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 \({{\mathbb {C}}}^4\). 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.

中文翻译：

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切片正则函数的全纯性

这项工作的目的是展示如何从全纯函数的相似性质完美地遵循关于切片正则函数的结果。为此，我们提供了一种通用策略，通过该策略，可以在切片正则函数的设置中转换全纯函数的属性。作为我们方法应用的一个例子，我们研究了切片正则函数的零与相应词干函数的值之间的关系，表明当且仅当相应词干函数在\（{{\ mathbb {C}}} ^ 4 \）的给定复杂分析子集。这使我们能够在这种设置下恢复全纯函数零点的许多属性。我们还将讨论如何在其他情况下调整我们的策略，例如研究零的分布，亚纯函数和表示公式。