Partition and Analytic Rank are Equivalent over Large Fields

TL;DR

This paper proves the equivalence of partition and analytic ranks of tensors over large finite fields, advancing understanding in additive combinatorics and polynomial inverse conjectures.

Contribution

It establishes the equality of partition and analytic ranks over large fields and introduces techniques for lifting polynomial decompositions and finding rational points.

Findings

Partition and analytic ranks are equal up to a constant over large fields.

A technique for lifting decompositions of multilinear polynomials is developed.

Proves the Polynomial Gowers Inverse Conjecture in the d vs. d-1 case.

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.