The Poincar\'e Problem, algebraic integrability and dicritical divisors

TL;DR

This paper addresses the Poincaré problem for plane foliations with a focus on cases with one dicritical divisor, providing algorithms to determine and compute rational first integrals, and establishing bounds based on geometric properties.

Contribution

It solves the Poincaré problem for foliations with one dicritical divisor and introduces algorithms for deciding and computing rational first integrals in various cases.

Findings

Algorithm for foliations with one dicritical divisor to decide and compute rational first integrals.

Bound on the degree of rational first integrals when there are two dicritical divisors.

Explicit bounds depending on genus, degree, and local singularity types for certain foliations.

Abstract

We solve the Poincar\'e problem for plane foliations with only one dicritical divisor. Moreover, in this case, we give an algorithm that decides whether a foliation has a rational first integral and computes it in the affirmative case. We also provide an algorithm to compute a rational first integral of prefixed genus $g\neq 1$ of any type of plane foliation $\cf$. When the number of dicritical divisors dic$(\cf)$ is larger than two, this algorithm depends on suitable families of invariant curves. When dic$(\cf) = 2$, it proves that the degree of the rational first integral can be bounded only in terms of $g$, the degree of $\cf$ and the local analytic type of the dicritical singularities of $\cf$.