On the typical rank of real binary forms

Abstract : We determine the rank of a general real binary form of degree d=4 or d=5. In the case d=5, the possible values of the rank of such general forms are 3, 4, 5. This is the first reported case, to our knowledge, where more than two typical ranks have been found. We prove that a real binary form of degree d with d real roots has rank d.
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Pierre Comon, Giorgio Ottaviani. On the typical rank of real binary forms. Linear and Multilinear Algebra, Taylor & Francis, 2012, 60 (6), pp.657-667. ⟨10.1080/03081087.2011.624097⟩. ⟨hal-00420415v2⟩

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