In this paper, we introduce geometric tools to study the families of rational vector fields of a given degree over \(\mathbb {C}\mathbb {P}^{1}\). To a generic vector field of such a parametric family, we associate several geometric objects: a periodgon, a star domain, and a translation surface. These objects generalize objects with the same name introduced in previous works on polynomial vector fields. They are used to describe the bifurcations inside the families. We specialize to the case of rational vector fields of degree 4.

中文翻译：

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关于ℂ1 $ \ mathbb {C} \ mathbb {P} ^ {1} $上的有理向量场的说明

在本文中，我们介绍了几何工具，以研究\（\ mathbb {C} \ mathbb {P} ^ {1} \）上给定度数的有理矢量场的族。对于这样一个参数族的通用矢量场，我们将几个几何对象相关联：一个时间轴，一个星域和一个平移表面。这些对象概括了在多项式矢量字段的先前工作中引入的具有相同名称的对象。它们用来描述家庭内部的分歧。我们专门研究4度有理向量场的情况。