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A new class of regular quaternionic functions, defined by power series in anatural fashion, has been introduced in recent years. Several results of thetheory recall the classical complex analysis, whereas other results reflect thepeculiarity of the quaternionic structure. A more recent paper identified alarger class of domains, on which the study of regular functions is mostnatural and not limited to the study of quaternionic power series. In thepresent paper we extend some basic results concerning the algebraic andtopological properties of the zero set to regular functions defined on thesedomains. We then use these results to prove the Maximum and Minimum ModulusPrinciples and a version of the Open Mapping Theorem in this new setting.

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