{
  "abstract": "We prove that Schumaker's lower bound gives the dimension of $S_{3r+1}^r(Δ)$ for every nondegenerate triangulation $Δ$ of a polygonal disk in $\\mathbb{R}^2$ and every integer $r\\ge3$. After accounting for the local vertex contributions, we reduce the remaining dimension defect to compatible vertex functionals. Apolarity associates a unique polynomial with each face, and compatibility forces differences of adjacent face polynomials to contain powers of both endpoint forms. We eliminate the leading residual coefficient using Hermite relations when $r$ is odd and an additional coefficient comparison when $r$ is even. A uniform induction then raises the endpoint powers until all face differences vanish; the zero boundary face potentials imply that every vertex functional vanishes and the global defect is zero. Together with the known cases $r=1,2$ and Hong's higher-degree theorem, this yields the dimension formula for all $r\\ge1$ and $d\\ge3r+1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08178",
  "author_total": 2,
  "authors": [
    {
      "name": "Xinyu Wu"
    },
    {
      "name": "Jiansong Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cvfgaz87cz57k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08178",
  "fulltext_url": null,
  "id": "cvfgaz87cz57k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08178",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08178",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cvfgaz87cz57k.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08178",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08178"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Universal Dimension Formula for Planar $C^r$ Splines at Degree $3r+1$",
  "title_text": "Universal Dimension Formula for Planar C^r Splines at Degree 3r+1",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:49Z",
  "url": "https://research.draconitic.com/paper/cvfgaz87cz57k",
  "venue": "arXiv",
  "versions": []
}