Slice regular functions as covering maps and global $\star$-roots

TL;DR

This paper demonstrates that many quaternionic slice regular functions act as ramified covering maps, enabling the determination of $k$-th $igstar$-roots and their solutions, with a detailed analysis of monodromy and root computation.

Contribution

It establishes conditions under which slice regular functions are covering maps and provides a method to compute all $k^2$ solutions for their roots.

Findings

Many slice regular functions are ramified covering maps.

Explicit computation of all $k^2$ roots is possible.

A technique for root computation from a single root is developed.

Abstract

The aim of this paper is to prove that a large class of quaternionic slice regular functions result to be (ramified) covering maps. By means of the topological implications of this fact and by providing further topological structures, we are able to give suitable natural conditions for the existence of $k$-th $\star$-roots of a slice regular function. Moreover, we are also able to compute all the solutions which, quite surprisingly, in the most general case, are in number of $k^2$. The last part is devoted to compute the monodromy and to present a technique to compute all the $k^2$ roots starting from one of them.