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

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.