{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08504",
  "author_total": 1,
  "authors": [
    {
      "name": "Benjamin Lovitz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7qdk2rq5ddsw5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08504",
  "fulltext_url": null,
  "id": "7qdk2rq5ddsw5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08504",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2605|Tensor decomposition and applications"
    ]
  ],
  "openalex": "W7221001434",
  "pdf_url": "https://arxiv.org/pdf/2610.08504",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7qdk2rq5ddsw5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08504",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08504"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Explicit tensors beyond the linear flattening barrier",
  "title_text": "Explicit tensors beyond the linear flattening barrier",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/7qdk2rq5ddsw5",
  "venue": "arXiv",
  "versions": []
}