A note on extensions of multilinear maps defined on multilinear varieties

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 subset $G_1 \times \cdots defined as the zero set forms, each which on some coordinates. map $\phi$ $B$ if for coordinate $c$ and all choices $x_i \in G_i$ $i\not =c$ restriction $y \mapsto \phi (x_1, x_{c-1}, y, x_{c+1}, x_k)$ linear where defined. In this note, we show that coincides $O_{k}(d^{O_{k}(1)})$ with whole $G_1\times Additionally, in case general finite fields, deduce similar (but slightly weaker) results.