For a polyhedral subdivision A of a region in Euclidean d-space, we consider the vector space C[(A) consisting of all C" piecewise polynomial functions over A of degree at most k. We consider the formal power series ~_,k~O dimR C~,(A) 2k, and show, under mild conditions on A, that this always has the form P(2)/(1-2) a+ 1, where P(2) is a polynomial in 2 with integral coefficients which satisfies P(0)= 1, P(1) = f~(A), and P'(1) = (r + 1)f~_ I(A). We discuss how the polynomial P(2) and bases for the spaces C[(A) can be effectively calculated by use of Gr6bner basis techniques of computational commutative algebra. A further application is given to the theory of hyperptane arrangements.

No abstract

No abstract

No abstract

The module of splines on a polyhedral complex can be viewed as the syzygy module of its dual graph with edges weighted by powers of linear forms. When the assignment of linear forms to edges meets certain conditions, we can decompose the graph into disjoint cycles without changing the isomorphism class of the syzygy module. Thus we can use this decomposition to compute the homological dimension and the Hilbert series of the module. We provide alternate proofs of some results of Schenck and Stillman, extending those results to the polyhedral case. We also provide examples which illustrate the role that geometry plays in determining the syzygy module.

No abstract

No abstract

We characterize module bases of spline spaces in terms of their determinants, degree sequences, and dimension series. These characterization also provide tests for freeness of the module. Applications are given to the basis and dimension problem for spline spaces.