On Generic Identifiability of 3-Tensors of Small Rank

We introduce an inductive method for the study of the uniqueness of of tensors, by means of tensors of rank 1. The method is based the geometric notion of weak defectivity. For three-dimensional tensors of (a, b, c), a\≤ b\≤ c, our method proves that the decomposition is unique(i.e. k-identifiability holds) for general tensors of rank k, as soon as k\≤(a+1)(b+1)/16. This improves considerably the known range for identifiability. method applies also to tensor of higher dimension. For tensors of small, we give a complete list of situations where identifiability does not. Among them, there are 4\×4\×4 tensors of rank 6, an interesting because of its connection with the study of DNA strings.