An effective criterion for the additive decompositions of forms

We give an effective criterion for the identifiability of additive decompositions of homogeneous forms of degree $d$ in a fixed number of variables. Asymptotically for large $d$ it has the same order of the Kruskal's criterion adapted to symmetric tensors given by L. Chiantini, G. Ottaviani and N. Vannieuwenhoven. We give a new case of indentifiability for $d=4$.