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.
Associated to the cohomology ring A of the complement X(A) of a hyperplane arrangement A in complex m-space are the resonance varieties R^k(A). The most studied of these is R^1(A), which is the union of the tangent cones at the origin to the characte
ristic varieties of the fundamental group of X. R^1(A) may be described in terms of Fitting ideals, or as the locus where a certain Ext module is supported. Both these descriptions give obvious algorithms for computation. In this note, we show that interpreting R^1(A) as the locus of decomposable two-tensors in the Orlik-Solomon ideal leads to a description of R^1(A) as the intersection of a Grassmannian with a linear space, determined by the quadratic generators of the Orlik-Solomon ideal. This method is much faster than previous alternatives.
