T. Sorokina: Computation of dimensions of multivariate spline spaces via Hilbert polynomials and Hilbert series.

posted Mar 17, 2012, 3:42 AM by Tony Shaska
A computational method to obtain explicit formulae for the dimension of spline spaces of smoothness $r$ and degree $d$ over simplicial partitions is described. We show how to derive these formulae in the form of a linear combination of binomial coefficients using computed values of this dimension for a finite number of parameters $r$ and $d$ to interpolate the Hilbert polynomial. Then we apply Hilbert series to obtain explicit formulae. The method is applied to conjecture the dimension formulae for the Alfeld split of an $n$-simplex and for several other tetrahedral partitions.