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, 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.

Keywords:

• tensor rank
• binary form
• symmetric tensor
• polynomial
• genericity

AMS Subject Classifications::

• 15A69
• 15A72
• 14L99

Acknowledgements

This work was begun at the workshop on Geometry and representation theory of tensors for computer science, statistics and other areas, held in July 2008 at the AIM, California, and presented at the Workshop on Tensors and Interpolation held in Nice in June 2009. We thank the AIM and J. Landsberg for useful talks. The work has been also supported in part by contract ANR-06-Blan-0074 ‘Decotes’.