We use methods of algebraic geometry to find new, effective methods for
detecting the identifiability of symmetric tensors. In particular, for ternary
symmetric tensors T of degree 7, we use the analysis of the Hilbert function of
a finite projective set, and the Cayley-Bacharach property, to prove that, when
the Kruskal's rank of a decomposition of T are maximal (a condition which holds
outside a Zariski closed set of measure 0), then the tensor T is identifiable,
i.e. the decomposition is unique, even if the rank lies beyond the range of
application of both the Kruskal's and the reshaped Kruskal's criteria
