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

TL;DR

This paper extends fundamental properties of zero sets and the open mapping theorem to a broader class of quaternionic regular functions defined on more general domains, enhancing the understanding of their algebraic and topological behavior.

Contribution

It introduces an extension of classical results to a larger class of domains for quaternionic regular functions, beyond power series.

Findings

Zero set algebraic and topological properties extended

Maximum and Minimum Modulus Principles proved

Open Mapping Theorem established in new setting

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.