Small Sunflowers and the Structure of Slice Rank Decompositions
Let d ≥ 3 be an integer. We show that whenever an order-d tensor admits d+1 decompositions according to Tao’s slice rank, if the linear subspaces spanned by their one-variable functions constitute a sunflower for each choice of special coordinate, then the tensor admits a decomposition where these linear subspaces are contained in the centers of these respective sunflowers. As an application, we deduce that for every nonnegative integer k and every finite field 𝔽 there exists an integer C(d,k,|𝔽|) such that every order-d tensor with slice rank k over 𝔽 admits at most C(d,k,|𝔽|) decompositions with length k, up to a class of transformations that can be easily described.
Slice rank
tensors
sunflowers
decompositions
Mathematics of computing~Discrete mathematics
Mathematics of computing~Combinatorics
67:1-67:22
Regular Paper
The author is supported by a European Research Council (ERC) Grant, number 883810.
https://arxiv.org/abs/2308.07101
The author thanks Timothy Gowers for discussions around Gowers’s paper [W. T. Gowers, 2021] that attracted the author’s attention to the idea of slice rank decompositions of the zero tensor, which was involved there and plays a role in the present paper as well. The author also thanks Jordan Ellenberg for encouraging him to prove results stating that even if the analogue of a "perfect" property which holds for the matrix rank does not hold for other notions of rank on tensors, some appropriate weakening of it does.
Thomas
Karam
Thomas Karam
Mathematical Institute, University of Oxford, UK
https://orcid.org/0009-0000-9983-7756
10.4230/LIPIcs.ITCS.2024.67
Edoardo Ballico, Alessandra Bernardi, and Pierpaola Santarsiero. Identifiability of rank-3 tensors, 2021.
Arthur Bik, Alessandro Danelon, Jan Draisma, and Rob H. Eggermont. Universality of high-strength tensors. Vietnam Journal of Mathematics, 50, 2021.
Arthur Bik, Jan Draisma, and Rob H. Eggermont. Polynomials and tensors of bounded strength. Communications in Contemporary Mathematics, 21, 2019.
Jop Briët, Harry Buhrman, Davi Castro-Silva, and Niels M. P. Neumann. Noisy decoding by shallow circuits with parities: classical and quantum, 2023.
Jop Briët and Davi Castro-Silva. Random restrictions of high-rank tensors and polynomial maps, 2022.
Ernie Croot, Vsevolod Lev, and Peter Pach. Progression-free sets in ℤ₄ⁿ are exponentially small. Annals of Mathematics, 185, 2016.
Jordan S. Ellenberg and Dion Gijswijt. On large subsets of 𝔽_qⁿ with no three-term arithmetic progression. Annals of Mathematics, 185, 2016.
W. T. Gowers. The slice rank of a direct sum, 2021.
W. T. Gowers and Thomas Karam. Equidistribution of high-rank polynomials with variables restricted to subsets of 𝔽_p, 2022.
Ben Green and Terence Tao. The distribution of polynomials over finite fields, with applications to the gowers norms. Contributions to Discrete Mathematics, 4, 2007.
Thomas Karam. High-rank subtensors of high-rank tensors, 2022.
David Kazhdan and Tamar Ziegler. Properties of high rank subvarieties of affine spaces. Geometric and Functional Analysis, 30, 2020.
Joseph B. Kruskal. Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra and its Applications, 18, 1977.
Eric Naslund. The partition rank of a tensor and k-right corners in 𝔽_qⁿ. Journal of Combinatorial Theory, Series A, 174, 2020.
Eric Naslund. The chromatic number of ℝⁿ with multiple forbidden distances, 2023.
Eric Naslund and Will Sawin. Upper bounds for sunflower-free sets. Forum of Mathematics, Sigma, 5, 2017.
Lisa Sauermann. Finding solutions with distinct variables to systems of linear equations over 𝔽_p. Mathematische Annalen, 386, 2021.
Will Sawin and Terence Tao. Notes on the "slice rank" of tensors, 2016.
Terence Tao. A symmetric formulation of the croot-lev-pach-ellenberg-gijswijt capset bound, 2016.
Thomas Karam
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