Inverse system of Gorenstein points in ${\mathbb{P}}_{\text{res}}^n$

Given a set of distinct points $X=\{P_1,\dots ,P_r\}$ in ${\mathbb{P}}_{\text{res}}^n,$ in this paper we characterize being $X$ arithmetically Gorenstein through the ``special" structure of the inverse system of the defining ideal $I(X)\subseteq R=\text{res}[X_0,\dots ,X_n]\mathrm{.}$ We describe the corresponding Zariski locally closed subset of $({\mathbb{P}}_{\text{res}}^n{)}^r$ 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.