E-Determinants of Tensors

TL;DR

This paper introduces E-determinants for general tensors, extending properties of matrix determinants to tensors, and explores their applications in polynomial system solvability and characteristic polynomials.

Contribution

It generalizes symmetric hyperdeterminants to E-determinants for all tensors, establishing key properties and applications.

Findings

E-determinant inherits matrix determinant properties

Triangular tensor systems with nonzero diagonals have solutions

Explicit formulas for characteristic polynomial coefficients in 2D

Abstract

We generalize the concept of the symmetric hyperdeterminants for symmetric tensors to the E-determinants for general tensors. We show that the E-determinant inherits many properties of the determinant of a matrix. These properties include: solvability of polynomial systems, the E-determinat of the composition of tensors, product formula for the E-determinant of a block tensor, Hadamard's inequality, Gersgrin's inequality and Minikowski's inequality. As a simple application, we show that if the leading coefficient tensor of a polynomial system is a triangular tensor with nonzero diagonal elements, then the system definitely has a solution. We investigate the characteristic polynomial of a tensor through the E-determinant. Explicit formulae for the coefficients of the characteristic polynomial are given when the dimension is two.