  Let $A$ be a local Artinian Gorenstein ring with algebraically closed residue
field $A/{\frak M}=k$ of characteristic 0, and let $P_A(z) :=
\sum_{p=0}^{\infty} ({\mathrm{ Tor}}_p^A(k,k))z^p $ be its Poincar\'e series.
We prove that $P_A(z)$ is rational if either $\dim_k({{\frak M}^2/{\frak M}^3})
\leq 4 $ and $ \dim_k(A) \leq 16,$ or there exist $m\leq 4$ and $c$ such that
the Hilbert function $H_A(n)$ of $A$ is equal to $ m$ for $n\in [2,c]$ and
equal to 1 for $n > c$. The results are obtained thanks to a decomposition of
the apolar ideal $\mathrm {Ann}(F)$ when $F=G+H$ and $G$ and $H$ belong to
polynomial rings in different variables.
