Quasi-linear relation between partition and analytic rank

2022/11/10 by Guy Moshkovitz, Moshkovitz, Guy, Daniel G. Zhu +1 · 8 citations
Computer Science · Mathematics · #11B30 #15A69 #68R05 #Coding theory and cryptography #Combinatorics (math.CO) #Error Correcting Code Techniques #FOS: Mathematics #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2211.05780

openalex publication_date 2022/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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