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 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.
Highlights
During the last 3 years one notices a significant further boost of interest in octonionic analysis both from mathematicians and from theoretical physicists, see for instance [17,18,19,21,28]
Many physicists currently believe that the octonions provide the adequate setting to describe the symmetries arising in a possible unified world theory combining the standard model of particle physics and aspects of supergravity
The octonions form an eight-dimensional real non-associative normed division algebra over the real numbers. They serve as a comfortable number system to describe the symmetries in recent unifying physical models connecting the standard model of particle physics and supergravity, see [4,11]
Summary
During the last 3 years one notices a significant further boost of interest in octonionic analysis both from mathematicians and from theoretical physicists, see for instance [17,18,19,21,28]. Many physicists currently believe that the octonions provide the adequate setting to describe the symmetries arising in a possible unified world theory combining the standard model of particle physics and aspects of supergravity. See [22] for the references therein. Already during the 1970s, but in the first decade of this century, a lot of effort has been made to carry over fundamental tools from Clifford analysis to the non-associative octonionic setting. This article is part of the Topical Collection on ISAAC 12 at Aveiro, July 29–August 2, 2019, edited by Swanhild Bernstein, Uwe Kaehler, Irene Sabadini, and Franciscus Sommen. ∗Corresponding author
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