Low-codimensional Subvarieties Inside Dense Multilinear Varieties

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 $\mathrm{\mid }V\mathrm{\mid }\ge c\mathrm{\mid }{G}_1\mathrm{\mid }\cdots \mathrm{\mid }{G}_k\mathrm{\mid }$, 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.