The cyclicity of polynomial centers via the reduced bautin depth

Abstract

We describe a method for bounding the cyclicity of the class of monodromic singularities of polyn omial planar families of vector fields X λ with an analytic Poincar e first return map having a polynomial Bautin ideal B in the ring of polynomials in the parameters λ 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 B is not radical by studying the stabilization of the integral closures of an ascending chain of polynomial ideals that stabilizes at B. The approach is based on computational algebra methods for determining a minimal basis of the integral closurē B of B. As far as we know, the obtained cyclicity bound is the minimum found in the literature.

The first author was partially supported by a MINECO grant number MTM2014-53703-P and by a CIRIT grant number 2014 SGR 1204.