Gröbner Basis Methods for Multivariate Splines

Ideals, Varieties, and Algorithms

“…Some references can be found on p. xvi of FARIN (1990). We note that there has also been some recent theoretical work by BILLERA and ROSE (1989) that applies Groebner basis methods to the problem of determining the vector space dimension of multivariate polynomial splines of a given degree on a given polyhedral decomposition of a region in n . See also Chapter 8 of COX, LITTLE and O'SHEA (1998).…”

Combinatorics and Commutative Algebra

“…The use of algebraic techniques to investigate splines was pioneered by Louis Billera, partially in collaboration with Lauren Rose [17], [18], [22], [23], [24]. Spline theory is of fundamental importance in numerical analysis and computer aided design.…”

Graphs, Syzygies, and Multivariate Splines

“…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 ( ).…”