Root arrangements of hyperbolic polynomial-like functions.

Kostov, Vladimir Petrov

Revista Matemática de la Universidad Complutense de Madrid, Tome 19 (2006), p. 197-225 / Harvested from Biblioteca Digital de Matemáticas

Résumé

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 xk (i) the roots of P(i), k = 1, ..., n-i, i = 0, ..., n-1. Then in the absence of any equality of the form

xi (j) = xk (i) (1)

one has

∀i < j xk (i) < xk (j) < xk+j-i (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 xk (i) 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 x1 (1) < x2 (1) < x1 (3) < x2 (3) < x3 (1) < x4 (1) 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).

Détail

Publié le : 2006-01-01

MR MR2219829 | Zbl 1155.12301

DMLE-ID : 3460