The zero sets of slice regular functions and the open mapping theorem

Abstract

A new class of regular quaternionic functions, defined by power series in a natural fashion, has been introduced in recent years. Several results of the theory recall the classical complex analysis, whereas other results reflect the peculiarity of the quaternionic structure. A more recent paper identified a larger class of domains, on which the study of regular functions is most natural and not limited to the study of quaternionic power series. In the present paper we extend some basic results concerning the algebraic and topological properties of the zero set to regular functions defined on these domains. We then use these results to prove the Maximum and Minimum Modulus Principles and a version of the Open Mapping Theorem in this new setting.

13 pages

References in corpus (2)

• Power series and analyticity over the quaternions
• Poles of regular quaternionic functions

Cited by in corpus (13)

• Twistor transforms of quaternionic functions and orthogonal complex structures
• A new series expansion for slice regular functions
• Regular Moebius transformations of the space of quaternions
• Some properties for quaternionic slice-regular functions on domains without real points
• Division algebras of slice functions
• Geometric function theory over quaternionic slice domains
• The Bohr Theorem for slice regular functions
• On the real differential of a slice regular function
• A Phragmén - Lindelöf principle for slice regular functions
• Slice starlike functions over quaternions
• A unified theory of regular functions of a hypercomplex variable
• Slice regular composition operators
• Quaternionic analogues of Bers's theorem and Iss'sa's theorem