Introduction

Let ∆ be a simplicial d-complex; i.e., ∆ is a pure, d-dimensional, strongly connected simplicial complex embedded in Rd. Let S = R[x1, x2, . . . , xd] be the polynomial ring in variables x1, x2, . . . , xd. For every integer k ≥ 0, let us define

Crk(∆) = {F : |∆| → R s.t. F |σ ∈ S is a polynomial of degree ≤ k for every d-simplex σ ∈ ∆ and F is globally smooth of order r on |∆|}

Then Crk(∆) is a finite dimensional vector space over R. To determine the R-dimension of the vector space Crk(∆) for a given r ≥ 0 and a given k ≥ 0 is fundamental unresolved problem of multivarate spline theory. Alfeld and Schumaker ([1],[2]) are the first approximation theorists to obtain basic results on this problem using, of course, the classical methods of a system of linear equations and determining the number of independent real constants. In the planer case they obtain a formula which essentially shows that for d = 2, the Hilbert polynomial and the Hilbert function are equal if k = 3r+1. The fact that this dimension depends upon the geometry of the triangulation of the region makes the problem more difficult. L. Billera [3] introduced the method of homological algebra to tackle this problem. He considers the following set for any r ≥ 0;

Cr(∆) = {F : |∆| → R s.t. F |σ ∈ S for every σ ∈ ∆ and F is globally smooth of order r on |∆|}

It is straightforward to see that Cr(∆) is a ring with pointwise operations and S is a subring of this ring. Thus Cr(∆) becomes a S-module and is called the spline module of ∆. Billera-Rose [5] embedded ∆ into the hyperplane xd+1 = 1 of the space Rd+1 and considered the simplicial complex ∆ˆ = v ∗∆ where v is the origin of Rd+1. Let R = R[x1, · · ·xd+1] be the polynomial ring in (d + 1) variables and note that Cr(∆ˆ) is a graded R-module. A useful observation is that, as a vector space over R,

Crk(∆) ∼= (Cr(∆ˆ))k i.e., the spline vector space Crk(∆) is isomorphic to the vector space of homogenous elements of degree k of the graded R-module Cr(∆ˆ). This result has proved very important for solving the dimension problem. They considered the Hilbert series of the graded module Cr(∆ˆ) which is always of the form P (Cr(∆ˆ), t)/(1−t)d+1 where P (Cr(∆ˆ), t) is a polynomial in Z[t]. This is indeed the generating function of the dimension series

∑ k(dimR C

k of the spline modules [5].