Abstract
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.
A generalization of Gaussian sums to vector spaces over finite fields