A family of ideals of minimal regularity and the Hilbert series of C^r(Δ)

For a simplicial subdivision Δ of a region inR², we analyze the dimension of the vector spaceC_k^r(Δ) ofC^rpiecewise polynomial functions (splines) on Δ of degree at mostk. We find an exact sequence which allows us to prove that the dimension series for splines given by5does indeed agree with the bounds on the dimension of the spline space given by Alfeld and Schumaker [1;2]. We give sufficient conditions for the Alfeld-Schumaker bounds to be attained in all degrees, where Δ is a two-dimensional simplicial complex. The conditions are satisfied by the class of complexes considered by6, but also by a much broader class of complexes. Furthermore, for conditions which involve only local geometric data, this result is the strongest possible.