论文标题
多线性和半偶联多项式在八元代数上的图像
The images of multilinear and semihomogeneous polynomials on the algebra of octonions
论文作者
论文摘要
广义的L'Vov-Kaplansky猜想指出，对于任何有限的简单代数$ a $ $ a $ a $ a $ a $ a $的图像是矢量空间。在本文中，我们证明了在满足某些指定条件的字段上的八元代数$ \ mathbb {o} $（特别是，我们证明了这是针对四型封闭的字段和字段$ \ mathbb {r} $）的。 实际上，我们证明该图像集必须为$ \ {0 \} $，$ f $，纯八元$ v $的空间或$ \ mathbb {o} $。我们讨论了对$ \ mathbb {O} $的半偶联多项式的可能评估以及对相应的Malcev代数上的任意多项式的评估。
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.
