Partition and Analytic Rank are Equivalent over Large Fields

Abstract

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.

arXiv commentAppeared in Duke Mathematical Journal

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Cited by in corpus (2)

• Ranges of polynomials control degree ranks of Green and Tao over finite prime fields
• A Tensor Restriction Theorem over Finite Fields