Pi$ defined by the arrangement of the roots of $P$ and $P^{(k)}$, where $2\le k\le n-1$. We prove that the strata are smooth contractible real algebraic varieties.