In Artés et al. (Geometric configurations of singularities of planar polynomial differential systems. A global classification in the quadratic case. Birkhäuser, Basel, 2019) the authors proved that there are 1765 different global geometrical configurations of singularities of quadratic differential systems in the plane. There are other 8 configurations conjectured impossible, all of them related with a single configuration of finite singularities. This classification is completely algebraic and done in terms of invariant polynomials and it is finer than the classification of quadratic systems according to the topological classification of the global configurations of singularities, the goal of this article. The long term project is the classification of phase portraits of all quadratic systems under topological equivalence. A first step in this direction is to obtain the classification of quadratic systems under topological equivalence of local phase portraits around singularities. In this paper we extract the local topological information around all singularities from the 1765 geometric equivalence classes. We prove that there are exactly 208 topologically distinct global topological configurations of singularities for the whole quadratic class. The 8 global geometrical configurations conjectured impossible do not affect this number of 208. From here the next goal would be to obtain a bound for the number of possible different phase portraits, modulo limit cycles.

中文翻译：

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整个二阶微分系统奇点的全局拓扑配置

在Artés等。（平面多项式微分系统的奇异性的几何构型。二次情况下的全局分类.Birkhäuser，巴塞尔，2019年）作者证明了平面上二次微分系统的奇异性存在1765种不同的全局几何构型。还有其他8种可能的构想是不可能的，所有这些都与单个奇点有关。这种分类是完全代数的，并且是根据不变多项式完成的，并且比根据奇异点整体配置的拓扑分类的二次系统的分类要好，这是本文的目标。长期项目是对拓扑等效下所有二次系统的相图进行分类。在该方向上的第一步是获得在奇异点附近局部相画像的拓扑等效下的二次系统分类。在本文中，我们从1765个几何等价类中提取了所有奇点周围的局部拓扑信息。我们证明，对于整个二次类，奇异点的正好有208个拓扑上不同的全局拓扑结构。推测为不可能的8个全局几何配置不会影响208的数量。从这里开始，下一个目标将是为可能的不同相像数量（模极限循环）获得一个边界。在本文中，我们从1765个几何等价类中提取了所有奇点周围的局部拓扑信息。我们证明，对于整个二次类，奇异点的正好有208个拓扑上不同的全局拓扑结构。推测为不可能的8个全局几何配置不会影响208的数量。从这里开始，下一个目标将是为可能的不同相像数量（模极限循环）获得一个边界。在本文中，我们从1765个几何等价类中提取了所有奇点周围的局部拓扑信息。我们证明对于整个二次类，奇异点的正好有208个拓扑上不同的全局拓扑结构。推测为不可能的8个全局几何配置不会影响208的数量。从这里开始，下一个目标将是为可能的不同相像数量（模极限循环）获得一个边界。