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)(∂/∂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 R 2 ≥0 that contains all the pairs (i, j) corresponding to monomials x i y j involved in the generic invariant curve of X.