The cyclicity of polynomial centers via the reduced Bautin depth

Author:
Isaac A. García

Journal:
Proc. Amer. Math. Soc. **144** (2016), 2473-2478

MSC (2010):
Primary 37G15, 37G10, 34C07

DOI:
https://doi.org/10.1090/proc/12896

Published electronically:
December 15, 2015

MathSciNet review:
3477063

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Abstract | References | Similar Articles | Additional Information

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

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Additional Information

**Isaac A. García**

Affiliation:
Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Spain

Email:
garcia@matematica.udl.cat

DOI:
https://doi.org/10.1090/proc/12896

Keywords:
Center,
polynomial vector fields,
Bautin ideal,
cyclicity,
limit cycle

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

Article copyright:
© Copyright 2015
American Mathematical Society