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 $X$.

Cited by in corpus (1)

• Algebraic integrability with bounded genus