Characteristic polynomials and eigenvalues of tensors

TL;DR

This paper explores the geometric properties of characteristic polynomials of tensors, revealing finiteness results for certain symmetric tensors and proposing conjectures on tensor varieties sharing the same characteristic polynomial.

Contribution

It establishes finiteness results for symmetric tensors of specific orders and dimensions and introduces conjectures on the dimension of tensor varieties with identical characteristic polynomials.

Findings

Finiteness of tensors sharing a characteristic polynomial for certain symmetric cases

Proposed conjectures on the dimension of tensor varieties with shared characteristic polynomials

Geometric foundations for the study of tensor characteristic polynomials

Abstract

We lay the geometric foundations for the study of the characteristic polynomial of tensors. For symmetric tensors of order $d \geq 3$ and dimension $2$ and symmetric tensors of order $3$ and dimension $3$, we prove that only finitely many tensors share any given characteristic polynomial, unlike the case of symmetric matrices and the case of non-symmetric tensors. We propose precise conjectures for the dimension of the variety of tensors sharing the same characteristic polynomial, in the symmetric and in the non-symmetric setting.