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Quasi-linear relation between partition and analytic rank
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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.
Forward citations
Cited by 3 Pith papers
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Slice rank and partition rank of the determinant
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.
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Algebraic aspects of the polynomial Littlewood-Offord problem
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.
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Strength and partition rank under limits and field extensions
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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