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\begin{document}$$G_1, \ldots , G_k$$\end{document} be vector spaces over a finite field F=Fq\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {F}} = {\mathbb {F}}_q$$\end{document} with a non-trivial additive character χ\documentclass[12pt]{minimal}
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\begin{document}$$\chi $$\end{document}. The analytic rank of a multilinear form α:G1×⋯×Gk→F\documentclass[12pt]{minimal}
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\begin{document}$$\alpha :G_1 \times \cdots \times G_k \rightarrow {\mathbb {F}}$$\end{document} is defined as arank(α)=-logqEx1∈G1,…,xk∈Gkχ(α(x1,…,xk))\documentclass[12pt]{minimal}
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\begin{document}$${\text {arank}}(\alpha ) = -\log _q \mathop {\mathbb {E}} _{x_1 \in G_1, \ldots , x_k\in G_k} \chi (\alpha (x_1,\ldots , x_k))$$\end{document}. The partition rank prank(α)\documentclass[12pt]{minimal}
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\begin{document}$${\text {prank}}(\alpha )$$\end{document} of α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is the smallest number of maps of partition rank 1 that add up to α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, 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 arank(α)≤O(prank(α))\documentclass[12pt]{minimal}
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\begin{document}$${\text {arank}}(\alpha ) \le O({\text {prank}}(\alpha ))$$\end{document} and it has been known that prank(α)\documentclass[12pt]{minimal}
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\begin{document}$${\text {prank}}(\alpha )$$\end{document} can be bounded from above in terms of arank(α)\documentclass[12pt]{minimal}
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\begin{document}$${\text {arank}}(\alpha )$$\end{document}. 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 prank(α)≤C(arank(α)D+1)\documentclass[12pt]{minimal}
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\begin{document}$${\text {prank}}(\alpha ) \le C ({\text {arank}}(\alpha )^D + 1)$$\end{document}. As a consequence, we prove a conjecture of Kazhdan and Ziegler. The same result was obtained independently and simultaneously by Janzer.
