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Division algebras of slice functions. (English) Zbl 1455.30040

Thanks to a result by Zorn, the only finite-dimensional alternative division algebras are the reals \(\mathbb{R}\), complex numbers \(\mathbb{C}\), quaternions \(\mathbb{H}\) and octonions \(\mathbb{O}\). The present paper aims to give a unified approach to the theory of slice regular functions over \(\mathbb{C},\mathbb{H}\) and \(\mathbb{O}\). As stated by the authors, some of the results were already obtained in other works and with other methods, while another set of results is completely new. In particular, after the introduction, the paper has 5 sections:
– Section 2 contains some background material on division algebras and on slice functions;
– Section 3 deals with an analysis of the zeroes of slice functions;
– Section 4 is about reciprocals: this section contains some genuine new result such as a representation formula for the reciprocal and some topological phenomena (several examples are given);
– In Section 5 the authors start to deal with global properties of slice regular functions and prove a maximum module principle together with a minimum one and an open mapping theorem;
– The last section studies singularities of slice regular functions and the authors prove a Casorati-Weierstrass type theorem and that the set of slice semi-regular functions (the analogue of meromorphic functions) form a division algebra.
All proofs are given in the case of octonions but they stay valid for \(\mathbb{C}\) or \(\mathbb{H}\).

MSC:

30G35 Functions of hypercomplex variables and generalized variables
17A35 Nonassociative division algebras
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30D30 Meromorphic functions of one complex variable (general theory)
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References:

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