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Quasi-linear relation between partition and analytic rank

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arxiv 2211.05780 v2 pith:PUYBIJ7A submitted 2022-11-10 math.CO

Guy Moshkovitz , Daniel G. Zhu This is my paper · ORCID
classification math.CO
keywords rankanalyticpartitionconjectureindependentpolynomialsadditivealgebraic
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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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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Lower bounds on the strength of the determinant

    math.AC 2026-07 accept novelty 8.0 of 10

    The determinant has strength p for prime p and partition rank n minus o(n), proved by a new Chern class obstruction.

  2. A weak regularity lemma for polynomials

    math.CO 2025-09 conditional novelty 8.0 of 10

    A weak regularity lemma decomposes polynomial maps into few approximately independent homogeneous parts, with bounds polynomial in the number of polynomials, and implies stronger quantitative versions of rank and arit...

  3. Slice rank and partition rank of the determinant

    math.CO 2025-09 conditional novelty 8.0 of 10

    The determinant has slice rank n, partition rank at least log2(n)+1, and the 4x4 determinant has partition rank 3, giving the first unbounded separation between partition rank and analytic rank.

  4. Algebraic aspects of the polynomial Littlewood-Offord problem

    math.CO 2025-05 accept novelty 8.0 of 10

    A corrected version of Costello's conjecture holds for multilinear polynomials with optimal exponent 1, complex quadratics get a 13/24 power saving, and the original conjecture is false for degree at least 3.

  5. Strength and partition rank under limits and field extensions

    math.AG 2025-02 conditional novelty 6.0 of 10

    For fixed degree d, strength and partition rank over any field are bounded by O(r^{d-1}) (plus a log factor on finite fields) in terms of their border rank analogues.

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