Upper bound of typical ranks of m x n x ((m-1)n-1) tensors over the real number field

Abstract: Let $3\leq m\leq n$. We study typical ranks of $m\times n\times ((m-1)n-1)$ tensors over the real number field. The number $(m-1)n-1$ is a minimal typical rank of $m\times n\times ((m-1)n-1)$ tensors over the real number field. We show that a typical rank of $m\times n\times ((m-1)n-1)$ tensors over the real number field is less than or equal to $(m-1)n$ and in particular, $m\times n\times ((m-1)n-1)$ tensors over the real number field has two typical ranks $(m-1)n-1, (m-1)n$ if $m\leq \rho(n)$, where $\rho$ is the Hurwitz-Radon function defined as $\rho(n)=2^b+8c$ for nonnegative integers $a,b,c$ such that $n=(2a+1)2^{b+4c}$ and $0\leq b<4$.