Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$. A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$, $i\not =c$, the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$. Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

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关于多线性变体上定义的多线性映射扩展的注释

让$G_1, \ldots , G_k$是素数域上的有限维向量空间$\mathbb {F}_p$. 至多是余维的多重线性变体$d$是的一个子集$G_1 \times \cdots \times G_k$定义为零集$d$形式，每个形式在坐标的某个子集上都是多线性的。一张地图$\phi$定义在多重线性变量上$B$如果对于每个坐标是多线性的$c$和所有的选择$x_i \in G_i$,$i\不 =c$, 限制图$y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$在定义的地方是线性的。在这篇笔记中，我们展示了一个多线性映射至多定义在一个多线性余维变体上$d$与余维的多重线性变体重合$O_{k}(d^{O_{k}(1)})$具有在整体上定义的多线性映射$G_1\times \cdots \times G_k$. 此外，在一般有限域的情况下，我们推导出相似（但稍弱）的结果。