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The algebra of slice functions


Authors: Riccardo Ghiloni, Alessandro Perotti and Caterina Stoppato
Journal: Trans. Amer. Math. Soc. 369 (2017), 4725-4762
MSC (2010): Primary 30G35; Secondary 17D05, 32A30, 30C15
DOI: https://doi.org/10.1090/tran/6816
Published electronically: November 28, 2016
MathSciNet review: 3632548
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Abstract: In this paper we study some fundamental algebraic properties of slice functions and slice regular functions over an alternative $ ^*$-algebra $ A$ over $ \mathbb{R}$. These recently introduced function theories generalize to higher dimensions the classical theory of functions of a complex variable. Slice functions over $ A$, which comprise all polynomials over $ A$, form an alternative $ ^*$-algebra themselves when endowed with appropriate operations. We presently study this algebraic structure in detail and we confront questions about the existence of multiplicative inverses. This study leads us to a detailed investigation of the zero sets of slice functions and of slice regular functions, which are of course of independent interest.


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Additional Information

Riccardo Ghiloni
Affiliation: Dipartimento di Matematica, Università di Trento, Via Sommarive 14, I-38123 Povo Trento, Italy
Email: riccardo.ghiloni@unitn.it

Alessandro Perotti
Affiliation: Dipartimento di Matematica, Università di Trento, Via Sommarive 14, I-38123 Povo Trento, Italy
Email: alessandro.perotti@unitn.it

Caterina Stoppato
Affiliation: Istituto Nazionale di Alta Matematica, Unità di Ricerca di Firenze c/o DiMaI “U. Dini” Università di Firenze, Viale Morgagni 67/A, I-50134 Firenze, Italy
Email: stoppato@math.unifi.it

DOI: https://doi.org/10.1090/tran/6816
Received by editor(s): February 27, 2015
Received by editor(s) in revised form: July 13, 2015
Published electronically: November 28, 2016
Additional Notes: This work was supported by GNSAGA of INdAM and by the grants FIRB “Differential Geometry and Geometric Function Theory” (RBFR12W1AQ) and PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica” (2010NNBZ78) of the Italian Ministry of Education. We warmly thank the anonymous referee, whose helpful suggestions have significantly improved the presentation
Article copyright: © Copyright 2016 American Mathematical Society