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On a Result of Darboux

Published online by Cambridge University Press:  01 February 2010

Javier Chavarriga ,
Jaume Llibre  and
Jean Moulin Ollagnier
Javier Chavarriga
Affiliation:
Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69. 25001 Lleida, Spain, chava@eup.uds.es
Jaume Llibre
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain,jllibre@mat.uab.es
Jean Moulin Ollagnier
Affiliation:
Laboratoire Gage, UMS CNRS 658 Medicis, École Polytechnique, F 91128 Palaiseau Cedex, France, Jean.Moulin-Ollagnier@polytechnique.fr

Abstract

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This paper is concerned with a relation of Darboux in enumerative geometry, which has very useful applications in the study of polynomial vector fields. The original statement of Darboux was not correct. The present paper gives two different elementary proofs of this relation. The first one follows the ideas of Darboux, and uses basic facts about the intersection index of two plane algebraic curves; the second proof is rather more sophisticated, and gives a stronger result, which should also be very useful. The power of the relation of Darboux is then illustrated by the provision of new, simple proofs of two known results. First, it is shown that an irreducible invariant algebraic curve of degree n > 1 without multiple points for a polynomial vector field of degree m satisfies n ≤ m + 1. Second, a proof is given that quadratic polynomial vector fields have no algebraic limit cycles of degree 3.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 4 , 2001 , pp. 197 - 210
Copyright
Copyright © London Mathematical Society 2001

References

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