{
  "abstract": "Membranes are sufficiently smooth vector-valued functions defined on the unit square, widely used in applications such as interpolation of image data. The signature of a membrane is a collection of nonlinear features that capture its essential geometry. We index these features in the graded Hopf algebra of two-parameter shuffle products and study the algebraic varieties for each graded component. At each truncation level, we obtain the dimension of this variety via a novel Witt-type formula. We propose a Chen-Chow theorem for membranes, provably at the second truncation level, and conjecturally for higher levels with extensive computational evidence in the computer algebra system OSCAR.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16724",
  "author_total": 3,
  "authors": [
    {
      "name": "Carlos Améndola"
    },
    {
      "name": "Gabriel Riffo"
    },
    {
      "name": "Leonard Schmitz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1kpf3qfrk01a3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16724",
  "fulltext_url": null,
  "id": "1kpf3qfrk01a3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16724",
  "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",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7221821736",
  "pdf_url": "https://arxiv.org/pdf/2610.16724",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1kpf3qfrk01a3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16724",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16724"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Universal Varieties for Signatures of Membranes",
  "title_text": "Universal Varieties for Signatures of Membranes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/1kpf3qfrk01a3",
  "venue": "arXiv",
  "versions": []
}