Images of Multilinear Polynomials on Generalized Quaternion Algebras

TL;DR

This paper extends previous work on multilinear polynomials to generalized quaternion algebras, classifying their images as vector spaces even when these algebras are not division rings.

Contribution

It generalizes the classification of polynomial images from division quaternion algebras to broader classes of generalized quaternion algebras.

Findings

Images of multilinear polynomials form vector spaces in generalized quaternion algebras

Provides a classification of possible polynomial images in these algebras

Extends previous results to non-division quaternion algebras

Abstract

The main goal of this paper is to extend [J. Algebra Appl. 20 (2021), 2150074] to generalized quaternion algebras, even when these algebras are not necessarily division rings. More precisely, in such cases, the image of a multilinear polynomial evaluated on a quaternion algebra is a vector space and we additionally provide a classification of possible images.