Differential topological aspects in octonionic monogenic function theory (Q2194022)

Let \(\mathbb{O}\) denote the real algebra of octonions, or Cayley numbers, \[x=x_0+\sum_{t=1}^7x_te_t,\] with \(x_t \in \mathbb{R}\) for all \(t \in \{0,\ldots,7\}\). The article studies {left (or right) octonionic monogenic} functions: namely, solutions of \(\mathcal{D}f=0\) (or \(f\mathcal{D}=0\)), where \[\mathcal{D}:=\frac{\partial}{\partial x_0}+\sum_{t=1}^7e_t\frac{\partial}{\partial x_t}\] is the \textit{octonionic Cauchy-Riemann operator}. This notion, attributed to [\textit{K. Imaeda} and \textit{M. Imaeda}, Appl. Math. Comput. 115, No. 2--3, 77--88 (2000; Zbl 1032.17003); \textit{X. Li} and \textit{L. Peng}, Approximation Theory Appl. 16, No. 2, 28--36 (2000; Zbl 0976.31008)], is distinct from the notion of octonionic regularity of \textit{G. Gentili} and \textit{D. C. Struppa} [Rocky Mt. J. Math. 40, No. 1, 225--241 (2010; Zbl 1193.30070)]. The author recalls several properties of octonionic monogenic functions known in literature and provides a proof of their Identity Principle. For each octonionic monogenic function \(f\) and each value \(a\) of \(f\), he defines and studies the {order} of \(f\) at an isolated point of the preimage \(f^{-1}(a)\). For the case when all points considered are isolated, he proves an Argument Principle and a version of Rouché's Theorem (whence a version of Hurwitz's Theorem). The final part of the article sets the framework to address the case of non-isolated points of the preimage \(f^{-1}(0)\) and generalizes the octonionic Argument Principle within this framework. The study of other properties within the same framework is left as an open problem.