Quasi-linear relation between partition and analytic rank
Guy Moshkovitz, Daniel G. Zhu
TL;DR
The paper tackles the central question of whether partition rank and analytic rank of k-tensors over finite fields are equivalent up to a constant, achieving a quasi-linear bound PR(T) ≤ O_k( AR(T) L_{\mathbb{F}}(AR(T)) ) by introducing a new local rank notion. The authors develop a comprehensive algebraic framework, including polynomial identities, a probabilistic random-walk analysis on zero sets, and derivative-based constructions, to bound local rank by analytic rank and partition rank by local rank. This yields several corollaries: improved polynomial equidistribution bounds, a near-polynomial Gowers inverse theorem for polynomial phases, and a precise relationship between ranks over finite fields and their algebraic closures. The results reduce reliance on heavy regularity lemmas and provide tools potentially useful in arithmetic geometry and higher-order Fourier analysis, advancing understanding of ranks for higher-degree tensors and their applications.
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.