{
  "abstract": "The Karpelevič theorem describes the complex numbers that occur as eigenvalues of stochastic matrices of a fixed order. We give a self-contained proof based on invariant polygons. Their geometry yields a product equation for an eigenvalue of maximal modulus at a fixed argument. Convexity determines this modulus, explicit stochastic matrices attain it, and a comparison between Farey intervals completes the boundary description. As the matrix order increases, the resulting equation gives a uniform relative asymptotic for the gap $1-|z|$ between a boundary point $z$ and the unit circle, even arbitrarily close to arc endpoints. We also determine the optimal contraction factor for a planar rotation and dilation when vector size is measured by scaling a polygon with at most $N$ vertices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26058",
  "author_total": 2,
  "authors": [
    {
      "name": "Brecht F. Verbeken"
    },
    {
      "name": "Vincent Ginis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2qk7ews2cd1jj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26058",
  "fulltext_url": null,
  "id": "2qk7ews2cd1jj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26058",
  "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.PR"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214125832",
  "pdf_url": "https://arxiv.org/pdf/2609.26058",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2qk7ews2cd1jj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2609.26058",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26058"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A proof of the Karpelevič theorem via invariant polygons",
  "title_text": "A proof of the Karpelevič theorem via invariant polygons",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/2qk7ews2cd1jj",
  "venue": "arXiv",
  "versions": []
}