The Galoisian envelope of a germ of foliation: the quasi-homogeneous case

TL;DR

This paper provides geometric and algorithmic criteria to determine the Galois closure of quasi-homogeneous foliation germs, advancing understanding of their algebraic and geometric structures.

Contribution

It introduces new geometric and algorithmic criteria for the Galois closure of quasi-homogeneous foliation germs, building on Malgrange's notion.

Findings

Criteria based on transverse analytic invariants

Algorithmic approach using formal normal forms

Characterization of Galois envelope in the quasi-homogeneous case

Abstract

We give geometric and algorithmic criterions in order to have of a proper Galois closure for a codimension one germ of quasi-homogeneous foliation. We recall this notion recently introduced by B. Malgrange, and describe the Galois envelope of a group of germs of analytic diffeomorphisms. The geometric criterions are obtained from transverse analytic invariants, whereas the algorithmic ones make use of formal normal forms.