TY - JOUR

T1 - The images of multilinear and semihomogeneous polynomials on the algebra of octonions

AU - Kanel-Belov, Alexei

AU - Malev, Sergey

AU - Pines, Coby

AU - Rowen, Louis

N1 - Publisher Copyright:
© 2022 Informa UK Limited, trading as Taylor & Francis Group.

PY - 2024

Y1 - 2024

N2 - 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 (Formula presented.) over a field F satisfying certain specified conditions (in particular, we prove it for quadratically closed fields, and for the field (Formula presented.)). In fact, letting V be the space of pure octonions in (Formula presented.), we prove that the image set must be either (Formula presented.), F, V or (Formula presented.). We discuss possible evaluations of semihomogeneous polynomials on (Formula presented.) and of arbitrary polynomials on the corresponding Malcev algebra.

AB - 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 (Formula presented.) over a field F satisfying certain specified conditions (in particular, we prove it for quadratically closed fields, and for the field (Formula presented.)). In fact, letting V be the space of pure octonions in (Formula presented.), we prove that the image set must be either (Formula presented.), F, V or (Formula presented.). We discuss possible evaluations of semihomogeneous polynomials on (Formula presented.) and of arbitrary polynomials on the corresponding Malcev algebra.

UR - http://www.scopus.com/inward/record.url?scp=85140420859&partnerID=8YFLogxK

U2 - 10.1080/03081087.2022.2158170

DO - 10.1080/03081087.2022.2158170

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AN - SCOPUS:85140420859

SN - 0308-1087

VL - 72

SP - 178

EP - 187

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

IS - 2

ER -