Hilbert functions and symmetric tensors identifiability

We study the Waring decompositions of a given symmetric tensor using tools of algebraic geometry for the study of finite sets of points. In particular we use the properties of the Hilbert functions and the Cayley-Bacharach property to study the uniqueness of a given decomposition (the identifiability problem), and its minimality, and show how, in some cases, one can effectively determine the uniqueness even in some range in which the Kruskal's criterion does not apply. We give also a more efficient algorithm that, under some hypothesis, certify the identifiability of a given symmetric tensor.