Boll. Unione Mat. Ital. (2018) 11:125–145
The number of real eigenvectors of a real polynomial
Received: 1 July 2016 / Accepted: 26 November 2016 / Published online: 9 January 2017
© Unione Matematica Italiana 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),ifd 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.
Keywords Eigenvectors · Eigenvalues · Real · Binary · Ternary · Forms
Given a real homogeneous polynomial f of degree d in n variables, its eigenvectors are
x ∈ C
such that ∇ f (x ) = λx ,forsomeλ ∈ C.
In an alternative way, the eigenvectors are the critical points of the Euclidean distance
function from f to the Veronese variety of polynomials of rank one (see ).
In the quadratic case (d = 2) the eigenvectors deﬁned in this way coincide with the
usual eigenvectors of the symmetric matrix associated to f . By the Spectral Theorem, the
eigenvectors of a quadratic polynomial are all real. So a natural question is to investigate
the reality of the eigenvectors of a polynomial f of any degree d. The number of complex
eigenvectors of a polynomial f of degree d in n variables, when it is ﬁnite, is given by
((d − 1)
− 1)/(d − 2), d ≥ 3
(d − 1)
+ (d − 1)
+ ...+ (d − 1)
= n, d = 2
The value in this formula has to be counted with multiplicities. The general polynomial has
all eigenvectors of multiplicity one. Formula (1) appears in  by Cartwright and Sturmfels.
Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze,
Viale Morgagni 67 A, Florence, Italy