In this work a theorical framework to apply the Poincaré compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the n-dimensional sphere, though the compactified vector field can be identically null in the equator. Moreover, for a fixed projection to the hemisphere, all the compactifications of a vector field, which are not identically null on the equator are equivalent. Also, the conditions determining the invariance of the equator for the compactified vector field are obtained. Up to the knowledge of the authors, this is the first time that the Poincaré compactification of locally Lipschitz continuous vector fields is studied. These results are illustrated applying them to some families of vector fields, like polynomial vector fields, vector fields defined as a sum of homogeneous functions and vector fields defined by piecewise linear functions.

中文翻译：

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非多项式矢量场的Poincaré压缩

在这项工作中，建立了将庞加莱压实技术应用于局部Lipschitz连续矢量场的理论框架。证明了这些向量场在n中是可压缩的维球体，尽管压缩后的矢量场在赤道中可以完全为零。此外，对于向半球的固定投影，矢量场的所有压缩（在赤道上并非相同为零）都是等效的。另外，获得确定紧缩矢量场的赤道不变性的条件。根据作者的知识，这是首次研究局部Lipschitz连续矢量场的Poincaré压缩。说明了将这些结果应用于某些矢量场族的情况，例如多项式矢量场，定义为齐次函数之和的矢量场和由分段线性函数定义的矢量场。