Typical ranks of semi-tall real 3-tensors

TL;DR

This paper investigates the typical ranks of semi-tall real 3-tensors, especially when p equals (m-1)(n-1)+1, revealing cases where the known relationship with nonsingular bilinear maps does not hold.

Contribution

It extends previous results by analyzing the case p=(m-1)(n-1)+1, showing that typical ranks can be p and p+1 even without the existence of certain nonsingular bilinear maps.

Findings

Typical ranks are p and p+1 in several cases including absence of nonsingular bilinear maps.

The 'only if' condition relating typical ranks to nonsingular bilinear maps does not always hold for p=(m-1)(n-1)+1.

The paper provides new insights into the structure of tensor ranks in semi-tall cases.

Abstract

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

Equations

$${\mathrm{fl}}_1(T):=(T_1,\dots ,T_n)\in K^{\ell \times mn}$$

$${\mathrm{fl}}_2(T):=\left(\begin{array}{c}{T}_1\\ ⋮\\ T_n\end{array}\right)\in K^{\ell n\times m}$$

$$\begin{array}[]{lcr}\mathbb{R}^{n\times p\times m}\supset\mathscr{V}\\ &\!\!\!\!\!\!\!\!\!\!\tau\nwarrow\!\!\searrow\sigma\!\!\!\!\!\!\!\!\\ &&\mathbb{R}^{u\times p}\\ &\!\!\!\!\!\!\!\!\!\!\mu\swarrow\!\!\nearrow\nu\!\!\!\!\!\!\!\!\\ \mathbb{R}^{u\times n\times m}\supset\mathscr{O}\end{array}$$

$$(\ast)\left\{\begin{array}[]{lll}D_{k}=\mathrm{Diag}(d_{1k},\ldots,d_{pk})&\mbox{for $1\leq k\leq m$,}\\ (d_{j1}W_{1}+\cdots+d_{jm}W_{m})\bm{b}_{j}=\bm{0}&\mbox{for $1\leq j\leq p$ and}\\ \mbox{$\begin{pmatrix}BD_{1}\\ \vdots\\ BD_{m-2}\\ B^{\leq n-\ell}D_{m-1}\end{pmatrix}$ is nonsingular.}\end{array}\right.$$

$$M({\mathbf{a}}_j^{\top },W){\mathbf{b}}_j=0$$

$$(\psi ({\mathbf{a}}_1,{\mathbf{b}}_1),\dots ,\psi ({\mathbf{a}}_p,{\mathbf{b}}_p))=\left(\begin{array}{c}B{D}_1\\ ⋮\\ B{D}_{m-2}\\ B^{\le n-\ell }{D}_{m-1}\end{array}\right)$$

$$A_k:=\left(\begin{array}{c}{O}_{(k-1)\times n}\\ E_n\\ O_{(m-k-1)\times n}\end{array}\right)\in {\mathbb{R}}^{u\times n}$$

$$A_m:=\left(\begin{array}{cc}{O}_{(m-1)\times (n-1)}& -{\mathbf{e}}_1\\ E_{n-1}& 0\end{array}\right)\in {\mathbb{R}}^{u\times n},$$

$${\lambda }_{m-1+s}=\det \left(\begin{array}{cccccc}{a}_{m-1}& a_{m-2}& \cdots & a_1& & \\ -1& \ddots & \ddots & & \ddots & \\ & \ddots & \ddots & \ddots & & a_1\\ & & \ddots & \ddots & \ddots & ⋮\\ & & & \ddots & \ddots & a_{m-2}\\ & & & & -1& a_{m-1}\end{array}\right)$$

$$N=M((a_1,\dots ,a_{m-1},-1),A)=\left(\begin{array}{ccccc}{a}_1& & & & 1\\ a_2& \ddots & & & \\ ⋮& \ddots & \ddots & & \\ a_{m-1}& & \ddots & \ddots & \\ -1& \ddots & & \ddots & a_1\\ & \ddots & \ddots & & a_2\\ & & \ddots & \ddots & ⋮\\ & & & -1& a_{m-1}\end{array}\right)$$

