The algebra of slice functions

Ghiloni, Riccardo; Perotti, Alessandro; Stoppato, Caterina

doi:10.1090/tran/6816

The algebra of slice functions
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by Riccardo Ghiloni, Alessandro Perotti and Caterina Stoppato PDF

Trans. Amer. Math. Soc. 369 (2017), 4725-4762 Request permission

Corrigendum: Trans. Amer. Math. Soc. (to appear).

Abstract:

In this paper we study some fundamental algebraic properties of slice functions and slice regular functions over an alternative $^*$-algebra $A$ over $\mathbb {R}$. These recently introduced function theories generalize to higher dimensions the classical theory of functions of a complex variable. Slice functions over $A$, which comprise all polynomials over $A$, form an alternative $^*$-algebra themselves when endowed with appropriate operations. We presently study this algebraic structure in detail and we confront questions about the existence of multiplicative inverses. This study leads us to a detailed investigation of the zero sets of slice functions and of slice regular functions, which are of course of independent interest.

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