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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Chebyshev criterion for Abelian integrals


Authors: M. Grau, F. Mañosas and J. Villadelprat
Journal: Trans. Amer. Math. Soc. 363 (2011), 109-129
MSC (2010): Primary 34C08, 41A50; Secondary 34C23
Published electronically: August 27, 2010
MathSciNet review: 2719674
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Abstract: We present a criterion that provides an easy sufficient condition in order for a collection of Abelian integrals to have the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be checked, in many cases, in a purely algebraic way. By using this criterion, several known results are obtained in a shorter way and some new results, which could not be tackled by the known standard methods, can also be deduced.


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Additional Information

M. Grau
Affiliation: Departament de Matemàtica, Universitat de Lleida, Lleida, Spain
Email: mtgrau@matematica.udl.cat

F. Mañosas
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Spain
Email: Francesc.Manosas@uab.cat

J. Villadelprat
Affiliation: Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Tarragona, Spain
Email: Jordi.Villadelprat@urv.cat

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05007-X
PII: S 0002-9947(2010)05007-X
Keywords: Planar vector field, Hamiltonian perturbation, limit cycle, Chebyshev system, Abelian integral
Received by editor(s): July 3, 2008
Published electronically: August 27, 2010
Additional Notes: The first author was partially supported by the MEC/FEDER grant MTM2005-06098-C02-02. The second author was supported by the MEC/FEDER grants MTM2005-02139 and MTM2005-06098 and the CIRIT grant 2005SGR-00550. The third author was supported by the MEC/FEDER grant MTM2005-06098-C02-01 and the CIRIT grant 2005SGR-00550.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.