Algebraic Integrability of Planar Polynomial Vector Fields by Extension to Hirzebruch Surfaces

Abstract

We study algebraic integrability of complex planar polynomial vector fields \(X=A (x,y)(\partial /\partial x) + B(x,y) (\partial /\partial y) \) through extensions to Hirzebruch surfaces. Using these extensions, each vector field X determines two infinite families of planar vector fields that depend on a natural parameter which, when X has a rational first integral, satisfy strong properties about the dicriticity of the points at the line \(x=0\) and of the origin. As a consequence, we obtain new necessary conditions for algebraic integrability of planar vector fields and, if X has a rational first integral, we provide a region in \({\mathbb {R}}_{\ge 0}^2\) that contains all the pairs (i, j) corresponding to monomials \(x^i y^j\) involved in the generic invariant curve of X.