Generalized identifiability of sums of squares

Let f be a homogeneous polynomial of even degree d.We study the decompositions f = r i=1 f 2 i where deg f i = d/2.The minimal number of summands r is called the 2-rank of f , so that the polynomials having 2-rank equal to 1 are exactly the squares.Such decompositions are never unique and they are divided into O(r)-orbits, the problem becomes counting how many different O(r)-orbits of decomposition exist.We say that f is O(r)-identifiable if there is a unique O(r)-orbit.We give sufficient conditions for generic and specific O(r)-identifiability.Moreover, we show the generic O(r)-identifiability of ternary forms.