On the typical rank of real binary forms

Pierre Comon (1) , Giorgio Ottaviani (2)

1 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe SIGNAL
2 Dipartimento di Matematica "Ulisse Dini"

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.

Mots clés: tensor rank; binary form; symmetric tensor; polynomial; genericity

Domaines: Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]; Géométrie algébrique [math.AG]

Pierre Comon, Giorgio Ottaviani. On the typical rank of real binary forms. Linear and Multilinear Algebra, 2012, 60 (6), pp.657-667. ⟨10.1080/03081087.2011.624097⟩. ⟨hal-00420415v2⟩