In this paper we apply a homologous version of the Cauchy integral formula for octonionic monogenic functions to introduce for this class of functions the notion of multiplicity of zeroes and a-points in the sense of the topological mapping degree. As a big novelty we also address the case of zeroes lying on certain classes of compact zero varieties. This case has not even been studied in the associative Clifford analysis setting so far. We also prove an argument principle for octonionic monogenic functions for isolated zeroes and for non-isolated compact zero sets. In the isolated case we can use this tool to prove a generalized octonionic Rouché’s theorem by a homotopic argument. As an application we set up a generalized version of Hurwitz theorem which is also a novelty even for the Clifford analysis case.

中文翻译：

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正离子单调函数理论中的差分拓扑方面

在本文中，我们对牛顿离子单基因函数应用了柯西积分公式的同源形式，以从拓扑映射度的角度为此类函数引入零和a点多重性的概念。作为一个新奇事物，我们还解决了某些紧凑型零变种上的零的情况。到目前为止，这种情况甚至都没有在关联的Clifford分析环境中进行过研究。我们还证明了独立零和非独立紧零集的八元单基因函数的论证原理。在孤立的情况下，我们可以使用该工具通过一个同位论证来证明广义的重音Rouché定理。作为应用程序，我们建立了Hurwitz定理的广义版本，即使对于Clifford分析案例，这也是一个新颖的事物。