eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-01-24
67:1
67:22
10.4230/LIPIcs.ITCS.2024.67
article
Small Sunflowers and the Structure of Slice Rank Decompositions
Karam, Thomas
1
https://orcid.org/0009-0000-9983-7756
Mathematical Institute, University of Oxford, UK
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol287-itcs2024/LIPIcs.ITCS.2024.67/LIPIcs.ITCS.2024.67.pdf
Slice rank
tensors
sunflowers
decompositions