  Let $G_1, \dots, G_k$ be finite-dimensional vector spaces over a finite field
$\mathbb{F}$. A multilinear variety of codimension $d$ is a subset of $G_1
\times \dots \times G_k$ 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 $d$ and all
choices of $x_i \in G_i$, $i\not=d$, the restriction map $y \mapsto \phi(x_1,
\dots, x_{d-1}, y, x_{d+1}, \dots, x_k)$ is linear where defined. In this note,
we show that a multilinear map defined on a multilinear variety of codimension
$d$ coincides on a multilinear variety of codimension $d^{O(1)}$ with a
multilinear map defined on the whole of $G_1\times\dots\times G_k$.
