The algebra of slice functions

TL;DR

This paper explores the algebraic structure of slice and slice regular functions over alternative $^*$-algebras, extending complex function theory to higher dimensions and analyzing their zero sets and invertibility properties.

Contribution

It provides a detailed analysis of the algebraic properties of slice functions over alternative $^*$-algebras, including their invertibility and zero set structures.

Findings

Slice functions form an alternative $^*$-algebra under suitable operations.

The paper characterizes the zero sets of slice and slice regular functions.

Conditions for the existence of multiplicative inverses are established.

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 with 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.