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

• Carlos Galindo ^[1] ; Francisco Monserrat ^[2] ; Elvira Pérez-Callejo ^[1]
  1. [1] Universitat Jaume I

     Universitat Jaume I

     Castellón, España

     
  2. [2] Universidad Politécnica de Valencia

     Universidad Politécnica de Valencia

     Valencia, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
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• Resumen

  • 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 xi y j involved in the generic invariant curve of X.