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. Previously, I worked on the computation of the real ranks of real binary forms of degree four and five with assigned complex rank.
Tensor rank and eigenvectors / Maccioni, Mauro. - (2017).
Tensor rank and eigenvectors
MACCIONI, MAURO
2017
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. Previously, I worked on the computation of the real ranks of real binary forms of degree four and five with assigned complex rank.File | Dimensione | Formato | |
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PHD Thesis.pdf
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