Generalized identifiability of sums of squares

Let f be a homogeneous polynomial of even degree d. We study the decompositions f=∑i=1rfi2 where deg⁡fi=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.