Slice regular functions on real alternative algebras

TL;DR

This paper develops a comprehensive theory of slice regular functions on real alternative algebras, unifying and extending existing theories for quaternions, octonions, and Clifford algebras, with new algebraic and integral results.

Contribution

It introduces a unified framework for slice regular functions on real alternative algebras, extending previous theories and establishing new fundamental algebraic and integral formulas.

Findings

A strong fundamental theorem of algebra for polynomials over real alternative algebras.

A Cauchy integral formula for slice functions of class C^1.

Extension of existing function theories to broader algebraic contexts.

Abstract

In this paper we develop a theory of slice regular functions on a real alternative algebra $A$. Our approach is based on a well--known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in $A$ and we prove a Cauchy integral formula for slice functions of class $C^1$.