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Examples of center cyclicity bounds using the reduced Bautin depth


Author: Isaac A. García
Journal: Proc. Amer. Math. Soc. 145 (2017), 4363-4370
MSC (2010): Primary 37G15, 37G10, 34C07
DOI: https://doi.org/10.1090/proc/13570
Published electronically: March 23, 2017
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Abstract: There is a method for bounding the cyclicity of non-degenerate monodromic singularities of polynomial planar families of vector fields $ \mathcal {X}_\lambda $ which can work even in the case that the Poincaré first return map has associated a non-radical Bautin ideal $ \mathcal {B}$. The method is based on the stabilization of the integral closures of an ascending chain of polynomial ideals in the ring of polynomials in the parameters $ \lambda $ of the family that stabilizes at $ \mathcal {B}$. In this work we use computational algebra methods to provide an explicit example in which the classical procedure to find the Bautin depth of $ \mathcal {B}$ fails but the new approach is successful.


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Additional Information

Isaac A. García
Affiliation: Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Spain
Email: garcia@matematica.udl.cat

DOI: https://doi.org/10.1090/proc/13570
Keywords: Center, polynomial vector fields, Bautin ideal, cyclicity, limit cycle
Received by editor(s): March 3, 2016
Received by editor(s) in revised form: October 27, 2016
Published electronically: March 23, 2017
Additional Notes: The author was partially supported by MINECO grant number MTM2014-53703-P and by CIRIT grant number 2014 SGR 1204.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2017 American Mathematical Society

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