The Poincare problem, algebraic integrability and dicritical divisors

We solve the Poincare 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 1 of any type of plane foliation F. When the number of dicritical divisors dic(F) is larger than 2, this algorithm depends on suitable families of invariant curves. When dic(F) =2, it proves that the degree of the rational first integral can be bounded only in terms of g, the degree of F. and the local analytic type of the dicritical singularities of F. (c) 2014 Elsevier Inc. All rights reserved.