Algebraic integrability of planar polynomial vector fields by extension to Hirzebruch surfaces

We study algebraic integrability of complex planar polynomial vector fields X=A(x,y)(∂/∂x)+B(x,y)(∂/∂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 R2≥0 that contains all the pairs (i, j) corresponding to monomials xiyj involved in the generic invariant curve of X