Semialgebraic splines

We study bivariate splines over partitions defined by arcs of irreducible algebraic curves, which we call semialgebraic splines. Such splines were first considered by Wang, Chui, and Stiller. We compute the dimension of the space of semialgebraic splines in two extreme cases when the cell decomposition has a single interior vertex. If the forms defining the edges span a two-dimensional space of forms of degree n, then we compute the dimension of the spline space in every degree. In the other extreme, the curves have distinct slopes at the central vertex and do not simultaneously vanish at any other point. In this case we give a formula for the dimension of the spline space in large degree and bound how large the degree must be for the formula to be correct. We also study the dimension of the spline space in the case of a single interior vertex in some examples where the curves do not satisfy either extreme. The results are derived using commutative and homological algebra.