Modules of piecewise polynomials and their freeness

An algorithm for constructing a basis for 𝐶^{𝑟}-spline modules over polynomial rings

“…We will use this representation of splines in constructing a basis for the R-module C r (2) whenever it is free. We must emphasize that as pointed out by Billera and Rose [4], the freeness of R-module C r (2) depends not only on the combinatorics of 2 but also on the geometry of 2. However, when d = 2, C r (2) is free iff 2 is a 2-dimensional manifold with boundary and therefore the freeness of C r (2) over R is independent of the geometry and is a combinatorial invariant.…”

“…It is easily seen that this module is finitely generated, torsion free and of rank equal to the number of d-faces of 2. The general question as to under what condition on d, r and 2, the R-module C r (2) is free, has been dealt with in [4]. The case when C r (2) is free is of practical importance in applications because in that case knowing a basis will easily determine general properties of a spline function on 2.…”

“…This would be especially useful when one can determine an algorithm for writing down basis elements of the R-modules C r (2) just by knowing the geometry of 2 alone. It is proved by Billera and Rose [4] that, when d = 2, C r (2) is free over R iff 2 is a manifold with boundary. However, even in this case there does not seem to be an algorithm for writing a basis even for especially simple rectangular grids.…”

On Projective Dimension of Spline Modules

“…Thus let g be a polyhedral d-complex (see preliminaries) embedded in R d and consider the set S r (g) of all piecewise polynomials on g which are continuously differentiable of order a given r 0 on whole of g. With respect to pointwise operations of addition and multiplication, the set S r (g) is a ring and the set R=R[x 1 , ..., x d ] considered as global polynomials on g forms a subring of S r (g). Hence S r (g) is naturally an R-module and is called the spline module of c r -splines on g. In a series of papers (see [5], [6], [7]), Billera and Rose initiated the study of this module and obtained some basic results. For example, in [6] and [7], the methods of commutative algebra have been used to study the dimension problem (mentioned earlier) of R-vector spaces S r k (q) of all c r -splines on q of degree at most k when g=q is a simplicial complex; in [5] the algebraic question viz.…”

“…Hence S r (g) is naturally an R-module and is called the spline module of c r -splines on g. In a series of papers (see [5], [6], [7]), Billera and Rose initiated the study of this module and obtained some basic results. For example, in [6] and [7], the methods of commutative algebra have been used to study the dimension problem (mentioned earlier) of R-vector spaces S r k (q) of all c r -splines on q of degree at most k when g=q is a simplicial complex; in [5] the algebraic question viz. under what conditions on g, r, and d the R-module S r (g) would be free, was studied; the case d=2 was completely solved (d=1 being trivial) for all polyhedral complexes g, and the case r=0 was solved for all d but for simplicial complexes q only.…”