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 $3°(P)-1$, then the difference of the two decompositions is completely determined either on a line or on a conic or two disjoint lines.

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