On fully real eigenconfigurations of tensors

TL;DR

This paper constructs generic real symmetric tensors that have only real eigenvectors, corresponding to real homogeneous polynomials with the maximum finite number of critical points on the sphere, advancing understanding of tensor eigenstructures.

Contribution

It introduces a method to construct generic real symmetric tensors with exclusively real eigenvectors, achieving the maximum number of critical points on the sphere.

Findings

Constructed tensors have only real eigenvectors.

Achieved maximum finite number of critical points on the sphere.

Provides a new approach to tensor eigenstructure analysis.

Abstract

We construct generic real symmetric tensors with only real eigenvectors or, equivalently, real homogeneous polynomials with the maximum possible finite number of critical points on the sphere.