Polynomial bound for partition rank in terms of analytic rank

Abstract

Let $G_1, \ldots , G_k$ 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 $\alpha :G_1 \times \cdots \times G_k \rightarrow {\mathbb {F}}$ is defined as ${\text {arank}}(\alpha ) = -\log _q \mathop {\mathbb {E}} _{x_1 \in G_1, \ldots , x_k\in G_k} \chi (\alpha (x_1,\ldots , x_k))$. The partition rank ${\text {prank}}(\alpha )$ 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 ${\text {arank}}(\alpha ) \le O({\text {prank}}(\alpha ))$ and it has been known that ${\text {prank}}(\alpha )$ can be bounded from above in terms of ${\text {arank}}(\alpha )$. 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 ${\text {prank}}(\alpha ) \le C ({\text {arank}}(\alpha )^D + 1)$. As a consequence, we prove a conjecture of Kazhdan and Ziegler. The same result was obtained independently and simultaneously by Janzer.