On the geometry of zero sets of central quaternionic polynomials II

TL;DR

This paper investigates the geometric structure of zero sets of central quaternionic polynomials, revealing they form products of spheres, and provides a new proof for a related conjecture, while also discussing limitations in general division algebras.

Contribution

It characterizes the algebraic hulls of points in quaternionic polynomial zero sets as products of spheres and proves a conjecture, extending previous work.

Findings

Zero sets form products of spheres

Provided a new proof for a conjecture of Gori, Sarfatti, and Vlacci

Main results do not extend to general division algebras

Abstract

Following the work of the first and last authors [2], we further analyze the structure of a zero set of a left ideal in the ring of central polynomials over the quaternion algebra H. We describe the "algebraic hull" of a point in H^n and prove it is a product of spheres. Using this description we give a new proof to a conjecture of Gori, Sarfatti and Vlacci. We also show that the main result of [2] does not extend to general division algebras.