Quasi-linear relation between partition and analytic rank

TL;DR

This paper demonstrates that the partition rank and analytic rank of tensors over finite fields are nearly equal, up to a logarithmic factor, using novel polynomial identities and random walk techniques.

Contribution

It proves the long-standing conjecture relating partition and analytic ranks up to a logarithmic factor, introducing a new vector-valued tensor rank for analysis.

Findings

Partition and analytic ranks are nearly equal up to a logarithmic factor.

Introduces a new vector-valued tensor rank called 'local rank'.

Employs recursive polynomial identities and random walks in the proof.

Abstract

An important conjecture in additive combinatorics, number theory, and algebraic geometry posits that the partition rank and analytic rank of tensors are equal up to a constant, over any finite field. We prove the conjecture up to a logarithmic factor. Our proof is largely independent of previous work, utilizing recursively constructed polynomial identities and random walks on zero sets of polynomials. We also introduce a new, vector-valued notion of tensor rank (``local rank''), which serves as a bridge between partition and analytic rank, and which may be of independent interest as a tool for analyzing higher-degree polynomials.