On the characterization of algebraically integrable plane foliations

TL;DR

This paper characterizes non-degenerated plane foliations with rational first integrals, linking their degree to the minimal number of points with infinitely many algebraic leaves, advancing understanding of foliation structure.

Contribution

It provides a new characterization theorem for such foliations and establishes a relation between their degree and algebraic leaves.

Findings

Degree r foliation has at least r+1 points with infinitely many algebraic leaves

Characterization theorem for non-degenerated plane foliations with rational first integrals

Degree r is minimal for the number of such points

Abstract

We give a characterization theorem for non-degenerated plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree $r$ of a non-degenerated foliation as above provides the minimum number, $r+1$, of points in the projective plane through which infinitely many algebraic leaves of the foliation go.