On the postulation of sd fat points in Pd

In connection with his counter-example to the fourteenth problem of Hilbert, Nagata formulated a conjecture concerning the postulation of r fat points of the same multiplicity in P2 and proved it when r is a square. Iarrobino formulated a similar conjecture in Pd. We prove Iarrobino's conjecture when r is a dth power. As a corollary, we obtain new counter-examples modeled on those by Nagata.