Pairs of foliations and Mattei-Moussu's theorem

TL;DR

This paper advances the understanding of pairs of foliations by proving a reduction of singularities and analyzing their classification, including a new proof of Mattei-Moussu's theorem and recent results on shared separatrices.

Contribution

It introduces a new proof of Mattei-Moussu's theorem and extends the classification to pairs of reduced foliations sharing separatrices.

Findings

Reduction of singularities for pairs of foliations achieved

New proof of Mattei-Moussu's theorem avoiding Gronwall's inequality

Recent results on pairs of reduced foliations sharing separatrices

Abstract

We prove a reduction of singularities for pairs of foliations by blowing-up, and then investigate the analytic classification of the reduced models. Those reduced pairs of regular foliations are well understood. The case of a regular and a singular foliation is dealt with Mattei-Moussu's Theorem for which we provide a new proof, avoiding Gronwall's inequality. We end-up announcing results recently obtained by the first author in the case of a pair of reduced foliations sharing the same separatrices.