We consider the problem of determining the dimension of the space of bivariate splines Ckr(Δ), for all k. This problem is closely related to the question of whether Cr(Δ̂) is a free R-module. The main result is that Cr(Δ̂) is free if and only if |Δ| has genus zero and Ckr(Δ) has the expected dimension for k = r + 1 (and hence for all k). We also obtain several interesting corollaries, including the following simple non-freeness criterion: given a fixed Δ having an edge with both vertices interior, and which does not extend to the boundary, there exists an r0, which can be determined by inspection, such that Cr(Δ̂) is not free for any r ≥ r0.
ASJC Scopus subject areas
- Algebra and Number Theory