Semialgebraic splines are bivariate splines over meshes whose edges are arcs of algebraic curves. They were first considered by Wang, Chui, and Stiller. We compute the dimension of the space of semialgebraic splines in two extreme cases. If the polynomials defining the edges span a three-dimensional space of polynomials, then we compute the dimensions from the dimensions for a corresponding rectilinear mesh. If the mesh is sufficiently generic, we give a formula for the dimension of the spline space valid in large degree and bound how large the degree must be for the formula to hold. We also study the dimension of the spline space in examples which do not satisfy either extreme. The results are derived using commutative and homological algebra.

半代数样条是网格上的二元样条，其边缘是代数曲线的圆弧。Wang，Chui和Stiller首先考虑了它们。我们在两种极端情况下计算半代数样条曲线的空间尺寸。如果定义边的多项式跨越了多项式的三维空间，则我们将从相应直线网格的尺寸中计算尺寸。如果网格足够通用，我们为样条空间的尺寸提供一个在很大程度上有效的公式，并限定该公式必须保持的程度。我们还将在不满足任何一个极端的示例中研究样条空间的维数。结果是使用可交换代数和同源代数得出的。