Normal forms of topologically quasi-homogeneous foliations on $(\C^2,0)$

TL;DR

This paper develops formal normal forms for topologically quasi-homogeneous foliations in two complex variables, decomposing them into specific components and linking free parameters to moduli space dimensions.

Contribution

It introduces a new formal normal form structure for these foliations, connecting the number of free coefficients to the moduli space of unfoldings.

Findings

Normal forms are composed of a quasi-homogeneous, hamiltonian, and radial term.

Number of free coefficients matches the dimension of Mattei's moduli space.

Provides a systematic way to classify these foliations.

Abstract

The aim of this paper is to construct formal normal forms for the class of topologically quasi-homogeneous foliations under generic conditions. Any such normal form is given as the sum of three terms: an initial generic quasi-homogeneous term, a hamiltonian term and a radial term. Moreover, we also show that the number of free coefficients in the hamiltonian part is consistent with the dimension of Mattei's moduli space of unfoldings.