The cyclicity of polynomial centers via the reduced Bautin depth
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- by Isaac A. García
- Proc. Amer. Math. Soc. 144 (2016), 2473-2478
- DOI: https://doi.org/10.1090/proc/12896
- Published electronically: December 15, 2015
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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.References
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Bibliographic Information
- Isaac A. García
- Affiliation: Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Spain
- Email: garcia@matematica.udl.cat
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2473-2478
- MSC (2010): Primary 37G15, 37G10, 34C07
- DOI: https://doi.org/10.1090/proc/12896
- MathSciNet review: 3477063