Differential Topological Aspects in Octonionic Monogenic Function Theory
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Advances in Applied Clifford Algebras
Differential Topological Aspects in Octonionic Monogenic Function Theory Rolf S¨oren Kraußhar∗ 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. Mathematics Subject Classification. 30G35. Keywords. Octonions, Winding numbers, Argument principle, Rouch´e’s theorem, Hurwitz theorem, Isolated and non-isolated zeroes.
1. Introduction Especially 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 [17–19,21,28]. In fact, 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 also [22] for the references therein. Already during the 1970s, but particularly 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
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Adv. Appl. Clifford Algebras
Many analogues of important theorems from Clifford analysis could also be established in the non-associative setting, such as for instance a Cauchy integral formula or Taylor and Laurent series representations involving direct analogues of the Fueter polynomials, see for example [16,24–27,29]. Of course, one carefully has to put parenthesis in order to take care of the non-associative nature. Although some of these fundamental theorems formally look very similar to those in the associative Clifford algebra setting(cf.[3]). Clifford analysis and octononic analysis are two different function theories. In [18,19] the authors describe a number of substantial and structural differences between the set of Clifford monogenic functions from R8 → Cl8 ∼ = R128 and the set of octonionic monogenic functions from O → O. This is not only reflected in the different mapping property, but also in the fact that unlike in the Clifford case, left octonionic monogeni
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