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Division algebras of slice functions

Part of: Geometric function theory Generalized function theory Entire and meromorphic functions, and related topics General nonassociative rings

Published online by Cambridge University Press:  15 March 2019

Riccardo Ghiloni ,
Alessandro Perotti  and
Caterina Stoppato
Riccardo Ghiloni
Affiliation:
Dipartimento di Matematica, Università di Trento, Via Sommarive 14, I-38123Povo, Italy (riccardo.ghiloni@unitn.it; alessandro.perotti@unitn.it)
Alessandro Perotti
Affiliation:
Dipartimento di Matematica, Università di Trento, Via Sommarive 14, I-38123Povo, Italy (riccardo.ghiloni@unitn.it; alessandro.perotti@unitn.it)
Caterina Stoppato
Affiliation:
Dipartimento di Matematica e Informatica ‘U. Dini’, Università di Firenze, Viale Morgagni 67/A, I-50134Firenze, Italy (stoppato@math.unifi.it)
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Abstract

This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.

Keywords

functions of one hypercomplex variabledivision algebrasgeometric function theorymeromorphic functions

MSC classification

Primary: 30G35: Functions of hypercomplex variables and generalized variables
Secondary: 17A35: Division algebras 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination 30D30: Meromorphic functions, general theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 2055 - 2082
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Copyright
Copyright © Royal Society of Edinburgh 2019

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