This article is cited in 6 papers

Brief communications

Root Configurations for Hyperbolic Polynomials of Degree 3, 4, and 5

V. P. Kostov

Université de Nice Sophia Antipolis

Abstract: A real polynomial of one real variable is (strictly) hyperbolic if it has only real (and distinct) roots. There are $10$ (resp. $116$) possible non-degenerate configurations between the roots of a strictly hyperbolic polynomial of degree $4$ (resp. $5$) and of its derivatives (i.e., configurations without equalities between roots). The standard Rolle theorem allows $12$ (resp. $286$) such configurations. The result is based on the study of the hyperbolicity domain of the family $P(x,a)=x^n+a_1x^{n-1}+\dots+a_n$ for $n=4,5$ (i.e., of the set of values of $a\in\mathbb{R}^n$ for which the polynomial is hyperbolic) and its stratification defined by the discriminant sets $\operatorname{Res}(P^{(i)},P^{(j)})=0$, $0\le i<j\le n-1$.

Keywords: hyperbolic polynomial, hyperbolicity domain, overdetermined stratum.

UDC: 512.622

Received: 12.11.2001

DOI: 10.4213/faa221

 English version: Functional Analysis and Its Applications, 2002, 36:4, 311–314

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