The images of multilinear and semihomogeneous polynomials on the algebra of octonions
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866916090243710976 |
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| author | Kanel-Belov, Alexei Malev, Sergey Pines, Coby Rowen, Louis |
| author_facet | Kanel-Belov, Alexei Malev, Sergey Pines, Coby Rowen, Louis |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_07139 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The images of multilinear and semihomogeneous polynomials on the algebra of octonions Kanel-Belov, Alexei Malev, Sergey Pines, Coby Rowen, Louis Algebraic Geometry Representation Theory 17D05 17D10 14R10 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. |
| title | The images of multilinear and semihomogeneous polynomials on the algebra of octonions |
| topic | Algebraic Geometry Representation Theory 17D05 17D10 14R10 |
| url | https://arxiv.org/abs/2204.07139 |