Document Open Access Logo

Small Sunflowers and the Structure of Slice Rank Decompositions

Author Thomas Karam

PDF

File

Thumbnail PDF
PDF
LIPIcs.ITCS.2024.67.pdf
  • Filesize: 0.67 MB
  • 22 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Mathematics of computing → Combinatorics
Keywords
  • Slice rank
  • tensors
  • sunflowers
  • decompositions

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

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.

Cite As Get BibTex

Thomas Karam. Small Sunflowers and the Structure of Slice Rank Decompositions. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 67:1-67:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITCS.2024.67

Author Details

Thomas Karam
  • Mathematical Institute, University of Oxford, UK

Funding

The author is supported by a European Research Council (ERC) Grant, number 883810.

Acknowledgements

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.

References

  1. Edoardo Ballico, Alessandra Bernardi, and Pierpaola Santarsiero. Identifiability of rank-3 tensors, 2021. Google Scholar
  2. Arthur Bik, Alessandro Danelon, Jan Draisma, and Rob H. Eggermont. Universality of high-strength tensors. Vietnam Journal of Mathematics, 50, 2021. Google Scholar
  3. Arthur Bik, Jan Draisma, and Rob H. Eggermont. Polynomials and tensors of bounded strength. Communications in Contemporary Mathematics, 21, 2019. Google Scholar
  4. Jop Briët, Harry Buhrman, Davi Castro-Silva, and Niels M. P. Neumann. Noisy decoding by shallow circuits with parities: classical and quantum, 2023. Google Scholar
  5. Jop Briët and Davi Castro-Silva. Random restrictions of high-rank tensors and polynomial maps, 2022. Google Scholar
  6. Ernie Croot, Vsevolod Lev, and Peter Pach. Progression-free sets in ℤ₄ⁿ are exponentially small. Annals of Mathematics, 185, 2016. Google Scholar
  7. Jordan S. Ellenberg and Dion Gijswijt. On large subsets of 𝔽_qⁿ with no three-term arithmetic progression. Annals of Mathematics, 185, 2016. Google Scholar
  8. W. T. Gowers. The slice rank of a direct sum, 2021. Google Scholar
  9. W. T. Gowers and Thomas Karam. Equidistribution of high-rank polynomials with variables restricted to subsets of 𝔽_p, 2022. Google Scholar
  10. Ben Green and Terence Tao. The distribution of polynomials over finite fields, with applications to the gowers norms. Contributions to Discrete Mathematics, 4, 2007. Google Scholar
  11. Thomas Karam. High-rank subtensors of high-rank tensors, 2022. Google Scholar
  12. David Kazhdan and Tamar Ziegler. Properties of high rank subvarieties of affine spaces. Geometric and Functional Analysis, 30, 2020. Google Scholar
  13. 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. Google Scholar
  14. Eric Naslund. The partition rank of a tensor and k-right corners in 𝔽_qⁿ. Journal of Combinatorial Theory, Series A, 174, 2020. Google Scholar
  15. Eric Naslund. The chromatic number of ℝⁿ with multiple forbidden distances, 2023. Google Scholar
  16. Eric Naslund and Will Sawin. Upper bounds for sunflower-free sets. Forum of Mathematics, Sigma, 5, 2017. Google Scholar
  17. Lisa Sauermann. Finding solutions with distinct variables to systems of linear equations over 𝔽_p. Mathematische Annalen, 386, 2021. Google Scholar
  18. Will Sawin and Terence Tao. Notes on the "slice rank" of tensors, 2016. Google Scholar
  19. Terence Tao. A symmetric formulation of the croot-lev-pach-ellenberg-gijswijt capset bound, 2016. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact