Division algebras of slice functions

TL;DR

This paper explores the properties of slice functions over finite-dimensional division algebras, revealing new phenomena in their zero sets and inverses, and applying these findings to classical function theory results.

Contribution

It extends the theory of slice regular functions from quaternions to general division algebras, establishing new properties and classical theorems in this broader context.

Findings

Zero sets of slice functions exhibit unexpected behaviors.

Derived a minimum modulus principle from the maximum modulus principle.

Proved a Casorati-Weierstrass type theorem for slice regular functions.

Abstract

This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.