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 give 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.