On the dimension of spline spaces on triangulations

We consider, from an algebro-geometric perspective, the problem of determining the dimension of the space of bivariate and trivariate piecewise polynomial functions (or splines) defined on triangular and tetrahedral partitions. Classical splines on planar rectangular grids play an important role in Computer Aided Geometric Design, and splines spaces over arbitrary subdivisions of planar domains are now considered for isogeometric analysis applications. Using the homological approach introduced by L. J. Billera, we establish upper and lower bounds on the dimension of the spline spaces; these formulas include terms that take into account the geometry of the faces surrounding the interior faces of the partition and, having no restriction on the orderings of the faces, these bounds yield more accurate approximations to the dimension than previous methods.