Root Configurations for Hyperbolic Polynomials of Degree 3, 4, and 5
Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 4, pp. 71-74
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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 $\operatorname{Res}(P^{(i)},P^{(j)})=0$, $0\le i$.
Keywords:
hyperbolic polynomial, hyperbolicity domain, overdetermined stratum.
@article{FAA_2002_36_4_a6,
author = {V. P. Kostov},
title = {Root {Configurations} for {Hyperbolic} {Polynomials} of {Degree} 3, 4, and 5},
journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
pages = {71--74},
publisher = {mathdoc},
volume = {36},
number = {4},
year = {2002},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/FAA_2002_36_4_a6/}
}
V. P. Kostov. Root Configurations for Hyperbolic Polynomials of Degree 3, 4, and 5. Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 4, pp. 71-74. http://geodesic.mathdoc.fr/item/FAA_2002_36_4_a6/