We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over any large enough finite field. The proof constructs rational maps computing a partition rank decomposition for successive derivatives of the tensor, on an open subset of the kernel variety associated with the tensor. This largely settles the main question in the "bias implies low rank" line of work in higher-order Fourier analysis, which was reiterated by Kazhdan and Ziegler, Lovett, and others.

中文翻译：

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最优逆定理

我们证明张量的分区秩和解析秩在任何足够大的有限域上都等于一个常数。证明在与张量相关的核变体的开放子集上构造有理图，计算有张量的连续导数的分区秩分解。这在很大程度上解决了高阶傅立叶分析中“偏见意味着低等级”工作中的主要问题，Kazhdan和Ziegler，Lovett等人重申了这一点。