A note on extensions of multilinear maps defined on multilinear varieties

TL;DR

This paper proves that multilinear maps defined on certain algebraic subsets called multilinear varieties can be extended to multilinear maps on the entire product space, with the extension's complexity depending polynomially on the original codimension.

Contribution

It establishes a polynomial bound on extending multilinear maps from multilinear varieties to the entire product space.

Findings

Multilinear maps on multilinear varieties can be extended to the whole space.

The extension's codimension grows polynomially with the original codimension.

Provides a theoretical foundation for extending multilinear structures in finite fields.

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 $G_1 \times \dots \times G_k$ 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 $G_1\times\dots\times G_k$.