Algebraic integrability of planar polynomial vector fields by extension to Hirzebruch surfaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866910427633418240 |
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| author | Galindo, Carlos Monserrat, Francisco Pérez-Callejo, Elvira |
| author_facet | Galindo, Carlos Monserrat, Francisco Pérez-Callejo, Elvira |
| contents | 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 $X$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2202_06134 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Algebraic integrability of planar polynomial vector fields by extension to Hirzebruch surfaces Galindo, Carlos Monserrat, Francisco Pérez-Callejo, Elvira Algebraic Geometry Dynamical Systems 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 $X$. |
| title | Algebraic integrability of planar polynomial vector fields by extension to Hirzebruch surfaces |
| topic | Algebraic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2202.06134 |