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A refinement of the Gauss-Lucas theorem
Author(s):
Dimitar
K.
Dimitrov
Journal:
Proc. Amer. Math. Soc.
126
(1998),
2065-2070.
MSC (1991):
Primary 30C15, 26C10
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Abstract:
The classical Gauss-Lucas Theorem states that all the critical points (zeros of the derivative) of a nonconstant polynomial  lie in the convex hull  of the zeros of  . It is proved that, actually, a subdomain of  contains the critical points of  .
References:
- 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.
- M. Marden, Geometry of Polynomials, Amer.Math.Soc.Surveys, no.3, Providence, R.I., 1966. MR 37:1562
- 3.
- M. Marden, Conjectures on the critical points of a polynomial, Amer.Math.Monthly 90(1983), 267-276. MR 84e:30007
- 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:
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
Copyright of article:
Copyright
1998,
American Mathematical Society
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