On the postulation of s^d fat points in P^d

In connection with his counterexample to the fourteenth problem of Hilbert, Nagata formulated a conjecture concerning the postulation of r fat points of the same multiplicity in P 2 and proved it when r is a square. Iarrobino formulated a similar conjecture in P d. We prove Iarrobino's conjecture when r is a d-th power. As a corollary, we obtain new counterexamples modeled on those by Nagata.