$$U=\left(\begin{array}{cccccc}{a}_{m-1}& a_{m-2}& \cdots & a_1& & 1\\ -1& \ddots & \ddots & & \ddots & \\ & \ddots & \ddots & \ddots & & a_1\\ & & \ddots & \ddots & \ddots & ⋮\\ & & & \ddots & \ddots & a_{m-2}\\ & & & & -1& a_{m-1}\end{array}\right)$$

$$0$$

$$\alpha(m,n)=\left\{\begin{array}[]{ll}{u/2\choose(m-1)/2}&\quad\mbox{if $m$ and $n$ are odd,}\\ {(u-1)/2\choose(m-2)/2}&\quad\mbox{if $m$ is even and $n$ is odd,}\\ {(u-1)/2\choose(m-1)/2}&\quad\mbox{if $m$ is odd and $n$ is even and}\\ 0&\quad\mbox{if $m$ and $n$ are even.}\end{array}\right.$$

Full text

Keywords: tensor rank, typical rank, tall tensor, semi-tall tensor, Bezout’s theorem, determinantal variety

MSC:15A69, 14P10, 14M12, 13C40

1 Introduction

Tensor rank is a subject which is widely studied in both pure and applied mathematics. A high dimensional array of datum is called a tensor in the field of data analysis. Precisely, let $N_{1}$, …, $N_{d}$ be positive integers. A $d$-dimensional array datum $T=(t_{i_{1}\cdots i_{d}})_{1\leq i_{j}\leq N_{j},1\leq j\leq d}$ is called a $d$-way tensor or simply a $d$-tensor of format $N_{1}\times\cdots\times N_{d}$. For a set $S$, the set of $N_{1}\times\cdots\times N_{d}$ tensors with entries in $S$ is denoted by $S^{N_{1}\times\cdots\times N_{d}}$.

Let $K$ be a field and $V_{j}$ an $N_{j}$-dimensional vector space over $K$ with fixed basis $v_{j1}$, …, $v_{jN_{j}}$ for $1\leq j\leq d$. Then there is a one to one correspondence between $K^{N_{1}\times\cdots\times N_{d}}$ and $V_{1}\otimes\cdots\otimes V_{d}$ by $(m_{i_{1}\cdots i_{d}})\leftrightarrow\sum_{i_{1}\cdots i_{d}}m_{i_{1}\cdots i_{d}}v_{1i_{1}}\otimes\cdots\otimes v_{di_{d}}$. A non-zero tensor corresponding to an element of $V_{1}\otimes\cdots\otimes V_{d}$ of the form $\bm{a}_{1}\otimes\cdots\otimes\bm{a}_{d}$ is called a rank 1 tensor. For a general tensor $T$ of format $N_{1}\times\cdots\times N_{d}$, the rank of $T$, denoted by $\mathrm{rank}T$, is by definition the minimum integer $r$ such that $T$ can be expressed as a sum of $r$ rank 1 tensors, where we set the empty sum to be zero. Thus, the rank is a measure of the complexity of a tensor. Further, for a 2-tensor, i.e., a matrix, the rank is identical with the one defined in linear algebra.

However, for the case where $d\geq 3$, the behavior of rank is much more complicated than the matrix case. In the matrix case, the rank is the maximum size of non-zero minors. Thus, if $K$ is an infinite field, the set of $m\times n$ matrices with rank $\min\{m,n\}$ form a Zariski dense open subset of $K^{m\times n}$. However, there are non-empty Euclidean open subsets of $\mathbb{R}^{2\times 2\times 2}$ such that one consists of rank 2 tensors and the other one consists of rank 3 tensors. In particular, it is not possible to characterize the rank of a tensor by vanishing and/or non-vanishing of polynomials.

