A generalization of the Gauss-Lucas theorem

J. L. Díaz-Barrero; J. J. Egozcue

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 2, page 481-486
  • ISSN: 0011-4642

Abstract

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Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.

How to cite

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Díaz-Barrero, J. L., and Egozcue, J. J.. "A generalization of the Gauss-Lucas theorem." Czechoslovak Mathematical Journal 58.2 (2008): 481-486. <http://eudml.org/doc/31223>.

@article{Díaz2008,
abstract = {Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.},
author = {Díaz-Barrero, J. L., Egozcue, J. J.},
journal = {Czechoslovak Mathematical Journal},
keywords = {polynomials; location of zeros; convex hull of the zeros; Gauss-Lucas theorem; polynomials; location of zeros; convex hull of the zeros; Gauss-Lucas theorem},
language = {eng},
number = {2},
pages = {481-486},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A generalization of the Gauss-Lucas theorem},
url = {http://eudml.org/doc/31223},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Díaz-Barrero, J. L.
AU - Egozcue, J. J.
TI - A generalization of the Gauss-Lucas theorem
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 2
SP - 481
EP - 486
AB - Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.
LA - eng
KW - polynomials; location of zeros; convex hull of the zeros; Gauss-Lucas theorem; polynomials; location of zeros; convex hull of the zeros; Gauss-Lucas theorem
UR - http://eudml.org/doc/31223
ER -

References

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  1. Collected Works, Leipzig, Teubner, 1900–1903, vol. 3, p. 112, and vol. 8, p. 32, and vol. 9, p. 187. Zbl0924.01032
  2. Propriétés géométriques des fractions rationelles, C. R. Acad. Sci. Paris 77 (1874), 431–433. (1874) 
  3. Characterization of Polynomials by Reflection Coefficients, PhD. Disertation (Advisor J. J. Egozcue), Universitat Politècnica de Catalunya, Barcelona, 2000. (2000) 
  4. The Geometry of the Zeros of a Polynomial in a Complex Variable, American Mathematical Society, Rhode Island, 1966. (1966) MR0031114

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