# Characterization of second type plane foliations using Newton polygons

## TL;DR

This paper characterizes second type plane foliations through Newton polygons, linking their structure to formal separatrices and Poincaré-Hopf index, with specific results for cuspidal foliations.

## Contribution

It provides a new characterization of second type foliations using Newton polygons and Poincaré-Hopf index, especially for cuspidal foliations.

## Key findings

- Characterization of second type foliations via Newton polygons.
- Precise conditions for cuspidal foliations with the same desingularization.
- Criteria for cuspidal foliations to be generalized curves or have a single separatrix.

## Abstract

In this article we characterize the foliations that have the same Newton polygon that their union of formal separatrices, they are the foliations called of the second type. In the case of cuspidal foliations studied by Loray, we precise this characterization using the Poincar\'e-Hopf index. This index also characterizes the cuspidal foliations having the same desingularization that the union of its separatrices. Finally we give necessary and sufficient conditions when these cuspidal foliations are generalized curves, and a characterization when they have only one separatrix.