The Poincaré problem, algebraic integrability and dicritical divisors

Abstract

We solve the Poincaré problem for plane foliations with only one dicritical divisor. Moreover, in this case, we give a simple 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≠1g≠1 of any type of plane foliation FF. When the number of dicritical divisors dic(F)dic(F) is larger than 2, this algorithm depends on suitable families of invariant curves. When dic(F)=2dic(F)=2, it proves that the degree of the rational first integral can be bounded only in terms of g , the degree of FF and the local analytic type of the dicritical singularities of FF.