A criterion for detecting the identifiability of symmetric tensors of size three

We prove a criterion for the identifiability of symmetric tensors $P$ of type $3\times ...\times 3$, $d$ times, whose rank $k$ is bounded by $(d^2+2d)/8$. The criterion is based on the study of the Hilbert function of a set of points $P_1,..., P_k$ which computes the rank of the tensor $P$.