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.

中文翻译：

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分区秩的多项式界于解析秩

令\（G_1，\ ldots，G_k \）是有限字段\（{\ mathbb {F}} = {\ mathbb {F}} _ q \）上具有非平凡加法字符\（\ chi \ ）。多线性形式\（\ alpha：G_1 \ times \ cdots \ times G_k \ rightarrow {\ mathbb {F}} \）的解析等级定义为\（{\ 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））\）。分区秩\（{\文本{恶作剧}}（\阿尔法）\）的\（\阿尔法\）是加起来分区秩1的地图数最小\（\阿尔法\），如果地图可以根据两种不同的坐标将其写为两种多线性形式的乘积，则该地图的分区等级为1。很容易看到\（{\ text {arank}}（\ alpha）\ le O（{\ text {prank}}（\ alpha））\），并且已经知道\（{\ text {prank }}（\ alpha）\）可以从上方限制为\（{\ text {arank}}（\ alpha} \）。在本文中，我们改进了后者与多项式的界限，即，我们证明仅存在依赖于k的量C，  D，从而\（{\ text {prank}}（\ alpha）\ le C（{\ text {arank }}（\ alpha）^ D + 1）\）。结果，我们证明了Kazhdan和Ziegler的猜想。Janzer独立且同时获得了相同的结果。