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# Small Sunflowers and the Structure of Slice Rank Decompositions

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LIPIcs.ITCS.2024.67.pdf
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## 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.

## Cite As

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

## 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.

## Subject Classification

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

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