On the typical rank of real binary forms

By Pierre Comon, Giorgio Ottaviani. 2009

80 citations recorded by the source corpus.

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, and 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.

Subfields: Computational Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bb75c9fff1c1ac3115d4e8cbefdeefadb98a093d

How researchers see this paper

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In Computational Mathematics

1. Bruce Reznick
2. Enrico Carlini
3. A. V. Geramita

In Discrete Mathematics and Combinatorics

1. M. Boij
2. Bruce Reznick
3. Enrico Carlini
4. A. V. Geramita

In Geometry and Topology

1. Bruce Reznick
2. M. Boij
3. Enrico Carlini
4. A. V. Geramita
5. Jiawang Nie
6. Pablo A. Parrilo
7. L. Fialkow
8. D. Wilhelm

Citations in the corpus

• Monomials as sums of powers: The real binary case (2010, result): “bers, the connection with real world problems is very direct. This has prompted a re-examination of the Waring problem for R = R[x0,x1], and a recent very suggestive paper of Comon and Ottaviani (see [CO09]) considered this very problem for degrees d ≤ 5. Our main result in this paper follows the li”
• Tensors : A brief introduction (2014, unknown): “The case of real symmetric tensors of dimension 2 is now well understood [67], [22], [13].”
• ON THE LENGTH OF BINARY FORMS (2010, methodology): “1 constant-counting value, with the four exceptions known since the 19th century – (n, d) = (3, 5), (4, 3), (4, 4), (4, 5) – in which the length is ⌈ 1 n ( n+d−1 n−1 ) ⌉ + 1.”