Geodesic
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. 
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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/

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