Differential topological aspects in octonionic monogenic function theory

TL;DR

This paper extends the theory of octonionic monogenic functions by introducing topological concepts like zero multiplicity and zero varieties, and proves generalized theorems including an argument principle and Hurwitz theorem.

Contribution

It introduces a novel topological framework for octonionic monogenic functions, including zero varieties and a generalized argument principle, advancing the understanding of their zero sets.

Findings

Defined multiplicity of zeros and a-points using topological degree.

Proved an argument principle for isolated and non-isolated zero sets.

Established a generalized Hurwitz theorem for octonionic functions.

Abstract

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 Rouch\'e'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.