On the generic and typical ranks of 3-tensors

TL;DR

This paper investigates the typical and generic ranks of 3-tensors, proposing a conjecture for the complex case and exploring multiple typical ranks over the real numbers through algebraic geometry and numerical verification.

Contribution

It introduces a conjecture for the exact generic rank of 3-tensors over complex numbers and provides examples of multiple typical ranks over real numbers, supported by numerical evidence.

Findings

Conjecture on the exact generic rank over complex numbers.

Numerical verification for tensors with dimensions up to 14.

Existence of multiple typical ranks over real numbers for specific tensor dimensions.

Abstract

We study the generic and typical ranks of 3-tensors of dimension l x m x n using results from matrices and algebraic geometry. We state a conjecture about the exact values of the generic rank of 3-tensors over the complex numbers, which is verified numerically for l,m,n not greater than 14. We also discuss the typical ranks over the real numbers, and give an example of an infinite family of 3-tensors of the form l=m, n=(m-1)^2+1, m=3,4,..., which have at least two typical ranks.