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 ₁ x ^n-1+...+a _n for n=4,5 (i.e., of the set of values of aⁿ for which the polynomial is hyperbolic) and its stratification defined by the discriminant sets Res(P ⁽ⁱ⁾,P ^(j))=0, 0 i < jn-1.