Some remarks on degeneracy of tridimensional tensors

TL;DR

This paper investigates the properties of complex tridimensional tensors using geometric methods, focusing on degeneracy, format, and tensor rank through determinantal schemes.

Contribution

It introduces a geometric approach to analyze tensor degeneracy, format, and rank, building on classical ideas and applying determinantal schemes.

Findings

Characterization of degeneracy conditions for complex tensors

Determination of tensor format and rank in specific cases

Application of geometric methods to tensor analysis

Abstract

We study tridimensional tensors on the complex field from the point of view of hypermatrices, taking into consideration the problem of determining whether they are degenerate or not, concise or not, what is their essential format if they are non-coincise, and, in some cases, their tensor rank. We use a geometrical approach to these problems which, in part, goes back to Schl\"{a}fli and consists in studying certain determinantal schemes associated to the hypermatrix.