We prove that the partition rank and the analytic rank of tensors are equivalent up to a constant, over any large enough finite field (independently of the number of variables). The proof constructs rational maps computing a partition rank decomposition for successive derivatives of the tensor, on a carefully chosen subset of the kernel variety associated with the tensor. Proving the equivalence between these two quantities is the main question in the "bias implies low rank" line of work in higher-order Fourier analysis, and was reiterated by multiple authors.

翻译：我们证明,对于任何足够有限的大字段(独立于变量数量)而言,隔段等级和分析等级等同于一个常数。 证据构建了合理的地图,计算了隔段等级分解的分解法,以计算与高压相关的内核品种中经过仔细选择的分层分解。 证明这两个数量之间的等值是高阶Fourier分析中“比值意味着低级”工作的主要问题,并得到了多个作者的重申。