Discriminant Sets of Families of Hyperbolic Polynomials of Degree 4 and 5

Abstract

A real polynomial of one real variable is hyperbolic (resp. strictly hyperbolic) if it has only real roots (resp. if its roots are real and distinct). We prove that there are 116 possible non-degenerate configurations between the roots of a degree 5 strictly hyperbolic polynomial and of its derivatives (i.e. configurations without equalities between roots). The standard Rolle theorem allows 286 such configurations. To obtain the result we study the hyperbolicity domain of the family P (x; a, b, c) = x^5 − x^3 + ax^2 + bx + c (i.e. the set of values of a, b, c ∈ R for which the polynomial is hyperbolic) and its stratification defined by the discriminant sets Res(P^(i) , P^(j) ) = 0, 0 ≤ i < j ≤ 4.