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.
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Comon, P., Ottaviani, G.: On the typical rank of real binary forms. (2009) available at arXiv:math/0909.4865
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Comas, G., Seiguier, M.: On the rank of a binary form. (2001) available at arXiv:math/0112311
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Causa, A., Re, R. On the maximum rank of a real binary form. Annali di Matematica 190, 55–59 (2011). https://doi.org/10.1007/s10231-010-0137-2
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DOI: https://doi.org/10.1007/s10231-010-0137-2