Typical ranks for 3-tensors, nonsingular bilinear maps and determinantal ideals

Abstract

Let $m,n\geq 3$, $(m-1)(n-1)+2\leq p\leq mn$, and $u=mn-p$. The set $\mathbb{R}^{u\times n\times m}$ of all real tensors with size $u\times n\times m$ is one to one corresponding to the set of bilinear maps $\mathbb{R}^m\times \mathbb{R}^n\to \mathbb{R}^u$. We show that $\mathbb{R}^{m\times n\times p}$ has plural typical ranks $p$ and $p+1$ if and only if there exists a nonsingular bilinear map $\mathbb{R}^m\times\mathbb{R}^n\to\mathbb{R}^{u}$. We show that there is a dense open subset $\mathscr{O}$ of $\mathbb{R}^{u\times n\times m}$ such that for any $Y\in\mathscr{O}$, the ideal of maximal minors of a matrix defined by $Y$ in a certain way is a prime ideal and the real radical of that is the irrelevant maximal ideal if that is not a real prime ideal. Further, we show that there is a dense open subset $\mathscr{T}$ of $\mathbb{R}^{ n\times p \times m}$ and continuous surjective open maps $\nu\colon\mathscr{O}\to\mathbb{R}^{u\times p}$ and $\sigma\colon\mathscr{T}\to\mathbb{R}^{u\times p}$, where $\mathbb{R}^{u \times p}$ is the set of $u\times p$ matrices with entries in $\mathbb{R}$, such that if $\nu(Y)=\sigma(T)$, then $\mathrm{rank} T=p$ if and only if the ideal of maximal minors of the matrix defined by $Y$ is a real prime ideal.