# Typical ranks of semi-tall real 3-tensors

## TL;DR

This paper investigates the typical ranks of semi-tall real 3-tensors, especially when p equals (m-1)(n-1)+1, revealing cases where the known relationship with nonsingular bilinear maps does not hold.

## Contribution

It extends previous results by analyzing the case p=(m-1)(n-1)+1, showing that typical ranks can be p and p+1 even without the existence of certain nonsingular bilinear maps.

## Key findings

- Typical ranks are p and p+1 in several cases including absence of nonsingular bilinear maps.
- The 'only if' condition relating typical ranks to nonsingular bilinear maps does not always hold for p=(m-1)(n-1)+1.
- The paper provides new insights into the structure of tensor ranks in semi-tall cases.

## Abstract

Let $m$, $n$ and $p$ be integers with $3\leq m\leq n$ and $(m-1)(n-1)+1\leq p\leq (m-1)m$. We showed in previous papers that if $p\geq (m-1)(n-1)+2$, then typical ranks of $p\times n\times m$-tensors over the real number field are $p$ and $p+1$ if and only if there exists a nonsingular bilinear map $\mathbb{R}^m\times \mathbb{R}^n\to\mathbb{R}^{mn-p}$. We also showed that the "if" part also valid in the case where $p=(m-1)(n-1)+1$. In this paper, we consider the case where $p=(m-1)(n-1)+1$ and show that the typical ranks of $p\times n\times m$-tensors over the real number field are $p$ and $p+1$ in several cases including the case where there is no nonsingular bilinear map $\mathbb{R}^m\times \mathbb{R}^n\to\mathbb{R}^{mn-p}$. In particular, we show that the "only if" part of the above mentioned fact does not valid for the case $p=(m-1)(n-1)+1$.