论文标题
平面多项式矢量场的代数一致性扩展到Hirzebruch表面
Algebraic integrability of planar polynomial vector fields by extension to Hirzebruch surfaces
论文作者
论文摘要
我们研究复杂平面多项式矢量场的代数集成性$ x = a（x，y）（\ partial/\ partial x） + b（x，y）（\ partial/\ partial y）$通过延伸到赫兹布鲁克表面。使用这些扩展名，每个矢量字段$ x $确定了两个无限的平面向量字段家族，这些家族依赖于自然参数，当$ x $具有合理的第一积分时，就满足了$ x = 0 $的点的分数，并满足了强大的属性。 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 $X$.
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 $X$.