Let $m$, $n$, $p$ be positive integers. If the set of rank $r$ tensors of format $p\times n\times m$ over $\mathbb{R}$ contains a non-empty Euclidean open subset of $\mathbb{R}^{p\times n\times m}$, we say that $r$ is a typical rank of $p\times n\times m$ tensors over $\mathbb{R}$. The set of typical ranks of $p\times n\times m$ tensors over $\mathbb{R}$ is denoted as ${\mathrm{trank}}_{\mathbb{R}}(p,n,m)$ or simply ${\mathrm{trank}}(p,n,m)$.

If the base field is $\mathbb{C}$, the set of tensors of format $p\times n\times m$ with rank at most $r$ contains a non-empty Zariski open set if and only if its Zariski closure is $\mathbb{C}^{p\times n\times m}$ (cf., Chevalley’s Theorem, see e.g., [Har92, p. 39]). Therefore, there exists exactly one “typical rank of $p\times n\times m$ tensors over $\mathbb{C}$”. This is called the generic rank of $p\times n\times m$ tensors over $\mathbb{C}$ and denoted as ${\mathrm{grank}}_{\mathbb{C}}(p,n,m)$ or simply ${\mathrm{grank}}(p,n,m)$.

It is fairly easy to show that ${\mathrm{grank}}(p,n,m)=\min{\mathrm{trank}}(p,n,m)$ (see e.g., [SSM16, Chapter 6]). Further $r\geq{\mathrm{grank}}(p,n,m)$ if and only if the $r$-th higher secant variety of the image of Segre embedding $\mathbb{P}_{\mathbb{C}}\mathbb{C}^{p}\times\mathbb{P}_{\mathbb{C}}\mathbb{C}^{n}\times\mathbb{P}_{\mathbb{C}}\mathbb{C}^{m}\to\mathbb{P}_{\mathbb{C}}\mathbb{C}^{p\times n\times m}$ is the whole space $\mathbb{P}_{\mathbb{C}}\mathbb{C}^{p\times n\times m}$, where $\mathbb{P}_{K}V$ denotes the projective space consisting of one dimensional subspaces of the $K$-vector space $V$. Thus, by counting the dimensions, we see that ${\mathrm{grank}}(p,n,m)\geq\lceil\frac{mnp}{m+n+p-2}\rceil$.

Suppose that $3\leq m\leq n\leq p$. Then $p\geq\lceil\frac{mnp}{m+n+p-2}\rceil$ if and only if $p\geq(m-1)(n-1)+1$. Catalisano, Geramita, and Gimigliano [CGG02] (see also [CGG08]) proved that if $(m-1)(n-1)+1\leq p\leq mn$, then ${\mathrm{grank}}(p,n,m)=p$. Thus, $\min{\mathrm{trank}}(p,n,m)=p$ in these cases. ten Berge [tB00] called a $p\times n\times m$-tensor with $(m-1)n<p<mn$ a tall array or a tall tensor and proved that ${\mathrm{trank}}(p,n,m)=\{p\}$ for these cases (see [SSM16, Chapter 6] for another proof). Here we define a $p\times n\times m$-tensor a semi-tall tensor if $(m-1)(n-1)+1\leq p\leq(m-1)n$. We [SSM13, SMS15, SMS17] studied the plurality of typical ranks of semi-tall tensors and proved that if $(m-1)(n-1)+2\leq p\leq(m-1)n$, then ${\mathrm{trank}}(p,n,m)=\{p,p+1\}$ if there exists a nonsingular  bilinear map $\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{mn-p}$ and ${\mathrm{trank}}(p,n,m)=\{p\}$ otherwise, where a bilinear map $\varphi\colon V_{1}\times V_{2}\to W$ is nonsingular  if $\varphi(x,y)=0$ implies $x=0$ or $y=0$.

