  The spline space $C_k^r(\Delta)$ attached to a subdivided domain $\Delta$ of
$\R^{d} $ is the vector space of functions of class $C^{r}$ which are
polynomials of degree $\le k$ on each piece of this subdivision. Classical
splines on planar rectangular grids play an important role in Computer Aided
Geometric Design, and spline spaces over arbitrary subdivisions of planar
domains are now considered for isogeometric analysis applications. We address
the problem of determining the dimension of the space of bivariate splines
$C_k^r(\Delta)$ for a triangulated region $\Delta$ in the plane. Using the
homological introduced by Billera (1988), we number the vertices and establish
a formula for an upper bound on the dimension. There is no restriction on the
ordering and we obtain more accurate approximations to the dimension than
previous methods and furthermore, in certain cases even an exact value can be
found. The construction makes also possible to get a short proof for the
dimension formula when $k\ge 4r+1$, and the same method we use in this proof
yields the dimension straightaway for many other cases.
