# Root arrangements of hyperbolic polynomial-like functions.

Revista Matemática Complutense (2006)

- Volume: 19, Issue: 1, page 197-225
- ISSN: 1139-1138

## Access Full Article

top## Abstract

top## How to cite

topKostov, Vladimir Petrov. "Root arrangements of hyperbolic polynomial-like functions.." Revista Matemática Complutense 19.1 (2006): 197-225. <http://eudml.org/doc/40885>.

@article{Kostov2006,

abstract = {A real polynomial P of degree n in one real variable is hyperbolic if its roots are all real. A real-valued function P is called a hyperbolic polynomial-like function (HPLF) of degree n if it has n real zeros and P(n) vanishes nowhere. Denote by xk(i) the roots of P(i), k = 1, ..., n-i, i = 0, ..., n-1. Then in the absence of any equality of the formxi(j) = xk(i) (1)one has∀i < j xk(i) < xk(j) < xk+j-i(i) (2)(the Rolle theorem). For n ≥ 4 (resp. for n ≥ 5) not all arrangements without equalities (1) of n(n+1)/2 real numbers xk(i) and compatible with (2) are realizable by the roots of hyperbolic polynomials (resp. of HPLFs) of degree n and of their derivatives. For n = 5 and when x1(1) < x2(1) < x1(3) < x2(3) < x3(1) < x4(1) we show that from the 40 arrangements without equalities (1) and compatible with (2) only 16 are realizable by HPLFs (from which 6 by perturbations of hyperbolic polynomials and none by hyperbolic polynomials).},

author = {Kostov, Vladimir Petrov},

journal = {Revista Matemática Complutense},

keywords = {Ceros de polinomios; Ceros de una función; hyperbolic polynomial; polynomial-like function; root arrangement; configuration vector},

language = {eng},

number = {1},

pages = {197-225},

title = {Root arrangements of hyperbolic polynomial-like functions.},

url = {http://eudml.org/doc/40885},

volume = {19},

year = {2006},

}

TY - JOUR

AU - Kostov, Vladimir Petrov

TI - Root arrangements of hyperbolic polynomial-like functions.

JO - Revista Matemática Complutense

PY - 2006

VL - 19

IS - 1

SP - 197

EP - 225

AB - A real polynomial P of degree n in one real variable is hyperbolic if its roots are all real. A real-valued function P is called a hyperbolic polynomial-like function (HPLF) of degree n if it has n real zeros and P(n) vanishes nowhere. Denote by xk(i) the roots of P(i), k = 1, ..., n-i, i = 0, ..., n-1. Then in the absence of any equality of the formxi(j) = xk(i) (1)one has∀i < j xk(i) < xk(j) < xk+j-i(i) (2)(the Rolle theorem). For n ≥ 4 (resp. for n ≥ 5) not all arrangements without equalities (1) of n(n+1)/2 real numbers xk(i) and compatible with (2) are realizable by the roots of hyperbolic polynomials (resp. of HPLFs) of degree n and of their derivatives. For n = 5 and when x1(1) < x2(1) < x1(3) < x2(3) < x3(1) < x4(1) we show that from the 40 arrangements without equalities (1) and compatible with (2) only 16 are realizable by HPLFs (from which 6 by perturbations of hyperbolic polynomials and none by hyperbolic polynomials).

LA - eng

KW - Ceros de polinomios; Ceros de una función; hyperbolic polynomial; polynomial-like function; root arrangement; configuration vector

UR - http://eudml.org/doc/40885

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.