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 x(k)((i)) the roots of P-(i), k = 1,..., n - i, i = 0,..., n - 1. Then in the absence of any equality of the form x(i)((j)) = x(k)((l)) (*) one has Vi < j, x(k)((i)) < x(k)((j)) < X-k+j-i((i)) (**) (the Rolle theorem). For n >= 4 (resp. for n >= 5) not all arrangements without equalities (*) of n(n + 1)/2 real numbers x(k)((i)) and compatible with (**) (we call them admissible) are realizable by the roots of hyperbolic polynok mials (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. (C) 2007 Elsevier Masson SAS. All rights reserved.