Real and complex rank for real symmetric tensors with low ranks. (English) Zbl 1327.14232
Summary: 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.
MSC:
| 14N05 | Projective techniques in algebraic geometry |
| 15A72 | Vector and tensor algebra, theory of invariants |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
References:
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