Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: December 27, 2001
MathSciNet review: 1896037
Full-text PDF

Abstract | References | Similar Articles | 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.

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

  • 1. A. Campillo and M.M. Carnicer, Proximity inequalities and bounds for the degree of invariant curves by foliations of $\mathbf{P}_{\mathbf{C}}^2$, 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. D. Cerveau and A. Lins Neto, Holomorphic foliations in CP(2) having an invariant algebraic curve, Ann. Inst. Fourier 41 (1991), 883-903. MR 93b:32050
  • 5. C. Christopher and J. Llibre, Integrability via Invariant Algebraic Curves for Planar Polynomial Differential Systems, Annals of Differential Equations 16 (2000), 5-19. MR 2001g:34001
  • 6. C. Christopher, P. Mardesic and C. Rousseau, Normalizable, integrable and linearizable saddle points in complex quadratic systems in $\mathbf{C}^2$, 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.
  • 8. J. P. Françoise, A. Fronville and H. Zoladek, Solution of the 1:-2 resonant center problem in the quadratic case, Fundamenta Mathematicae 157 (1998), 191-207. MR 99g:34063
  • 9. S. D. Furta, On non-integrability of general systems of differential equations, Z. angew Math. Phys. 47 (1996), 112-131. MR 97f:34003
  • 10. J .P. Jouanolou, Equations de Pfaff algébriques, Lectures Notes in Mathematics 708, Springer-Verlag, 1979. MR 81k:14008
  • 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.
  • 13. Yiu-Kwong Man and M. A. H. Maccallum, A Rational Approach to the Prelle-Singer Algorithm, Journal of Symbolic Computation 24 (1997), 31-43. MR 98g:12009
  • 14. J. Moulin-Ollagnier, About a conjecture on quadratic vector fields, preprint, 2000.
  • 15. J. M. Pearson, N. G. Lloyd and C. J. Christopher, Algorithmic derivation of centre conditions, SIAM Review 38 no.4 (1996), 619-636. MR 98c:34050
  • 16. H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré I and II, Rendiconti del circolo matematico di Palermo 5 (1891), 161-191; 11 (1897), 193-239.
  • 17. M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983), 215-229. MR 85d:12008
  • 18. M. F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673-688. MR 92m:12014

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34C05, 34A34, 34C14

Retrieve articles in all journals with MSC (2000): 34C05, 34A34, 34C14

Additional Information

Colin Christopher
Affiliation: School of Mathematics and Statistics, University of Plymouth, Plymouth, Devon PL4 8AA, United Kingdom

Jaume Llibre
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

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
Published electronically: December 27, 2001
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society