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, P-A(z) := Sigma(infinity)(p=0) (Tor(p)(A)(k, k))z(p) its Poicare series. We prove that P-A (z) is rational if either dim(k) (m(2)/m(3)) <= 4 and dim(k) (A) <= 16; or there exist m <= 4 and c such that the Hilbert function H-A(n) of A is equal to m for n is an element of [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.