{
  "abstract": "Determining the dimension of the $C^1$ cubic spline space $S_3^1(\\mathcal{T})$ on an arbitrary nondegenerate planar triangulation has remained unresolved since the 1970s. Schumaker's lower bound includes a local correction $σ$ for singular interior four-stars, and it was conjectured that this bound is always attained. We disprove this conjecture by constructing a one-parameter family of nondegenerate realizations of a fixed 18-triangle complex, with only the central vertex moving as $v_6(t)=(t,0)$ on the admissible interval $I=(-3/4,24/55)$. The family exhibits three distinct cases. For $t\\in I\\setminus\\{1/5,3/83\\}$, the lower bound is attained and $\\dim S_3^1(\\mathcal{T}(t))=33$. At $t=3/83$, the central four-star is singular, $σ=1$, and the resulting dimension 34 is exactly accounted for by the classical local correction. At $t=1/5$, however, all interior vertices are nonsingular and $σ=0$, yet $\\dim S_3^1(\\mathcal{T}(1/5))=34>P_{\\mathcal{T}(1/5)}(1,3)=33$. The smoothing-cofactor calculation shows that the dependence at $t=3/83$ is confined to the central vertex block, whereas the dependence at $t=1/5$ couples all seven interior vertex cycles even though every individual block has full row rank. A complementary Bernstein--Bézier calculation gives the same dimension profile. Thus the singular-four-star correction does not capture every geometry-dependent contribution to $\\dim S_3^1(\\mathcal{T})$; genuinely global compatibility must also be taken into account.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02424",
  "author_total": 2,
  "authors": [
    {
      "name": "Xinyu Wu"
    },
    {
      "name": "Jiansong Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aqv85zf8xr6e6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02424",
  "fulltext_url": null,
  "id": "aqv85zf8xr6e6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02424",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Analysis Techniques"
    ]
  ],
  "openalex": "W7207574047",
  "pdf_url": "https://arxiv.org/pdf/2609.02424",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aqv85zf8xr6e6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02424",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02424"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Geometry-dependent rank defect in $C^1$ cubic spline space",
  "title_text": "Geometry-dependent rank defect in C^1 cubic spline space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/aqv85zf8xr6e6",
  "venue": "arXiv",
  "versions": []
}