We also showed in [SMS17] that the former part of the above mentioned result also valid in the case where $p=(m-1)(n-1)+1$. Therefore, the latter part of the above mentioned result in the case where $p=(m-1)(n-1)+1$ is left open. In this paper, we treat the case where $p=(m-1)(n-1)+1$ and show that ${\mathrm{trank}}(p,n,m)=\{p,p+1\}$ in several cases. In particular, we show that the latter part does not valid in the case where $p=(m-1)(n-1)+1$.

2 Preliminaries

Let $K$ be a field and $T=(t_{ijk})\in K^{\ell\times m\times n}$. For $1\leq k\leq n$, we set $T_{k}=(t_{ijk})\in K^{\ell\times m}$ and denote $T=(T_{1};\cdots;T_{n})$. For $P\in{\mathrm{GL}}(\ell,K)$ and $Q\in{\mathrm{GL}}(m,K)$, we set $PTQ=(PT_{1}Q;\cdots;PT_{n}Q)$. Note $\mathrm{rank}PTQ=\mathrm{rank}T$ by the definition of rank.

We first state the definition of the typical rank over $\mathbb{R}$.

Definition 2.1

If the set of rank $r$ tensors over $\mathbb{R}$ of format $\ell\times m\times n$ contains a non-empty Euclidean open subset of $\mathbb{R}^{\ell\times m\times n}$, then we say $r$ is a typical rank of $\ell\times m\times n$ tensors over $\mathbb{R}$. We denote the set of typical ranks of $\ell\times m\times n$ tensors over $\mathbb{R}$ by ${\mathrm{trank}}_{\mathbb{R}}(\ell,m,n)$ or simply ${\mathrm{trank}}(\ell,m,n)$.

By the definition of the rank, we see the following fact.

Lemma 2.2

Let $n_{1}$, $n_{2}$ and $n_{3}$ be positive integers. Then ${\mathrm{trank}}(n_{i_{1}},n_{i_{2}},n_{i_{3}})={\mathrm{trank}}(n_{1},n_{2},n_{3})$ for any permutation $i_{1}$, $i_{2}$, $i_{3}$ of 1, 2, 3.

Definition 2.3

For $T=(T_{1};\cdots;T_{n})\in K^{\ell\times m\times n}$, we set

[TABLE]

and

[TABLE]

and call flattenings of $T$.

By the correspondence $K^{\ell\times m\times n}\leftrightarrow V_{1}\otimes V_{2}\otimes V_{3}$, where $V_{1}$ (resp. $V_{2}$, $V_{3}$) is a vector space over $K$ of dimension $\ell$ (resp. $m$, $n$) with fixed basis, flattenings correspond to natural isomorphisms $V_{1}\otimes V_{2}\otimes V_{3}\to V_{1}\otimes(V_{2}\otimes V_{3})$ and $V_{1}\otimes V_{2}\otimes V_{3}\to(V_{1}\otimes V_{3})\otimes V_{2}$. In particular, $\mathrm{rank}(\mathrm{fl}_{1}(T))\leq\mathrm{rank}T$ and $\mathrm{rank}(\mathrm{fl}_{2}(T))\leq\mathrm{rank}T$.

Definition 2.4

For $M=(\bm{m}_{1},\ldots,\bm{m}_{n})=\begin{pmatrix}\bm{m}^{(1)}\\ \vdots\\ \bm{m}^{(\ell)}\end{pmatrix}\in K^{\ell\times n}$, we set $M_{\leq j}\mathrel{:=}(\bm{m}_{1},\ldots,\bm{m}_{j})$, ${}_{j<}M\mathrel{:=}(\bm{m}_{j+1},\ldots,\bm{m}_{n})$, $M^{\leq i}\mathrel{:=}\begin{pmatrix}\bm{m}^{(1)}\\ \vdots\\ \bm{m}^{(i)}\end{pmatrix}$ and ${}^{i<}M\mathrel{:=}\begin{pmatrix}\bm{m}^{(i+1)}\\ \vdots\\ \bm{m}^{(\ell)}\end{pmatrix}$.

