Poincar\'e series and deformations of Gorenstein local algebras with low socle degree

Let $K$ be an algebraically closed field of characteristic $0$, and let $A$ be an Artinian Gorenstein local commutative and Noetherian $K$–algebra, with maximal ideal $M$. In the present paper we prove a structure theorem describing such kind of $K$–algebras satisfying $M^4=0$. We use this result in order to prove that such a $K$–algebra $A$ has rational Poincar'e series and it is always smoothable in any embedding dimension, if ${\dim }_K{M}^2\mathrm{/}{M}^3\le 4$. We also prove that the generic Artinian Gorenstein local $K$–algebra with socle degree three has rational Poincar'e series, in spite of the fact that such algebras are not necessarily smoothable.