Degree of the First Integral of a Foliation in the Pencil $\mathcal{P}_4$

TL;DR

This paper explicitly calculates the degree of the rational first integral for a dense set of degree 4 foliations in the projective plane, revealing how this degree varies with the parameters.

Contribution

It provides an explicit formula for the degree of the first integral as a function of parameters within a specific family of foliations.

Findings

Degree of the first integral varies with parameters

Explicit formula for the degree as a function of parameters

Dense set of foliations with known first integral degree

Abstract

Let $\mathcal{P}_4$ be the linear family of foliations of degree 4 in $\mathbb{P}^2$ given by A. Lins Neto, whose set of parameters with first integral $I_p(\mathcal{P}_4)$ is dense and countable. In this work, we will calculate explicitly the degree of the rational first integral of the foliations in this linear family, as a function of the parameter.