# Discriminant Sets of Families of Hyperbolic Polynomials of Degree 4 and 5

• Volume: 28, Issue: 2, page 117-152
• ISSN: 1310-6600

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## Abstract

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∗ Research partially supported by INTAS grant 97-1644A real polynomial of one real variable is hyperbolic (resp. strictly hyperbolic) if it has only real roots (resp. if its roots are real and distinct). We prove that there are 116 possible non-degenerate configurations between the roots of a degree 5 strictly hyperbolic polynomial and of its derivatives (i.e. configurations without equalities between roots). The standard Rolle theorem allows 286 such configurations. To obtain the result we study the hyperbolicity domain of the family P (x; a, b, c) = x^5 − x^3 + ax^2 + bx + c (i.e. the set of values of a, b, c ∈ R for which the polynomial is hyperbolic) and its stratification defined by the discriminant sets Res(P^(i) , P^(j) ) = 0, 0 ≤ i < j ≤ 4.

## How to cite

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Kostov, Vladimir. "Discriminant Sets of Families of Hyperbolic Polynomials of Degree 4 and 5." Serdica Mathematical Journal 28.2 (2002): 117-152. <http://eudml.org/doc/11551>.

@article{Kostov2002,
abstract = {∗ Research partially supported by INTAS grant 97-1644A real polynomial of one real variable is hyperbolic (resp. strictly hyperbolic) if it has only real roots (resp. if its roots are real and distinct). We prove that there are 116 possible non-degenerate configurations between the roots of a degree 5 strictly hyperbolic polynomial and of its derivatives (i.e. configurations without equalities between roots). The standard Rolle theorem allows 286 such configurations. To obtain the result we study the hyperbolicity domain of the family P (x; a, b, c) = x^5 − x^3 + ax^2 + bx + c (i.e. the set of values of a, b, c ∈ R for which the polynomial is hyperbolic) and its stratification defined by the discriminant sets Res(P^(i) , P^(j) ) = 0, 0 ≤ i < j ≤ 4.},
author = {Kostov, Vladimir},
journal = {Serdica Mathematical Journal},
keywords = {Hyperbolic Polynomial; Hyperbolicity Domain; Overdetermined Stratum; hyperbolic polynomial; hyperbolicity domain; overdetermined stratum},
language = {eng},
number = {2},
pages = {117-152},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Discriminant Sets of Families of Hyperbolic Polynomials of Degree 4 and 5},
url = {http://eudml.org/doc/11551},
volume = {28},
year = {2002},
}

TY - JOUR
AU - Kostov, Vladimir
TI - Discriminant Sets of Families of Hyperbolic Polynomials of Degree 4 and 5
JO - Serdica Mathematical Journal
PY - 2002
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 28
IS - 2
SP - 117
EP - 152
AB - ∗ Research partially supported by INTAS grant 97-1644A real polynomial of one real variable is hyperbolic (resp. strictly hyperbolic) if it has only real roots (resp. if its roots are real and distinct). We prove that there are 116 possible non-degenerate configurations between the roots of a degree 5 strictly hyperbolic polynomial and of its derivatives (i.e. configurations without equalities between roots). The standard Rolle theorem allows 286 such configurations. To obtain the result we study the hyperbolicity domain of the family P (x; a, b, c) = x^5 − x^3 + ax^2 + bx + c (i.e. the set of values of a, b, c ∈ R for which the polynomial is hyperbolic) and its stratification defined by the discriminant sets Res(P^(i) , P^(j) ) = 0, 0 ≤ i < j ≤ 4.
LA - eng
KW - Hyperbolic Polynomial; Hyperbolicity Domain; Overdetermined Stratum; hyperbolic polynomial; hyperbolicity domain; overdetermined stratum
UR - http://eudml.org/doc/11551
ER -

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