  Given a set of distinct points $X=\{P_1, \dots,P_r\} $ in $ \mathbb
P^n_{\res}, $ in this paper we characterize being $X$ arithmetically Gorenstein
through the ``special" structure of the inverse system of the defining ideal
$I(X) \subseteq R= \res[X_0, \dots, X_n].$ We describe the corresponding
Zariski locally closed subset of $\mathbb (\mathbb P_{\res}^{n})^r$
parametrized by the coordinates of the points. Several examples are given to
show the effectiveness of the results. As a consequence of the main result we
characterize the Artinian Gorenstein rings which are the Artinian reduction of
Gorenstein points. The problem is of interest in several areas of research,
among others the $G$-linkage, the Waring decomposition, the smoothability of
Artinian algebras and the identifiability of symmetric tensors.
