We show that a real binary form f of degree n has n distinct real roots if and only if for any (alpha, beta) is an element of R(2)\{0} all the forms alpha f(x) + beta f(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 a distinct real roots.
On the maximum rank of a real binary form
Re, R.
2011-01-01
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) is an element of R(2)\{0} all the forms alpha f(x) + beta f(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 a distinct real roots.
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