Planar splines on a triangulation with a single totally interior edge

2023/06/29 by Michael DiPasquale, DiPasquale, Michael, Beihui Yuan +1
Computer Science · Engineering · Mathematics · #11P21 #13P10 #13P25 #41A15 #52C05 #Advanced Numerical Analysis Techniques #Advanced Optimization Algorithms Research #Commutative Algebra (math.AC) #FOS: Mathematics #Numerical Analysis (math.NA) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2306.16825

openalex publication_date 2023/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive an explicit formula, valid for all integers r,d≥ 0, for the dimension of the vector space C^r_d(Δ) of piecewise polynomial functions continuously differentiable to order r and whose constituents have degree at most d, where Δ is a planar triangulation that has a single totally interior edge. This extends previous results of Tohǎneanu, Mináč, and Sorokina. Our result is a natural successor of Schumaker's 1979 dimension formula for splines on a planar vertex star. Indeed, there has not been a dimension formula in this level of generality (valid for all integers d,r≥ 0 and any vertex coordinates) since Schumaker's result. We derive our results using commutative algebra.

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