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A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals
Authors:
Colin Christopher and Jaume Llibre
Journal:
Proc. Amer. Math. Soc. 130 (2002), 2025-2030
MSC (2000):
Primary 34C05, 34A34, 34C14
Posted:
December 27, 2001
MathSciNet review:
1896037
Full-text PDF Free Access
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Additional Information
Abstract: We give a class of quadratic systems without rational first integral which contains irreducible algebraic solutions of arbitrarily high degree. The construction gives a negative answer to a conjecture of Lins Neto and others.
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- 1.
- A. Campillo and M.M. Carnicer, Proximity inequalities and bounds for the degree of invariant curves by foliations of
 , Trans. Amer. Math. Soc. 349 (1997), 2211-2228. MR 97h:32051
- 2.
- M. M. Carnicer, The Poincaré problem in the nondicritical case, Annals of Math. 140 (1994), 289-294. MR 95k:32031
- 3.
- C. Christopher, Invariant algebraic curves and conditions for a centre, Proc. Royal Soc. Edinburgh 124A (1994), 1209-1229. MR 95m:34052
- 4.
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- 5.
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- 6.
- C. Christopher, P. Mardesic and C. Rousseau, Normalizable, integrable and linearizable saddle points in complex quadratic systems in , Preprint, 2000.
- 7.
- G. Darboux, Mémoire sur les équations différentielles algébrique du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. 2ème série, 2 (1878), 60-96; 123-144; 151-200.
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- 10.
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- 11.
- Weigu Li, J. Llibre and Xiang Zhang, Planar analytic vector fields with generalized rational first integrals, submitted to Bulletin des sciences mathématiques.
- 12.
- A. Lins Neto, Some examples for Poincaré problem and a question of Brunella, preprint.
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- 18.
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Additional Information
Colin Christopher
Affiliation:
School of Mathematics and Statistics, University of Plymouth, Plymouth, Devon PL4 8AA, United Kingdom
Email:
cchristopher@plymouth.ac.uk
Jaume Llibre
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Email:
jllibre@mat.uab.es
DOI:
http://dx.doi.org/10.1090/S0002-9939-01-06253-0
PII:
S 0002-9939(01)06253-0
Keywords:
Rational first integral,
invariant algebraic curve,
quadratic systems
Received by editor(s):
November 1, 2000
Received by editor(s) in revised form:
January 25, 2001
Posted:
December 27, 2001
Communicated by:
Carmen C. Chicone
Article copyright:
© Copyright 2001 American Mathematical Society
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