On the maximum rank of a real binary form

Abstract

We show that a real binary form f of degree n has n distinct real roots if and only if for any $${(\alpha,\beta)\in\mathbb{R}^2{\setminus}\{0\}}$$ all the forms αf x + βf y have n − 1 distinct real roots. This answers to a question of Comon and Ottaviani (On the typical rank of real binary forms, available at arXiv:math/0909.4865, 2009), and allows to complete their argument to show that f has symmetric rank n if and only if it has n distinct real roots.