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

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 $P(x,a)=x^n+a_1x^{n-1}+... +a_n$, $a_i,x\in {\bf R}$, the set $\{a\in {\bf 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 $2\leq k\leq n-1$. We prove that the strata are smooth contractible real algebraic varieties.