Bivariate splines on a triangulation with a single totally interior edge (Q6594421)

Denote by \(C^r_d(\Delta)\) the vector space of \(r\) times continuously differentiable polynomial splines of degree at most \(d\) on the planar triangulation \(\Delta\). A key problem of spline theory is to determine the dimension \(\dim C^r_d(\Delta)\).\N\NA classical result by [\textit{L. L. Schumaker}, Multivariate approximation theory, Proc. Conf. math. Res. Inst. Oberwolfach 1979, ISNM Vol. 51, 396--412 (1979; Zbl 0461.41006)] provides a dimension formula for the case when \(\Delta\) is a planar vertex star. Good results for the dimension of spline spaces over general planar triangulations are known for \(r\) being sufficiently small or large when compared to \(d\) but formulas that hold for arbitrary values of \(r\) and \(d\) are very rare.\N\NThis article provides a dimension formula for the case when \(\Delta\) has exactly one interior edge. It comes in a lattice point formulation (Theorem~4.1) and an explicit formula (Theorem~6.1) which, besides \(r\) and \(d\) also involves the numbers of non-interior edges of different slopes through the interior edge's vertices. In their rather technical proof the authors use methods of commutative algebra.\N\NIt has been observed that there exist certain pairs \((r,d)\) for which the dimension of \(C^r_d(\Delta)\) does not change if the interior edge is removed from \(\Delta\). This phenomenon the authors elucidate and extend in their article as well.