## 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 (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.

Original language | English |
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Pages (from-to) | 178-187 |

Number of pages | 10 |

Journal | Linear and Multilinear Algebra |

Volume | 72 |

Issue number | 2 |

DOIs | |

State | Published - 2024 |

### Bibliographical note

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

### Funding

This work was supported by Israel Science Foundation[1623/16]. We would like to thank I. Shestakov and E. Plotkin for interesting and fruitful discussions regarding this paper.

Funders | Funder number |
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Israel Science Foundation | 1623/16 |