# The number of real eigenvectors of a real polynomial

The number of real eigenvectors of a real polynomial 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\$\$ 2 c + 1 , if d is odd, and t is greater or equal than \$\$\max (3,2c+1)\$\$ max ( 3 , 2 c + 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bollettino dell'Unione Matematica Italiana Springer Journals

# The number of real eigenvectors of a real polynomial

, Volume 11 (2) – Jan 9, 2017
21 pages

/lp/springer_journal/the-number-of-real-eigenvectors-of-a-real-polynomial-u0QrTXd0Ag
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1972-6724
eISSN
2198-2759
D.O.I.
10.1007/s40574-016-0112-y
Publisher site
See Article on Publisher Site

### 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\$\$ 2 c + 1 , if d is odd, and t is greater or equal than \$\$\max (3,2c+1)\$\$ max ( 3 , 2 c + 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.

### Journal

Bollettino dell'Unione Matematica ItalianaSpringer Journals

Published: Jan 9, 2017

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