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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A refinement of the Gauss-Lucas theorem


Author: Dimitar K. Dimitrov
Journal: Proc. Amer. Math. Soc. 126 (1998), 2065-2070
MSC (1991): Primary 30C15, 26C10
MathSciNet review: 1452801
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Abstract | References | Similar Articles | Additional Information

Abstract: The classical Gauss-Lucas Theorem states that all the critical points (zeros of the derivative) of a nonconstant polynomial $p$ lie in the convex hull $\Xi$ of the zeros of $p$. It is proved that, actually, a subdomain of $\Xi$ contains the critical points of $p$.


References [Enhancements On Off] (What's this?)

  • 1. F. Lucas, Propriétés géométriques des fractions rationnelles, C. R. Acad. Sci. Paris 77(1874), 431-433; 78(1874), 140-144; 78(1874), 180-183; 78(1874), 271-274.
  • 2. Morris Marden, Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972 (37 #1562)
  • 3. Morris Marden, Conjectures on the critical points of a polynomial, Amer. Math. Monthly 90 (1983), no. 4, 267–276. MR 700266 (84e:30007), http://dx.doi.org/10.2307/2975758
  • 4. G. Szeg\H{o}, Bemerkungen zu einen Satz von J.H.Grace über die Wurzeln algebraischer Gleichungen, Math.Z. 13(1922), 28-55.
  • 5. J. L. Walsh, On the location of the roots of the derivative of a polynomial, C. R. Congr. Internat. des Mathématiciens, Strasbourg, 1920, pp. 339-342.

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Additional Information

Dimitar K. Dimitrov
Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil
Email: dimitrov@nimitz.dcce.ibilce.unesp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04381-0
PII: S 0002-9939(98)04381-0
Keywords: Nontrivial critical point of a polynomial
Received by editor(s): December 29, 1996
Additional Notes: Research supported by the Brazilian foundation CNPq under Grant 300645/95-3 and the Bulgarian Science Foundation under Grant MM-414.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1998 American Mathematical Society