Finding inverse systems from coordinates

Let I be a homogeneous ideal in R=K[x0,…,xn], such that R/I is an Artinian Gorenstein ring. A famous theorem of Macaulay says that in this instance I is the ideal of polynomial differential operators with constant coefficients that cancel the same homogeneous polynomial F. A major question related to this result is to be able to describe F in terms of the ideal I. In this note we give a partial answer to this question, by analyzing the case when I is the Artinian reduction of the ideal of a reduced (arithmetically) Gorenstein zero-dimensional scheme Γ⊂Pn. We obtain F from the coordinates of the points of Γ.