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The number of real eigenvectors of a real polynomial

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Abstract

I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, I prove that t is greater or equal than \(2c+1\), if d is odd, and t is greater or equal than \(\max (3,2c+1)\), if d is even, where c is the number of ovals in the zero locus of f. About binary forms, I prove that t is greater or equal than the number of real roots of f. Moreover, the above inequalities are sharp for binary forms of any degree and for cubic and quartic ternary forms.

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Acknowledgements

This work has been partially supported by G.N.S.A.G.A. of INDAM and by MIUR. I thank the referee for the careful revision.

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  1. Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67 A, Florence, Italy

    Mauro Maccioni

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  1. Mauro Maccioni
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Maccioni, M. The number of real eigenvectors of a real polynomial. Boll Unione Mat Ital 11, 125–145 (2018). https://doi.org/10.1007/s40574-016-0112-y

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