The Hilbert functions of ACM sets of points in

If is a set of points in , then the associated coordinate ring is an -graded ring. The Hilbert function of , defined by for all , is studied. Since the ring may or may not be Cohen–Macaulay, we consider only those that are ACM. Generalizing the case of k=1 to all k, we show that a function is the Hilbert function of an ACM set of points if and only if its first difference function is the Hilbert function of a multi-graded Artinian quotient. We also give a new characterization of ACM sets of points in , and show how the graded Betti numbers (and hence, Hilbert function) of ACM sets of points in this space can be obtained solely through combinatorial means.