Inverse systems of zero-dimensional schemes in Pn

The authors construct the global Macaulay inverse system LZ for a zero-dimensional subscheme Z of projective n-space Pn over an algebraically closed field k, from the local inverse systems of the irreducible components of Z. They show that when Z is locally Gorenstein a generic element F of degree d apolar to Z determines Z if d is larger than an invariant β(Z). As a consequence of this globalization, they show that a natural upper bound for the Hilbert function of Gorenstein Artin quotients of the coordinate ring of Z is achieved for large socle degree. They also show the uniqueness of generalized additive decompositions of a homogeneous form into powers of linear forms, under suitable hypotheses.The main tools are elementary, but delicate. They involve a careful study of how to homogenize a local inverse system and of the behavior of the homogenization under a change of coordinates.