Root Arrangements of Hyperbolic Polynomial-like Functions

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 (i) k the roots of P , k = 1, . . . , n− i, i = 0, . . . , n− 1. Then in the absence of any equality of the form x (j) i = x (l) k (1) one has ∀i < j x (i) k < x (j) k < x (i) k+j−i (2) (the Rolle theorem). For n ≥ 4 (resp. for n ≥ 5) not all arrangements without equalities (1) of n(n+ 1)/2 real numbers x (i) k and compatible with (2) are realizable by the roots of hyperbolic polynomials (resp. of HPLFs) of degree n and of their derivatives. For n = 5 and when x (1) 1 < x (1) 2 < x (3) 1 < x (3) 2 < x (1) 3 < x (1) 4 we show that from the 40 arrangements without equalities (1) and compatible with (2) only 16 are realizable by HPLFs (from which 6 by perturbations of hyperbolic polynomials and none by hyperbolic polynomials).