  In this paper we study the isomorphism classes of Artinian Gorenstein local
rings with socle degree three by means of Macaulay's inverse system. We prove
that their classification is equivalent to the projective classification of the
hypersurfaces of $\mathbb P ^{n }$ of degree three. This is an unexpected
result because it reduces the study of this class of local rings to the
homogeneous case. The result has applications in problems concerning the
punctual Hilbert scheme $Hilb_d (\mathbb P^n)$ and in relation to the problem
of the rationality of the Poincar\'e series of local rings.
