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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 $P$.

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