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 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 versus case.
Citation
Alex Cohen. Guy Moshkovitz. "Partition and analytic rank are equivalent over large fields." Duke Math. J. 172 (12) 2433 - 2470, 1 September 2023. https://doi.org/10.1215/00127094-2022-0086
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