A geometric criterion for the dimension of Morgan-Scott spline spaces S2rr(ΔMS)
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≥1, concurrence of the three auxiliary lines joining corresponding interior and boundary vertices implies that dim S2rr(ΔMS) exceeds the Schumaker lower bound by one. The proof constructs a quadratic form through the six dual spoke points and uses its rth 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.
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BibTeX
@misc{wu2026geometric,
title = {{A geometric criterion for the dimension of Morgan-Scott spline spaces $S_2r^r(Δ_\mathrmMS)$}},
author = {Xinyu Wu and Yi Zheng and Jiansong Deng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21192},
eprint = {2610.21192},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21192},
}
RIS
TY - UNPB
TI - A geometric criterion for the dimension of Morgan-Scott spline spaces S_2r^r(Δ_MS)
AU - Xinyu Wu
AU - Yi Zheng
AU - Jiansong Deng
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21192
UR - https://arxiv.org/abs/2610.21192
AB - 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≥1, concurrence of the three auxiliary lines joining corresponding interior and boundary vertices implies that dim S_2r^r(Δ_MS) exceeds the Schumaker lower bound by one. The proof constructs a quadratic form through the six dual spoke points and uses its rth 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.
ER -