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

摘要

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 divisorEof the canonical blow-upB(I)DOUBLE-STRUCK CAPITAL C(n)of the Bautin idealI. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcsArc(BIDOUBLE-STRUCK CAPITAL C-n,E). The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.