A geometric criterion for the dimension of Morgan-Scott spline spaces $S_{2r}^r(Δ_{\mathrm{MS}})$
Year: 2026
Type: preprint
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 $r>2$.
Source: arXiv (Cornell University)
Authors: Wu, Xinyu, Zheng, Yi, Deng, Jiansong
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Awards: G18429415, G4937366425
OpenAlex ID: W7222452806