Planar splines on a triangulation with a single totally interior edge

06/29/2023

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by   Michael DiPasquale, et al.

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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.