Poincar\ue8 series and deformations of Gorenstein local algebras

Let $(A, \m, K) $ be an Artinian Gorenstein local ring with $K$ an algebraically closed field of characteristic $0. $ In the present paper we prove a structure theorem describing the Artinian Gorenstein local $K$-algebras satisfying $\m^4=0$. We use this result in order to prove that such a $K$-algebra has rational Poincar{\'e} series and it is smoothable in any embedding dimension, provided $\dim_K \m^2/\m^3 \le 4$. We also prove that the generic Artinian Gorenstein local $K$-algebra with $\m^4=0$ has rational Poincar{\'e} series