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

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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.

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Authors and Affiliations

  1. Universitat Jaume I, Campus de Riu Sec, Departamento de Matemáticas & Institut Universitari de Matemàtiques i Aplicacions de Castelló, 12071, Castellón de la Plana, Spain

    Carlos Galindo & Elvira Pérez-Callejo

  2. Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia, Spain

    Francisco Monserrat

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  1. Carlos Galindo
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  2. Francisco Monserrat
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Correspondence to Francisco Monserrat.

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Partially supported by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, grant PGC2018-096446-B-C22, by MCIN/AEI/10.13039/501100011033 and by “ESF Investing in your future”, grant PRE2019-089907, as well as by Universitat Jaume I, grant UJI-B2021-02. No datasets were generated or analyzed.

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Galindo, C., Monserrat, F. & Pérez-Callejo, E. Algebraic Integrability of Planar Polynomial Vector Fields by Extension to Hirzebruch Surfaces. Qual. Theory Dyn. Syst. 21, 126 (2022). https://doi.org/10.1007/s12346-022-00655-1

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