Gorde:
| Egile Nagusiak: | , , |
|---|---|
| Formatua: | Preprint |
| Argitaratua: |
2022
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| Gaiak: | |
| Sarrera elektronikoa: | https://arxiv.org/abs/2202.06134 |
| Etiketak: |
Etiketa erantsi
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Aurkibidea:
- 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 $X$.