Combinatorial and Topological Invariants of Modules of Piecewise Polynomials

Module Bases for Multivariate Splines

“…In this case, a basis is determined by the degrees of the basis elements. [8]. Thus, a basis for C r (2) will never be reduced.…”

“…By studying the algebraic properties of C r (2), we have been able to glean information about the C r k (2)'s, in many cases simultaneously for all k. In [5], we showed that a reduced module basis for C r (2) will give rise to bases of the spline spaces C r k (2). In [5,8] we have gave combinatorial and topological conditions for C r (2) to be free and for a reduced basis to exists. In [3], we showed that the generating function of the dimension sequence of the spline spaces is the Hilbert series of the graded version or homogenisation of C r (2).…”

Graphs, Syzygies, and Multivariate Splines

“…This is true because each cycle in a basis for the cycle space of G will lower the rank of Syz r (G) by 1. Also, dim The following result is Theorem 4.2 in [9]. Let hd(A) denote the homological dimension of the S-module A.…”

“…In [4] and [5] we showed that the Hilbert Series of C r (ˆ ) is the generating function of the dimensions of the C r m ( )'s, whereˆ is the join of with a point in R d+1 outside the affine span of , i.e., the homogenization of . In [6] we were concerned with finding combinatorial and topological conditions on for C r ( ) to be a free module, and in [9] this study was extended to finding the homological dimension of C r ( ). Since C r ( ) is in general neither combinatorially nor topologically determined, one of the motivations of this work is to explore how the particular embedding of affects the algebraic structure of C r ( ).…”

“…The use of syzygies to describe splines first appeared in [14], and a vector space analog of B r ( ) appears in [16]. We introduced B r ( ) in [9] as an alternative way of representing elements of C r ( ), and we used this syzygy module to characterize the homological dimension of C r ( ) when the dual graph of has a basis of disjoint cycles.…”