Real and complex rank for real symmetric tensors with low ranks

We study the case of a real homogeneous polynomial P whose minimal real and complex decompositions in terms of powers of linear forms are different. We prove that if the sum of the complex and the real ranks of P is at most 3 deg(P) − 1, then the difference of the two decompositions is completely determined either on a line or on a conic or two disjoint lines.