RMC - Vol. 19, Num. 1 - Vladimir P. Kostov

Full paper in PDF:
$%V. P. Kostov, Root arrangements of hyperbolic polynomial-like functions, Rev. Mat. Complut. 19 (2006), no. 1, 197–225.%$

Root Arrangements of Hyperbolic Polynomial-like Functions

Vladimir Petrov KOSTOV

Université de Nice
Laboratoire de Mathématiques
Parc Valrose
06108 Nice Cedex 2 — France

kostov@math.unice.fr

Received: March 3, 2005
Accepted: December 12, 2005

To Prof. A. Galligo

ABSTRACT

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

x(j)= x(l) i k

(1)

one has

(i) (j) (i) A i< j xk < xk <xk+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 $x(ki)$ 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 $x(11)< x(12)< x(31)< x(23) <x(31)< x(14)$ 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).

Key words: hyperbolic polynomial, polynomial-like function, root arrangement, configuration vector.
2000 Mathematics Subject Classification: 12D10.