{
  "abstract": "We study graph hypersurfaces associated with higher-dimensional ($n>2$) Cobb--Douglas production functions with variable returns to scale. We prove a rigidity theorem showing that every flat model in this class is necessarily homogeneous. The proof relies on an affine dimension-reduction principle that transfers the higher-dimensional flatness condition to suitable two-dimensional restrictions. We then obtain a complete classification in the non-degree-one case: every such flat model is a power of a positive linear form. For degree one, we give a local structural description in terms of envelopes on the rank-one set, with linearity on open rank-zero regions. These results extend the two-input analysis recently developed in [8] to the higher-dimensional setting, where the degree-one case exhibits a genuinely different behavior.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27646",
  "author_total": 5,
  "authors": [
    {
      "name": "Bang-Yen Chen"
    },
    {
      "name": "Sharief Deshmukh"
    },
    {
      "name": "Nasser Bin Turki"
    },
    {
      "name": "Alina-Daniela Vîlcu"
    },
    {
      "name": "Gabriel-Eduard Vîlcu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/31er34b8wpyhd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27646",
  "fulltext_url": null,
  "id": "31er34b8wpyhd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27646",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214232499",
  "pdf_url": "https://arxiv.org/pdf/2609.27646",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/31er34b8wpyhd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27646",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27646"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Rigidity Theorem for Flat Graph Hypersurfaces of Variable-Returns-to-Scale Cobb--Douglas Production Functions",
  "title_text": "A Rigidity Theorem for Flat Graph Hypersurfaces of Variable-Returns-to-Scale Cobb--Douglas Production Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/31er34b8wpyhd",
  "venue": "arXiv",
  "versions": []
}