Quasi-linear relation between partition and analytic rank
Combinatorics
2024-11-04 v2
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.
Keywords
Cite
@article{arxiv.2211.05780,
title = {Quasi-linear relation between partition and analytic rank},
author = {Guy Moshkovitz and Daniel G. Zhu},
journal= {arXiv preprint arXiv:2211.05780},
year = {2024}
}
Comments
Various small updates, improved logarithmic term using result by Chen-Ye