In this paper, we study the probability of occurrence of phase portraits in the set of random planar homogeneous polynomial vector fields, of degree n. In particular, for \(n=1,2,3,\) we give the complete solution of the problem; that is, we either give the exact value of each probability of occurrence or we estimate it by using the Monte Carlo method. Remarkably is that all but two of these phase portraits are characterized by the index at the origin and by the number of invariant straight lines through this point.

中文翻译：

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平面随机齐次矢量场的相图

在本文中，我们研究了度为n的一组随机平面齐次多项式矢量场中相像出现的可能性。特别是，对于\（n = 1,2,3，\），我们给出了问题的完整解决方案；也就是说，我们要么给出每种发生概率的确切值，要么使用蒙特卡洛方法进行估算。值得注意的是，除两个相图外，所有相图都以原点处的索引和通过该点的不变直线的数量为特征。