Piecewise polynomials on polyhedral complexes

For a d-dimensional polyhedral complex P, the dimension of the space of piecewise polynomial functions (splines) on P of smoothness r and degree k is given, for k sufficiently large, by a polynomial f(P,r,k) of degree d. When d=2 and P is simplicial, Alfeld and Schumaker determined a formula for all three coefficients of f. However, in the polyhedral case, no formula is known. Using localization techniques and specialized dual graphs associated to codimension--2 linear spaces, we obtain the first three coefficients of f(P,r,k), giving a complete answer when d=2.