  The authors construct the global Macaulay inverse system for a
zero-dimensional subscheme Z of projective n-space P^n, from the local inverse
systems of the irreducible components of Z. They show that when Z is locally
Gorenstein a generic homogeneous form F of degree d apolar to Z determines Z
when d is larger than an invariant b(Z). They also show that a natural upper
bound for the Hiilbert function of Gorenstein Artin quotient of the coordinate
ring is achieved for large socle degree. They show the uniqueness of
generalized additive decompositions of a homogeneous form F into powers of
linear forms, under suitable hypotheses. They include many examples.
