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The cyclicity of polynomial centers via the reduced Bautin depth

Author: Isaac A. García
Journal: Proc. Amer. Math. Soc. 144 (2016), 2473-2478
MSC (2010): Primary 37G15, 37G10, 34C07
Published electronically: December 15, 2015
MathSciNet review: 3477063
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Abstract: We describe a method for bounding the cyclicity of the class of monodromic singularities of polynomial planar families of vector fields $ \mathcal {X}_\lambda $ with an analytic Poincaré first return map having a polynomial Bautin ideal $ \mathcal {B}$ in the ring of polynomials in the parameters $ \lambda $ of the family. This class includes the nondegenerate centers, generic nilpotent centers and also some degenerate centers. This method can work even in the case in which $ \mathcal {B}$ is not radical by studying the stabilization of the integral closures of an ascending chain of polynomial ideals that stabilizes at $ \mathcal {B}$. The approach is based on computational algebra methods for determining a minimal basis of the integral closure $ \bar {\mathcal {B}}$ of $ \mathcal {B}$. As far as we know, the obtained cyclicity bound is the minimum found in the literature.

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Additional Information

Isaac A. García
Affiliation: Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Spain

Keywords: Center, polynomial vector fields, Bautin ideal, cyclicity, limit cycle
Received by editor(s): June 3, 2015
Received by editor(s) in revised form: July 3, 2015
Published electronically: December 15, 2015
Additional Notes: The first author was partially supported by a MINECO grant number MTM2014-53703-P and by a CIRIT grant number 2014 SGR 1204.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society

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