On a stratification defined by real roots of polynomials

Abstract

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