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Explicit tensors beyond the linear flattening barrier — arXiv:2610.08504

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.08504
License (as archived): CC0-1.0
Archived: 2026-10-07T13:40:35.752Z
SHA-256 of the archived text: ad9dc3ee1c49495f1096a1d6a089a1673bf1ce54e3e0635b95fdd11974b546e2
Stable URL of this page: https://metraipolis.com/documento/dc_e5845efffddb

How to cite

Explicit tensors beyond the linear flattening barrier — arXiv:2610.08504 — arXiv (en), https://arxiv.org/abs/2610.08504 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_e5845efffddb (Metraipolis archive dc_e5845efffddb, sha256 ad9dc3ee1c49495f1096a1d6a089a1673bf1ce54e3e0635b95fdd11974b546e2)

BibTeX

@misc{dce5845efffddb,
  title = {Explicit tensors beyond the linear flattening barrier — arXiv:2610.08504},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.08504},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_e5845efffddb; sha256 ad9dc3ee1c49495f1096a1d6a089a1673bf1ce54e3e0635b95fdd11974b546e2},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.08504. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We construct explicit n x n x n tensors of border rank at least 3n-o(n), improving the previous record of (2 + $\varepsilon$)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.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv cs.DS, nuovi lavori del giorno).
Categorie arXiv: Data Structures and Algorithms (cs.DS) ; Algebraic Geometry (math.AG).
Pagina del lavoro: https://arxiv.org/abs/2610.08504 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T13:13:03.925Z dalla pagina https://arxiv.org/list/cs.DS/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).