Curves of tangencies of foliation pairs and normalizing transformations

TL;DR

This paper provides a comprehensive description of the curves of tangencies in foliation pairs induced by analytic differential equations, utilizing local models, normalizing transformations, and normal forms.

Contribution

It introduces k-normal forms for normalizing transformations and characterizes the realization of analytic curves as tangency curves under generic conditions.

Findings

Complete classification of tangency curves for foliation pairs.

Introduction of k-normal forms for normalizing transformations.

Realization of generic analytic curves as tangency curves.

Abstract

In this work we give a complete description of the collection of curves of tangencies induced by germs of foliation pairs -- non dicritical and dicritical -- given by analytic differential equations with degenerated non dicritical and dicritical singularities, satisfying some genericity assumptions. To this purpose we use local models and analytic normalizing transformations. Moreover, for each natural number $k$ we obtain $k$-normal forms for the normalizing transformations. These normal forms are used to give parametrizations, up to a finite jet, of the branches of the curves of tangencies. We also prove that under natural genericity assumptions any germ of analytic curve having pairwise transversal smooth branches is realized as curve of tangencies of a -- non dicritical and dicritical -- foliation pair.