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

2016/10/24 by Françoise, Jean-Pierre, Lubomir Gavrilov, Dongmei Xiao +2
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1610.07582

Abstract

This article 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 B_I\mathbb 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(B_I\mathbb Cⁿ,E). The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.

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