On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence

Let $A$ be a local Artinian Gorenstein algebra with maximal ideal $\fM$, $P_A(z) := \sum_{p=0}^{\infty} (\tor_p^A(k,k))z^p $ its Poicar\'{e} series. We prove that $P_A(z)$ is rational if either $\dim_k({\fM^2/\fM^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 $n =c+1$. The results are obtained thanks to a decomposition of the apolar ideal $\Ann(F)$ when $F=G+H$ and $G$ and $H$ belong to polynomial rings in different variables.