On fiber bundles and quaternionic slice regular functions

TL;DR

This paper extends the application of fiber bundle theory to quaternionic slice regular functions, simplifying previous computations and providing new insights into the zero sets of these functions.

Contribution

It introduces a simplified fiber bundle framework for quaternionic slice regular functions and offers new interpretations of their zero sets.

Findings

Elements of total space are defined from harmonic functions and orthogonal vectors.

Simplified computational framework compared to previous work.

Provides fiber bundle interpretations of zero sets of quaternionic polynomials.

Abstract

The papers \cite{O1,O2} are the first works to apply the theory of fiber bundles in the study of the quaternionic slice regular functions. The main goal of the present work is to extend the results given in \cite{O1}, where the quaternionic right linear space of quaternionic slice regular functions was presented as the base space of a fiber bundle. When the quaternionic right linear space of quaternionic slice regular functions is associated to certain domains then this paper shows that the elements of total space, given in \cite{O1}, are defined from a pair of harmonic functions and a pair of orthogonal vectors. Simplifying the computations presented in \cite{O1}, where each element of the total space is formed by two pair of conjugate harmonic functions and a pair of orthogonal unit vectors. This work also gives some interpretations of the behavior of the zero sets of some quaternionic slice regular polynomials in terms of the theory of fiber bundles.