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Algebraic invariant curves and first integrals for Riccati polynomial differential systems


Authors: Jaume Llibre and Clàudia Valls
Journal: Proc. Amer. Math. Soc. 142 (2014), 3533-3543
MSC (2010): Primary 34C05, 34A34, 34C14
DOI: https://doi.org/10.1090/S0002-9939-2014-12085-5
Published electronically: June 23, 2014
MathSciNet review: 3238428
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Abstract: We study the algebraic invariant curves and first integrals for the Riccati polynomial differential systems of the form $ x'=1$, $ y' =a(x) y^2 +b(x)y +c(x)$, where $ a(x)$, $ b(x)$ and $ c(x)$ are polynomials. We characterize them when $ c(x)=\kappa (b(x)-\kappa a(x))$ for some $ \kappa \in \mathbb{C}$. We conjecture that algebraic invariant curves and first integrals for these Riccati polynomial differential systems only exist if $ c(x)=\kappa (b(x)-\kappa a(x))$ for some $ \kappa \in \mathbb{C}$.


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Additional Information

Jaume Llibre
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra, Barcelona, Catalonia, Spain
Email: jllibre@mat.uab.cat

Clàudia Valls
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal
Email: cvalls@math.ist.utl.pt

DOI: https://doi.org/10.1090/S0002-9939-2014-12085-5
Keywords: Algebraic first integrals, algebraic invariant curves, Riccati polynomial differential equations.
Received by editor(s): November 18, 2011
Received by editor(s) in revised form: December 13, 2011, May 1, 2012, September 6, 2012, and October 28, 2012
Published electronically: June 23, 2014
Additional Notes: The first author was partially supported by the MICINN/FEDER grant MTM2008–03437, AGAUR grant 2009SGR-410, ICREA Academia and FP7-PEOPLE-2012-IRSES-316338
The second author was partially supported by the FCT through CAMGDS, Lisbon
Communicated by: Yingfei Yi
Article copyright: © Copyright 2014 American Mathematical Society

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