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Hilbert’s 16th problem on a period annulus and Nash space of arcs

Part of: Birational geometry Qualitative theory

Published online by Cambridge University Press:  12 July 2019

JEAN–PIERRE FRANÇOISE ,
LUBOMIR GAVRILOV  and
DONGMEI XIAO
JEAN–PIERRE FRANÇOISE
Affiliation:
Université P.-M. Curie, Paris 6, Laboratoire Jacques–Louis Lions UMR 7598 CNRS, 4 Place Jussieu, 75252, Paris, France, and School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, PR China. e-mail: Jean-Pierre.Francoise@upmc.fr
LUBOMIR GAVRILOV
Affiliation:
Institut de Mathématiques de Toulouse; UMR 5219 Université de Toulouse; CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France. e-mail: lubomir.gavrilov@math.univ-toulouse.fr
DONGMEI XIAO
Affiliation:
School of Mathematical Sciences Shanghai Jiao Tong University, Shanghai, 200240, PR China. e-mail: xiaodm@sjtu.edu.cn
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Abstract

This paper introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert’s 16th problem on a period annulus. Each possible bifurcation function is in one-to-one correspondence with a point in the exceptional divisor E of the canonical blow-up BI ℂn of the Bautin ideal I. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcs Arc(BI ℂn, E). The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.

MSC classification

Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications)
Secondary: 14E18: Arcs and motivic integration
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 2 , September 2020 , pp. 377 - 409
Copyright
© Cambridge Philosophical Society 2019

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