Quaternionic analogues of Bers's theorem and Iss'sa's theorem

TL;DR

This paper extends classical complex analysis theorems, Bers's and Iss'sa's, into the quaternionic setting by analyzing algebraic structures of slice preserving semiregular functions.

Contribution

It provides the first quaternionic analogues of Bers's and Iss'sa's theorems, connecting algebraic properties of semiregular functions in quaternionic analysis.

Findings

Quaternionic Bers's theorem established

Quaternionic Iss'sa's theorem proved

Algebraic structures of slice preserving functions characterized

Abstract

In their recent work, Gentili and Struppa proposed a different quaternionic analogue of the notion of holomorphic functions in the complex plane, called \textit{slice regular functions}, which has led to several analogues of classical theorems in complex function theory in the quaternionic setting. The quaternionic analogue of meromorphic functions is called \textit{semiregular functions}. Such a function is said to be \textit{slice preserving} if it maps each domain in a complex line to a domain in the same complex line. In this work, we deal with the relation between analytic and algebraic properties of certain fields of semiregular functions in symmetric slice domains of the real quaternions. More precisely, we prove quaternionic analogues of Bers's theorem and Iss'sa's theorem that deal with the algebraic structures of the ring of slice preserving regular functions and of certain fields of slice preserving semiregular functions on symmetric slice domains in the real quaternions.