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Division algebras of slice functions

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

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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References

1Altavilla, A.. Some properties for quaternionic slice regular functions on domains without real points. Complex Var. Elliptic Equ. 60 (2015), 59–77.CrossRefGoogle Scholar
2Baez, J. C.. The octonions. Bull. Amer. Math. Soc. (N.S.) 39 (2002), 145–205.CrossRefGoogle Scholar
3Colombo, F., Gentili, G., Sabadini, I. and Struppa, D.. Extension results for slice regular functions of a quaternionic variable. Adv. Math. 222 (2009), 1793–1808.CrossRefGoogle Scholar
4Colombo, F., Gentili, G. and Sabadini, I.. A Cauchy kernel for slice regular functions. Ann. Global Anal. Geom. 37 (2010), 361–378.CrossRefGoogle Scholar
5Ebbinghaus, H.-D., Hermes, H., Hirzebruch, F., Koecher, M., Mainzer, K., Neukirch, J., Prestel, A. and Remmert, R.. Numbers, volume 123 of Graduate Texts in Mathematics (New York: Springer-Verlag, 1990). With an introduction by K. Lamotke, Translated from the second German edition by H. L. S. Orde, Translation edited and with a preface by J. H. Ewing, Readings in Mathematics.Google Scholar
6Gentili, G. and Struppa, D. C.. A new approach to Cullen-regular functions of a quaternionic variable. C. R. Math. Acad. Sci. Paris 342 (2006), 741–744.CrossRefGoogle Scholar
7Gentili, G. and Struppa, D. C.. A new theory of regular functions of a quaternionic variable. Adv. Math. 216 (2007), 279–301.CrossRefGoogle Scholar
8Gentili, G. and Stoppato, C.. Zeros of regular functions and polynomials of a quaternionic variable. Michigan Math. J. 56 (2008), 655–667.CrossRefGoogle Scholar
9Gentili, G. and Stoppato, C.. The open mapping theorem for regular quaternionic functions. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) VIII (2009), 805–815.Google Scholar
10Gentili, G. and Struppa, D. C.. Regular functions on the space of Cayley numbers. Rocky Mountain J. Math. 40 (2010), 225–241.CrossRefGoogle Scholar
11Gentili, G. and Stoppato, C.. The zero sets of slice regular functions and the open mapping theorem. In Hypercomplex analysis and applications (eds. Sabadini, I. and Sommen, F.). Trends in Mathematics, pp. 95–107 (Basel: Birkhäuser Verlag, 2011).CrossRefGoogle Scholar
12Gentili, G., Stoppato, C., Struppa, D. C. and Vlacci, F.. Recent developments for regular functions of a hypercomplex variable. In Hypercomplex analysis (ed. Sabadini, I., Shapiro, M. V. and Sommen, F.). Trends Math., pp. 165–185 (Basel, Birkhäuser Verlag, 2009).Google Scholar
13Gentili, G., Stoppato, C. and Struppa, D. C.. Regular functions of a quaternionic variable (Heidelberg: Springer Monographs in Mathematics, Springer, 2013).CrossRefGoogle Scholar
14Ghiloni, R. and Perotti, A.. On a class of orientation-preserving maps of ℝ4, arxiv.org/abs/1902.11227 (Submitted).Google Scholar
15Ghiloni, R. and Perotti, A.. Slice regular functions on real alternative algebras. Adv. Math. 226 (2011), 1662–1691.CrossRefGoogle Scholar
16Ghiloni, R. and Perotti, A.. Zeros of regular functions of quaternionic and ctonionic variable: a division lemma and the camshaft effect. Ann. Mat. Pura Appl. (4) 190 (2011), 539–551.CrossRefGoogle Scholar
17Ghiloni, R., Moretti, V. and Perotti, A.. Continuous slice functional calculus in quaternionic Hilbert spaces. Rev. Math. Phys. 25 (2013), 1350006, 83.CrossRefGoogle Scholar
18Ghiloni, R., Perotti, A. and Stoppato, C.. Singularities of slice regular functions over real alternative *-algebras. Adv. Math. 305 (2017), 1085–1130.CrossRefGoogle Scholar
19Ghiloni, R., Perotti, A. and Stoppato, C.. The algebra of slice functions. Trans. Amer. Math. Soc. 369 (2017), 4725–4762.CrossRefGoogle Scholar
20Klimek, M.. Pluripotential theory, volume 6 of London Mathematical Society Monographs. New Series (New York: The Clarendon Press, Oxford University Press, 1991). Oxford Science Publications.Google Scholar
21Stoppato, C.. Poles of regular quaternionic functions. Complex Var. Elliptic Equ. 54 (2009), 1001–1018.CrossRefGoogle Scholar
22Stoppato, C.. Singularities of slice regular functions. Math. Nachr. 285 (2012), 1274–1293.CrossRefGoogle Scholar
23Wang, X.. On geometric aspects of quaternionic and octonionic slice regular functions. J. Geom. Anal. 27 (2017), 2817–2871.CrossRefGoogle Scholar
24Ward, J. P.. Quaternions and Cayley numbers, volume 403 of Mathematics and its Applications (Dordrecht: Kluwer Academic Publishers Group, 1997). Algebra and applications.CrossRefGoogle Scholar