Geometry-dependent rank defect in C1 cubic spline space
Abstract
Determining the dimension of the C1 cubic spline space S31(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 v6(t)=(t,0) on the admissible interval I=(−3/4,24/55). The family exhibits three distinct cases. For t∈ I∖{1/5,3/83}, the lower bound is attained and dim S31(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 S31(𝒯(1/5))=34>P𝒯(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 S31(𝒯); genuinely global compatibility must also be taken into account.
Related
- Sturm-Liouville-Type Parity and Oscillation of a Cubic Spline Eigenbasis
- Bilinear Strichartz estimates on the hyperbolic space and the cubic NLS
- Hierarchical hyperbolicity, ball quotients, and the moduli space of smooth cubic surfaces
- The moduli space of cubic surfaces and its orbifold fundamental group
- The Moduli Space of Determinantal Representations of Cubic Surfaces and Invariant Theory of Root Systems
BibTeX
@misc{wu2026geometrydependent,
title = {{Geometry-dependent rank defect in $C^1$ cubic spline space}},
author = {Xinyu Wu and Jiansong Deng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02424},
eprint = {2609.02424},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02424},
}
RIS
TY - UNPB
TI - Geometry-dependent rank defect in C^1 cubic spline space
AU - Xinyu Wu
AU - Jiansong Deng
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02424
UR - https://arxiv.org/abs/2609.02424
AB - Determining the dimension of the C^1 cubic spline space S_3^1(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∈ I∖{1/5,3/83}, the lower bound is attained and dim S_3^1(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(𝒯(1/5))=34>P_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(𝒯); genuinely global compatibility must also be taken into account.
ER -