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Algebraic Integrability of Foliations of the Plane

C. Galindo·F. Monserrat
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Abstract

We give an algorithm to decide whether an algebraic plane foliation F has a rational first integral and to compute it in the affirmative case. The algorithm runs whenever we assume the polyhedrality of the cone of curves of the surface obtained after blowing-up the set B_F of infinitely near points needed to get the dicritical exceptional divisors of a minimal resolution of the singularities of F. This condition can be detected in several ways, one of them from the proximity relations in B_F and, as a particular case, it holds when the cardinality of B_F is less than 9.

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C. Galindo, F. Monserrat. 2005-12-07. Algebraic Integrability of Foliations of the Plane. https://arxiv.org/abs/math/0511416

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