# Root arrangements of hyperbolic polynomial-like functions.

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

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## Abstract

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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 &lt; j xk(i) &lt; xk(j) &lt; 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) &lt; x2(1) &lt; x1(3) &lt; x2(3) &lt; x3(1) &lt; 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).

## How to cite

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Kostov, 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 &lt; j xk(i) &lt; xk(j) &lt; 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) &lt; x2(1) &lt; x1(3) &lt; x2(3) &lt; x3(1) &lt; 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).},
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
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 &lt; j xk(i) &lt; xk(j) &lt; 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) &lt; x2(1) &lt; x1(3) &lt; x2(3) &lt; x3(1) &lt; 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 -

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