Polynomial bound for partition rank in terms of analytic rank

Abstract

Let $G_1, \dots, G_k$ be vector spaces over a finite field $\mathbb{F} = \mathbb{F}_q$ with a non-trivial additive character $\chi$. The analytic rank of a multilinear form $\alpha \colon G_1 \times \dots \times G_k \to \mathbb{F}$ is defined as $\operatorname{arank}(\alpha) = -\log_q \mathbb{E}_{x_1 \in G_1, \dots, x_k\in G_k} \chi\big(\alpha(x_1,\dots, x_k)\big)$. The partition rank $\operatorname{prank}(\alpha)$ of $\alpha$ is the smallest number of maps of partition rank 1 that add up to $\alpha$, where a map is of partition rank 1 if it can be written as a product of two multilinear forms, depending on different coordinates. It is easy to see that $\operatorname{arank}(\alpha) \leq O\Big(\operatorname{prank}(\alpha)\Big)$ and it has been known that $\operatorname{prank}(\alpha)$ can be bounded from above in terms of $\operatorname{arank}(\alpha)$. In this paper, we improve the latter bound to polynomial, i.e. we show that there are quantities $C, D$ depending on $k$ only such that $\operatorname{prank}(\alpha) \leq C (\operatorname{arank}(\alpha)^D + 1)$. As a consequence, we prove a conjecture of Kazhdan and Ziegler. The same result was obtained independently and simultaneously by Janzer.