Root Configurations for Hyperbolic Polynomials of Degree 3, 4, and 5

A real polynomial of one real variable is (strictly) hyperbolic if it has only real (and distinct) roots. There are $10$ (resp. $116$) possible non-degenerate configurations between the roots of a strictly hyperbolic polynomial of degree $4$ (resp. $5$) and of its derivatives (i.e., configurations without equalities between roots). The standard Rolle theorem allows $12$ (resp. $286$) such configurations. The result is based on the study of the hyperbolicity domain of the family $P(x,a)=x^n+a_1x^{n-1}+\dots+a_n$ for $n=4,5$ (i.e., of the set of values of $a\in\mathbb{R}^n$ for which the polynomial is hyperbolic) and its stratification defined by the discriminant sets $\operatorname{Res}(P^{(i)},P^{(j)})=0$, $0\le i<j\le n-1$.