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DOI: 10.4230/LIPIcs.ITCS.2024.67

Small Sunflowers and the Structure of Slice Rank Decompositions

Authors: Thomas Karam

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

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.

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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)

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@InProceedings{karam:LIPIcs.ITCS.2024.67,
  author =	{Karam, Thomas},
  title =	{{Small Sunflowers and the Structure of Slice Rank Decompositions}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{67:1--67:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.67},
  URN =		{urn:nbn:de:0030-drops-195953},
  doi =		{10.4230/LIPIcs.ITCS.2024.67},
  annote =	{Keywords: Slice rank, tensors, sunflowers, decompositions}
}

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Small Sunflowers and the Structure of Slice Rank Decompositions

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