On a stratification defined by real roots of polynomials

We consider the family of polynomials $P(x,a)=x^n+a_1{x}^{n-1}+\mathrm{.}\mathrm{.}\mathrm{.}+a_n$, $x,a_i\in \mathbf{R}$, and the stratification of ${\bf R}^n\cong {(a_1,… ,a_n)|a_i\in {\bf R}}$definedbythemultiplicityvectoroftherealrootsof$P$. We prove smoothness of the strata and a transversality property of their tangent spaces.