A note on extensions of multilinear maps defined on multilinear varieties

Abstract

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 $G_1\times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\varphi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $d$ and all choices of $x_i\in G_i$, $i\ne d$, the restriction map $y\mapsto \varphi (x_1,\dots ,x_{d-1},y,x_{d+1},\dots ,x_k)$ 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 $G_1\times \cdots \times G_k$.

18 pages

References in corpus (1)

• A quantitative inverse theorem for the $U^4$ norm over finite fields