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

TL;DR

This paper investigates the typical ranks of 3-tensors, the existence of nonsingular bilinear maps, and properties of determinantal ideals, establishing conditions for multiple typical ranks and the primeness of certain ideals.

Contribution

It characterizes when multiple typical ranks occur for 3-tensors and links this to the existence of nonsingular bilinear maps, also analyzing properties of determinantal ideals.

Findings

Plural typical ranks p and p+1 occur iff a nonsingular bilinear map exists.

Dense open subsets where maximal minors define prime ideals are identified.

Continuous surjective maps relate tensor ranks to primeness of 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.