Explicit tensors beyond the linear flattening barrier
Abstract
We construct explicit n x n x n tensors of border rank at least 3n-o(n), improving the previous record of (2 + ε)n due to (Landsberg and Michalek 2025). We also prove that the linear flattening method cannot be used to establish lower bounds on border rank beyond 2n-1. When n is odd, we prove that this bound is achieved by Koszul flattenings. When n is even, a result of (Landsberg 2015) shows that Koszul flattenings can achieve 2n-2, leaving an open gap of size one. Our 2n-1 bound improves the best known 6n-4 linear flattening barrier due to (Garg et al. 2019) and (Buczyński 2026). Combined, these results show that our 3n-o(n) construction, as well as the construction of Landsberg and Michalek, provide explicit examples of tensors with higher border rank than any linear flattening can achieve.
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BibTeX
@misc{lovitz2026explicit,
title = {{Explicit tensors beyond the linear flattening barrier}},
author = {Benjamin Lovitz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08504},
eprint = {2610.08504},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08504},
}
RIS
TY - UNPB
TI - Explicit tensors beyond the linear flattening barrier
AU - Benjamin Lovitz
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08504
UR - https://arxiv.org/abs/2610.08504
AB - We construct explicit n x n x n tensors of border rank at least 3n-o(n), improving the previous record of (2 + ε)n due to (Landsberg and Michalek 2025). We also prove that the linear flattening method cannot be used to establish lower bounds on border rank beyond 2n-1. When n is odd, we prove that this bound is achieved by Koszul flattenings. When n is even, a result of (Landsberg 2015) shows that Koszul flattenings can achieve 2n-2, leaving an open gap of size one. Our 2n-1 bound improves the best known 6n-4 linear flattening barrier due to (Garg et al. 2019) and (Buczyński 2026). Combined, these results show that our 3n-o(n) construction, as well as the construction of Landsberg and Michalek, provide explicit examples of tensors with higher border rank than any linear flattening can achieve.
ER -