Examples of center cyclicity bounds using the reduced Bautin depth

García, Isaac

doi:10.1090/proc/13570

Examples of center cyclicity bounds using the reduced Bautin depth
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by Isaac A. García PDF

Proc. Amer. Math. Soc. 145 (2017), 4363-4370 Request permission

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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