Holomorphicity of Slice-Regular Functions

The aim of this work is to show how a number of results about slice-regular functions follow flawlessly from the analogous properties of holomorphic functions. For this purpose, we provide a general strategy by which properties of holomorphic functions can be translated in the setting of slice-regular functions. As an example of application of our method, we study the relation between the zeroes of a slice-regular function and the values of the corresponding stem function, showing that a slice-regular function vanishes if and only if the corresponding stem function takes values in a given complex analytic subset of C4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {C}}}^4$$\end{document}. This allows us to recover in this setting a number of properties of the zeroes of holomorphic functions. We also discuss how our strategy can be adapted in some other contexts, like the study of the distribution of zeroes, of meromorphic functions, of representation formulas.