We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over finite fields of any characteristic and any large enough cardinality depending on the analytic rank. Moreover, we show that a plausible improvement of our field cardinality requirement would imply that the ranks are equal up to 1+o(1) in the exponent over every finite field. At the core of the proof is a technique for lifting decompositions of multilinear polynomials in an open subset of an algebraic variety, and a technique for finding a large subvariety that retains all rational points such that at least one of these points satisfies a finite-field analogue of genericity with respect to it. Proving the equivalence between these two ranks, ideally over fixed finite fields, is a central question in additive combinatorics, and was reiterated by multiple authors. As a corollary we prove, allowing the field to depend on the value of the norm, the Polynomial Gowers Inverse Conjecture in the d vs. d-1 case.

翻译：此外,我们证明,根据分析等级,分层等级和分析等级等于一个常数、超限的字段,任何特性和任何足够大的基本程度取决于分析等级。此外,我们证明,我们实地最基本要求的合理改进将意味着,每个有限字段的排量最高等于1+o(1)。证据的核心是将多线性多元分子分解在一个开放的代数种类子集中的一种技术,以及找到一个保留所有合理点的大型次等技术,这些点中至少有一个至少符合一个通用性的限定场类比。证明这两级之间的等同性,最好是超过固定的定数字段,是混合组合法中的一个核心问题,并得到多个作者的重申。我们证明,允许该字段依赖规范价值的必然结果是,在d. d-1案中,多线性戈韦尔s Inversective Conjecture(Polynomial Gowers Inpjecture)。