On root arrangements for hyperbolic polynomial-like functions and their derivatives

A real polynomial P of degree n in one real variable is hyperbolic if its roots are all real. A real-valued function P is called a hyperbolic polynomial-like function (HPLF) of degree n if it has n real zeros and P(n) vanishes nowhere. Denote by the roots of P(i), k=1,…,n−i, i=0,…,n−1. Then in the absence of any equality of the form one has ∀i<j, (the Rolle theorem). For n?4 (resp. for n?5) not all arrangements without equalities (∗) of n(n+1)/2 real numbers and compatible with (∗∗) (we call them admissible) are realizable by the roots of hyperbolic polynomials (resp. of HPLFs) of degree n and of their derivatives. For n=5 we show that from 286 admissible arrangements, exactly 236 are realizable by HPLFs; from these 236 arrangements, 116 are realizable by hyperbolic polynomials and 24 by perturbations of such.