{
  "abstract": "In this paper, we show that the border Waring rank of $x_1x_2\\cdots x_n$ over fields of characteristic zero is exactly $2^{n-1}$. As a consequence, the classical polarization identity is an optimal Waring decomposition even if we allow limits. As a symmetric tensor, this monomial is identified with the $n\\times n$ permanent tensor. However, the lower bound is proved via the higher-order Koszul flattening of the $n\\times n$ determinant tensor, which is not symmetric.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09141",
  "author_total": 1,
  "authors": [
    {
      "name": "Jong In Han"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0hkr071h21czq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09141",
  "fulltext_url": null,
  "id": "0hkr071h21czq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09141",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2605|Tensor decomposition and applications"
    ]
  ],
  "openalex": "W7212021343",
  "pdf_url": "https://arxiv.org/pdf/2609.09141",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0hkr071h21czq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.09141",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09141"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The border Waring rank of $x_1\\cdots x_n$ is $2^{n-1}$",
  "title_text": "The border Waring rank of x_1⋯ x_n is 2^n−1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/0hkr071h21czq",
  "venue": "arXiv",
  "versions": []
}