{
  "abstract": "Problem 73 on Green's list of open problems asks whether a smooth submanifold of codimension two in $\\mathbb{R}^n$ must meet some affine two-plane in five points when $n$ is large. In this paper, we answer this question for every $n \\geq 6$. Every nonempty relatively open subset of such a submanifold contains the vertices of an affinely regular pentagon, and the bound $n \\geq 6$ is sharp for affinely regular pentagons. More generally, for every prime $p \\geq 5$, the graph of a continuous map from a ball in $\\mathbb{R}^{p-1}$ to $\\mathbb{R}^2$ contains an affinely regular $p$-gon. A second argument gives six coplanar points when $n \\geq 7$. Both proofs find the points by making the low Fourier modes of the graph map vanish along a circle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14032",
  "author_total": 2,
  "authors": [
    {
      "name": "Sohail"
    },
    {
      "name": "Sarkar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a7eeem3sazdv7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14032",
  "fulltext_url": null,
  "id": "a7eeem3sazdv7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14032",
  "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.CO"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7221763649",
  "pdf_url": "https://arxiv.org/pdf/2610.14032",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a7eeem3sazdv7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14032",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14032"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Polygons and coplanar points on submanifolds of codimension 2",
  "title_text": "Polygons and coplanar points on submanifolds of codimension 2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/a7eeem3sazdv7",
  "venue": "arXiv",
  "versions": []
}