  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.
