The number of real eigenvectors of a real polynomial

TL;DR

This paper establishes lower bounds on the number of real eigenvectors of real symmetric tensors, relating them to geometric properties of associated polynomials, with proven optimal bounds for specific cases.

Contribution

It provides new lower bounds on the count of real eigenvectors based on polynomial zero loci, extending known results and proving their optimality in certain cases.

Findings

Lower bounds depend on the degree and zero locus of the polynomial.

Bounds are proven to be optimal for binary forms and certain ternary forms.

Results connect algebraic geometry with tensor eigenvector counts.

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), is d is even, where c is the number of ovals of the locus of zeros of f; for binary forms, i prove that t is greater or equal than the number of real roots of f. Moreover, the above inequalities are the best possible for binary forms of any degree and for ternary cubic and quartic forms.