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

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.

Theorems & Definitions (17)

• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 3.1: Sp
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4