Fat Points, Inverse Systems, and Piecewise Polynomial Functions

We explore the connection between ideals of fat points (which correspond to subschemes of Pn obtained by intersecting (mixed) powers of ideals of points), and piecewise polynomial functions (splines) on a d-dimensional simplicial complex ∆ embedded in R. Using the inverse system approach introduced by Macaulay [11], we give a complete characterization of the free resolutions possible for ideals in k[x, y] generated by powers of homogeneous linear forms (we allow the powers to differ). We show how ideals generated by powers of homogeneous linear forms are related to the question of determining, for some fixed ∆, the dimension of the vector space of splines on ∆ of degree less than or equal to k. We use this relationship and the results above to derive a formula which gives the number of planar (mixed) splines in sufficiently high degree. ∗Partially supported by the Natural Sciences and Engineering Research Council of Canada †Partially supported by the National Security Agency through grant MDA 904-95-H-1035