The cyclicity of polynomial centers via the reduced bautin depth
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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.
Is part ofProceedings of the American Mathematical Society, 2016, vol. 144, núm. 6, p. 2473–2478
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García, I. A. (Isaac A.); Llibre, Jaume; Maza Sabido, Susanna (Elsevier, 2015-01-20)We consider families of planar polynomial vector fields having a singularity with purely imaginary eigenvalues for which a basis of its Bautin ideal B is known. We provide an algorithm for computing an upper bound of the ...
García, I. A. (Isaac A.) (American Mathematical Society, 2017-09-01)There is a method for bounding the cyclicity of nondegenerate monodromic singularities of polynomial planar families of vector fields Xλ which can work even in case that the Poincar´e first return map has associated a ...
García, I. A. (Isaac A.); Llibre, Jaume; Maza Sabido, Susanna (Springer, 2016-09-01)In this paper we study the cyclicity of the centers of the quartic polynomial family written in complex notation as z = i z + z z (A z^2 + B z z + C z^2 ), where A,B,C. We give an upper bound for the cyclicity of any ...