Low-codimensional Subvarieties Inside Dense Multilinear Varieties

TL;DR

This paper proves that dense multilinear varieties over finite fields contain structured subvarieties defined by a bounded number of multilinear forms, advancing understanding in additive combinatorics.

Contribution

It establishes an optimal bound on the size of subvarieties within dense multilinear varieties, linking their density to the existence of structured subvarieties.

Findings

Dense multilinear varieties contain structured subvarieties.

The size of these subvarieties is bounded by a logarithmic function of the inverse density.

Result is optimal up to a constant factor.

Abstract

Let $G_1, \dots, G_k$ be finite-dimensional vector spaces over a prime field $\mathbb{F}_p$. Let $V$ be a variety inside $G_1 \times \cdots \times G_k$ defined by a multilinear map. We show that if $|V| \geq c |G_1| \cdots |G_k|$, then $V$ contains a subvariety defined by at most $K(\log_{p} c^{-1} + 1)$ multilinear forms, where $K$ depends on $k$ only. This result is optimal up to multiplicative constant and is relevant to the partition vs. analytic rank problem in additive combinatorics.