We prove that the slice rank of a 3-tensor (a combinatorial notion introduced by Tao in the context of the cap-set problem), the analytic rank (a Fourier-theoretic notion introduced by Gowers and Wolf), and the geometric rank (a recently introduced algebro-geometric notion) are all equivalent up to an absolute constant. As a corollary, we obtain strong trade-offs on the arithmetic complexity of a biased bililnear map, and on the separation between computing a bilinear map exactly and on average. Our result settles open questions of Haramaty and Shpilka [STOC 2010], and of Lovett [Discrete Anal., 2019] for 3-tensors.

中文翻译：

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双线性映射的结构与随机性

我们证明了3张量的切片等级（在套集问题的背景下Tao引入的组合概念），分析等级（Gowers和Wolf引入的Fourier理论概念）和几何等级（ （最近引入的代数几何概念）都等于一个绝对常数。作为推论，我们在有偏向的胆管图的算术复杂性以及精确计算双线性图和平均计算双线性图之间的分离上获得了很强的权衡。我们的结果解决了Haramaty和Shpilka [STOC 2010]以及Lovett [Discrete Anal。，2019]针对三张量的公开问题。