The cyclicity of polynomial centers via the reduced Bautin depth

García, Isaac

doi:10.1090/proc/12896

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.

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