Classifying Slice-Regular Polynomials via Group Actions on the Twistor Space

TL;DR

This paper investigates the classification of slice-regular polynomials over quaternions using group actions on the twistor space, providing a geometric perspective on their equivalence classes.

Contribution

It introduces a novel approach employing twistor geometry to classify slice-regular functions and polynomials under quaternionic projective linear group actions.

Findings

Characterization of slice-regular functions with planar twistor lifts.

Identification of normal classes of slice-regular polynomials under subgroup actions.

Application of twistor construction to classify polynomial equivalence classes.

Abstract

We study the equivalence classes of slice-regular functions $f:\Omega\to\mathbb{H}$ on a symmetric slice domain $\Omega$, and of their subclass made of polynomial slice-regular functions, with respect to the natural action of $\mathrm{PGL}(2,\mathbb{H})$ and its subgroups, by employing the twistor construction. In particular, we characterize slice--regular functions whose twistor lift is planar and belongs to a given orbit, and we find normal classes of slice-regular polynomials with respect to the action of a parabolic subgroup of $\mathrm{GL}(2,\mathbb{H})$.