Dimensions of Bivariate Spline Spaces and Algebraic Geometry

Splines are piecewise polynomial functions of a given order of smoothness r. Given complex delta the set of splines of degree less than or equal to d forms a vector space and is denoted by Sr d(delta). For a simplicial complex delta, Strang conjectured a lower bound on the dimension of spline space Srd(delta) and it is known that the equality holds for sufficiently large d. It is called the dimension formula. In this dissertation, we approach the study of splines from the viewpoint of algebraic geometry. This dissertation follows the works of Lau and Stiller. They introduced the conformality conditions which lead to the machinery of sheaves and cohomology which provided a powerful type of generalization of linear algebra. First, we try to analyze effects in the dimensions of spline spaces when we remove or add certain faces in the given complex. We define the cofactor spaces and cofactor maps from the given complexes and use them to interpret the changes in the dimensions of spline spaces. Second, given polyhedral complex delta, we break it into two smaller complexes delta1 and delta2 which are usually easier to handle. We will find conditions for delta1 and delta2 which guarantee that the dimension formula holds for the original complex delta. Next, we use the previous splitting method on certain types of triangulations. We explain how to break the given triangulation and show what kind of simple complexes we end up with. Finally, we study the "2r+1" conjecture on a certain triangulation. The "2r+1" conjecture is that the dimension formula holds on any triangulation for d >/= 2r + 1. We know that the conjecture is sharp because the dimension formula fails on a certain triangulation for d = 2r, but we do not know if it holds on the same triangulation when d = 2r + 1. It is related to a Toeplitz matrix