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

Abstract

Let $3\le m\le n$. 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 $m\le \rho (n)$, 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 $0\le b<4$.

References in corpus (1)

• Typical rank of $m\times n\times (m-1)n$ tensors with $3\le m\le n$ over the real number field