  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 the projective plane and proved it when r is a square.
Iarrobino formulated a similar conjecture in any projective space P^d. We prove
Iarrobino's conjecture when r is a d-th power. As a corollary, we obtain new
counter-examples modeled on those by Nagata.
