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

Let A be a local Artinian Gorenstein algebra with maximal ideal M, PA(z):=∑p=0∞( orAp(k,k))zp its Poicar'{e} series. We prove that PA(z) is rational if either dimk(M2/M3)≤4 and dimk(A)≤16, or there exist m≤4 and c such that the Hilbert function HA(n) of A is equal to m for n∈[2,c] and equal to 1 for n=c+1. The results are obtained due to a decomposition of the apolar ideal Ann(F) when F=G+H and G and H belong to polynomial rings in different variables