{
  "abstract": "We establish a geometric dimension criterion for spline spaces on nondegenerate standard Morgan-Scott triangulations with four distinct edge directions at each interior vertex. For every integer $r\\ge1$, concurrence of the three auxiliary lines joining corresponding interior and boundary vertices implies that $\\dim S_{2r}^r(Δ_{\\mathrm{MS}})$ exceeds the Schumaker lower bound by one. The proof constructs a quadratic form through the six dual spoke points and uses its $r$th power to define a nonzero functional annihilating the image of the smoothing-cofactor map. Diener's upper bound then gives the dimension formula. We also give a matrix representation of this functional and an explicit example in the quadratic case. Combined with the known necessary conditions and the negative-branch exclusion of Deng, Feng, and Kozak, the result yields a necessary and sufficient concurrence criterion for $r>2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21192",
  "author_total": 3,
  "authors": [
    {
      "name": "Xinyu Wu"
    },
    {
      "name": "Yi Zheng"
    },
    {
      "name": "Jiansong Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/946amefqqp0nh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21192",
  "fulltext_url": null,
  "id": "946amefqqp0nh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21192",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Analysis Techniques"
    ]
  ],
  "openalex": "W7222452806",
  "pdf_url": "https://arxiv.org/pdf/2610.21192",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/946amefqqp0nh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21192",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21192"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A geometric criterion for the dimension of Morgan-Scott spline spaces $S_{2r}^r(Δ_{\\mathrm{MS}})$",
  "title_text": "A geometric criterion for the dimension of Morgan-Scott spline spaces S_2r^r(Δ_MS)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/946amefqqp0nh",
  "venue": "arXiv",
  "versions": []
}