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

TL;DR

This paper develops an algebro-geometric framework for analyzing bifurcation functions related to Hilbert's 16th problem on a period annulus, linking bifurcations to Nash spaces of arcs and the Bautin ideal.

Contribution

It introduces a novel algebro-geometric approach to study bifurcation functions, connecting them with Nash spaces of arcs and the blow-up of the Bautin ideal, providing new insights into Hilbert's 16th problem.

Findings

Establishes a correspondence between bifurcation functions and points on the exceptional divisor.

Defines essential perturbations via irreducible components of Nash spaces of arcs.

Discusses the approach with planar quadratic vector fields in the Kapteyn normal form.

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 $B_I{\mathbb C}^n$ 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 C^n,E)$. The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.