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Let $K$ be an algebraically closed field of characteristic $0$, and let $A$be an Artinian Gorenstein local commutative and Noetherian $K$--algebra, withmaximal ideal $M$. In the present paper we prove a structure theorem describingsuch kind of $K$--algebras satisfying $M^4=0$. We use this result in order toprove that such a $K$--algebra $A$ has rational Poincar\'e series and it isalways smoothable in any embedding dimension, if $\dim_K M^2/M^3 \le 4$. Wealso prove that the generic Artinian Gorenstein local $K$--algebra with socledegree three has rational Poincar\'e series, in spite of the fact that suchalgebras are not necessarily smoothable.

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