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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - V. P. Kostov
TI  - Root Configurations for Hyperbolic Polynomials of Degree 3, 4, and 5
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2002
SP  - 71
EP  - 74
VL  - 36
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2002_36_4_a6/
LA  - ru
ID  - FAA_2002_36_4_a6
ER  - 
%0 Journal Article
%A V. P. Kostov
%T Root Configurations for Hyperbolic Polynomials of Degree 3, 4, and 5
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2002
%P 71-74
%V 36
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2002_36_4_a6/
%G ru
%F 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/