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×⋯×33×⋯×3, d times, whose rank k   is bounded by (d2+2d)/8(d2+2d)/8. The criterion is based on the study of the Hilbert function of a set of points P1,…,PkP1,…,Pk which computes the rank of the tensor P.