Real and complex rank for real symmetric tensors with low ranks

Abstract

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.

arXiv comment8 pages

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Cited by in corpus (3)

• Semialgebraic decomposition of real binary forms of a given degree's space
• On real Waring decompositions of real binary forms
• On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank