Polynomial bound for partition rank in terms of analytic rank

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.