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.
Categories: math.AC math.AG