Piecewise Polynomial Functions . . .

Splines are piecewise polynomial functions of a given order of smoothness r on a triangulated region ∆ (or polyhedrally subdivided region �) of R d. The set of splines of degree at most k forms a vector space C r k (∆). Moreover, a nice way to study C r k (∆) is to embed ∆ in Rd+1, and form the cone � ∆ of ∆ with the origin. It turns out that the set of splines on � ∆ is a graded module C r ( � ∆) over the polynomial ring R[x1,..., xd+1], and the dimension of C r k (∆) is the dimension of Cr ( � ∆)k. This dissertation follows the works of Billera and Rose, as well as Schenck and Stillman, who each approached the study of splines from the viewpoint of homological and commutative algebra. They both defined chain complexes of modules such that C r ( � ∆) appeared as the top homology module. First, we analyze the effects of gluing planar simplicial complexes. Suppose ∆1, ∆2, and ∆ = ∆1 ∪ ∆2 are all planar simplicial complexes which triangulate pseudomanifolds. When ∆1 ∩ ∆2 is also a planar simplicial complex, we use the Mayer-Vietoris sequence to obtain a natural relationship between the spline module