The images of multilinear and semihomogeneous polynomials on the algebra of octonions

TL;DR

This paper proves the generalized L'vov-Kaplansky conjecture for the algebra of octonions, showing that the image of multilinear polynomials is always a specific vector space, and discusses evaluations of semihomogeneous polynomials.

Contribution

It establishes the conjecture for octonions over certain fields, identifying possible polynomial image sets and analyzing evaluations on octonions and related Malcev algebra.

Findings

Image of multilinear polynomials on octonions is {0}, F, pure octonions, or all octonions.

Proves the conjecture for quadratically closed fields and the real numbers.

Discusses evaluations of semihomogeneous polynomials and Malcev algebra evaluations.

Abstract

The generalized L'vov-Kaplansky conjecture states that for any finite-dimensional simple algebra $A$ the image of a multilinear polynomial on $A$ is a vector space. In this paper we prove it for the algebra of octonions $\mathbb{O}$ over a field satisfying certain specified conditions (in particular, we prove it for quadratically closed field and for field $\mathbb{R}$). In fact, we prove that the image set must be either $\{0\}$, $F$, the space of pure octonions $V$, or $\mathbb{O}$. We discuss possible evaluations of semihomogeneous polynomials on $\mathbb{O}$ and of arbitrary polynomials on the corresponding Malcev algebra.