Poincaré series and deformations of Gorenstein local algebras

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