Typical ranks of certain 3-tensors and absolutely full column rank tensors

TL;DR

This paper investigates the typical ranks of certain 3-tensors over the real numbers, revealing multiple typical ranks under specific dimensional and congruence conditions, expanding understanding of tensor rank variability.

Contribution

It identifies new cases where multiple typical ranks occur for 3-tensors, especially involving specific relationships between tensor dimensions and modular conditions.

Findings

Multiple typical ranks exist for certain 3-tensors over R.

Specific dimension and modular conditions determine the occurrence of multiple typical ranks.

The results extend the classification of tensor ranks in higher dimensions.

Abstract

In this paper, we study typical ranks of 3-tensors and show that there are plural typical ranks for m\times n\times p tensors over R in the following cases: (1) 3\leq m\leq \rho(n) and (m-1)(n-1)+1\leq p\leq (m-1)n, where \rho\ is the Hurwitz-Radon function, (2) m=3, n\equiv 3\pmod 4 and p=2n-1, (3) m=4, n\equiv 2\pmod 4, n\geq 6 and p=3n-2, (4) m=6, n\equiv 4\pmod 8, n\geq 12 and p=5n-4. (5) m=10, n\equiv 24\pmod{32} and p=9n-8.