Universal Dimension Formula for Planar Cr Splines at Degree 3r+1
Abstract
We prove that Schumaker's lower bound gives the dimension of S3r+1r(Δ) for every nondegenerate triangulation Δ of a polygonal disk in ℝ2 and every integer r≥3. 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≥1 and d≥3r+1.
Related
BibTeX
@misc{wu2026universal,
title = {{Universal Dimension Formula for Planar $C^r$ Splines at Degree $3r+1$}},
author = {Xinyu Wu and Jiansong Deng},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08178},
eprint = {2611.08178},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08178},
}
RIS
TY - UNPB
TI - Universal Dimension Formula for Planar C^r Splines at Degree 3r+1
AU - Xinyu Wu
AU - Jiansong Deng
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08178
UR - https://arxiv.org/abs/2611.08178
AB - 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 ℝ^2 and every integer r≥3. 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≥1 and d≥3r+1.
ER -