Decompositions of the Hilbert function of a set of points in P-n

Let H be the Hilbert function of some set of distinct points in P-n and let alpha = alpha (H) be the least degree of a hypersurface of P-n containing these points. Write alpha = d(s) + d(s-1) +...+ d(1) (where d(i) > 0). We c anonically decompose H into s other Hilbert functions H <----> (H-s',...,H- 1') and show how to find sets of distinct points Y-s,..., Y-1, lying on red uced hypersurfaces of degrees d(s),...,d(1) (respectively) such that the Hi lbert function of Y-i is H-i' and the Hilbert function of Y = boolean OR (s )(i=1) Y-i is H. Some extremal properties of this canonical decomposition a re also explored.