INVERSE SYSTEM OF A SYMBOLIC POWER .3. THIN ALGEBRAS AND FAT POINTS

We state a conjectural upper bound for the Hilbert function of the ide al J(p)((a)) of functions vanishing to order at least 'a' at a set P o f s generic points of P-n, and verify the bound in some cases. We show that if 3 equal to or less than n, s < 2(n), and a is sufficiently la rge, then J(p)((a)) is never in CL-generic position (Theorem 1). R. Fr oberg has given conjectural lower bounds on the Hilbert function of id eals generated by generic homogeneous polynomials, and thus also for i deals of powers of linear forms; our method is to translate these boun ds to the vanishing problem, using Macaulay's inverse systems. We give an application to bounding the dimensions of spline functions for cer tain polyhedra in R-n, using a result of L. Rose relating these dimens ions to the number of syzygies of power algebras.