Typical ranks of semi-tall real 3-tensors

Abstract

Let $m$, $n$ and $p$ be integers with $3\le m\le n$ and $(m-1)(n-1)+1\le p\le (m-1)m$. We showed in previous papers that if $p\ge (m-1)(n-1)+2$, 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 $p=(m-1)(n-1)+1$.