In this paper one determines the Macaulay inverse system for the vanishing ideal at \(s\) points of the projective space \(\mathbb{P}^{r-1}\) and for the symbolic powers of the graded ideal vanishing at a union of irreducible varieties in the projective space.
The paper has very interesting further applications, mentioned in the introduction. We note here only the solution to the Waring problem for forms, due to the second author in part II of this paper [A. Iarrobino, J. Algebra 174, No. 3, 1091-1110 (1995; see the review in Zbl 0842.11016)].