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

TL;DR

This paper explores the algebraic integrability of complex planar polynomial vector fields by extending them to Hirzebruch surfaces, deriving new necessary conditions and characterizing invariant curves when a rational first integral exists.

Contribution

It introduces a novel approach using Hirzebruch surface extensions to analyze integrability and provides new criteria and regions for invariant monomials in rational first integrals.

Findings

New necessary conditions for algebraic integrability.

Identification of a region in parameter space for invariant monomials.

Analysis of dicriticity properties at specific points.

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 $\mathbb{R}_{\geq 0}^2$ that contains all the pairs $(i,j)$ corresponding to monomials $x^i y^j$ involved in the generic invariant curve of $X$.