English

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

R2 v1 2026-06-28T05:37:35.636Z