In this paper, we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, “mixed smoothness” refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from homological algebra. These tools were first applied to the study of splines by Billera (Trans. Am. Math. Soc. 310(1), 325–340, 1988). Using them, estimation of the spline space dimension amounts to the study of the Billera-Schenck-Stillman complex for the spline space. In particular, when the homology in positions 1 and 0 of this complex is trivial, the dimension of the spline space can be computed combinatorially. We call such spline spaces “lower-acyclic.” In this paper, starting from a spline space which is lower-acyclic, we present sufficient conditions that ensure that the same will be true for the spline space obtained after relaxing the smoothness requirements across a subset of the mesh edges. This general recipe is applied in a specific setting: meshes of arbitrary topologies. We show how our results can be used to compute the dimensions of spline spaces on triangulations, polygonal meshes, and T-meshes with holes.

在本文中，我们研究了多边形网格上混合光滑度的二元多项式样条的维数。在此，“混合平滑度”是指在网格的不同边缘上选择不同阶的平滑度。为了研究此类样条的空间尺寸，我们使用了同源代数中的工具。这些工具首先由Billera施加到花键的研究（反。PM。数学。SOC。310（1），325-340，1988）。使用它们，样条空间维数的估计相当于对样条空间的Billera-Schenck-Stillman复合体的研究。尤其是，当此复合体的位置1和0的同源性很小时，可以组合计算样条空间的维数。我们称这样的样条空间为“下无环”。在本文中，从低无环的样条空间开始，我们提供了充分的条件，以确保在对网格边缘的子集的平滑度要求放宽后，对于获得的样条空间也将适用。此通用配方适用于特定设置：任意拓扑的网格。我们展示了如何将我们的结果用于计算三角剖分，多边形网格和带孔的T形网格上的样条空间的尺寸。