The border of the Hilbert function of a set of points in Pn1×⋯×Pnk

We describe the eventual behaviour of the Hilbert function of a set of distinct points in Pn1×⋯×Pnk. As a consequence of this result, we show that the Hilbert function of a set of points in Pn1×⋯×Pnk can be determined by computing the Hilbert function at only a finite number of values. Our result extends the result that the Hilbert function of a set of points in Pn stabilizes at the cardinality of the set of points. Motivated by our result, we introduce the notion of the border of the Hilbert function of a set of points. By using the Gale–Ryser Theorem, a classical result about (0,1)-matrices, we characterize all the possible borders for the Hilbert function of a set of distinct points in P1×P1.