Definition 2.5

Let $R$ be a commutative ring and $M\in R^{\ell\times n}$. We denote by $I_{t}(M)$ the ideal of $R$ generated by $t$-minors of $R$.

3 A condition of an $n\times p\times m$-tensor to be of rank $p$

From now on, let $m$, $n$ and $p$ be integers with $3\leq m\leq n$ and $(m-1)(n-1)+1\leq p\leq mn$. We set $u=mn-p$.

Fact 3.1

1. (1)

${\mathrm{grank}}(p,n,m)=p$. In particular, $\min{\mathrm{trank}}(p,n,m)=p$ **[CGG02]**. 2. (2)

If $p>(m-1)n$, then ${\mathrm{trank}}(p,n,m)=\{p\}$ **[tB00]**. 3. (3)

Suppose $p\leq(m-1)n$. If there exists a nonsingular  bilinear map $\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{u}$, then ${\mathrm{trank}}(p,n,m)=\{p,p+1\}$. Moreover, if $(m-1)(n-1)+2\leq p$, then the converse also hold true **[SSM13, SMS15, SMS17]**.

Therefore, the case where $p=(m-1)(n-1)+1$ and there is no nonsingular bilinear map $\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{u}$ is still left open. In the following, we consider the case where $p=(m-1)(n-1)+1$ and study if there are plural typical ranks of $p\times n\times m$ tensors over $\mathbb{R}$.

Before concentrating on the case where $p=(m-1)(n-1)+1$, we state notations and a criterion of an $n\times p\times m$ tensor to be of rank $p$ in the case where $(n-1)(m-1)+1\leq p\leq(m-1)n$. Note that ${\mathrm{trank}}(p,n,m)={\mathrm{trank}}(n,p,m)$.

Definition 3.2

We set $\mathscr{V}\mathrel{:=}\{T\in\mathbb{R}^{n\times p\times m}\mid\mathrm{fl}_{2}(T)^{\leq p}$ is nonsingular.$\}$, $\mathscr{O}\mathrel{:=}\{Y\in\mathbb{R}^{u\times n\times m}\mid{}_{p<}\mathrm{fl}_{1}(Y)$ is nonsingular.$\}$, $\sigma\colon\mathscr{V}\to\mathbb{R}^{u\times p}$, $\sigma(T)=({}^{p<}\mathrm{fl}_{2}(T))(\mathrm{fl}_{2}(T)^{\leq p})^{-1}$, $\nu\colon\mathscr{O}\to\mathbb{R}^{u\times p}$, $\nu(Y)=-({}_{p<}\mathrm{fl}_{1}(Y))^{-1}(\mathrm{fl}_{1}(Y)_{\leq p})$, $\tau\colon\mathbb{R}^{u\times p}\to\mathscr{V}$, $\tau(W)=\mathrm{fl}_{2}^{-1}\begin{pmatrix}E_{p}\\ W\end{pmatrix}$ and $\mu\colon\mathbb{R}^{u\times p}\to\mathscr{O}$, $\mu(W)=\mathrm{fl}_{1}^{-1}(W,-E_{u})$.

Remark 3.3

[TABLE]

and $\sigma(\tau(W))=\nu(\mu(W))=W$ for $W\in\mathbb{R}^{u\times p}$.

Definition 3.4

Let $\bm{x}=(x_{1},\ldots,x_{m})$ be a row vector of indeterminates, i.e., $x_{1}$, …, $x_{m}$ are independent indeterminates. For $A=(A_{1};\cdots;A_{m})\in\mathbb{R}^{u\times n\times m}$, we set $M(\bm{x},A)\mathrel{:=}x_{1}A_{1}+\cdots+x_{m}A_{m}\in\mathbb{R}[x_{1},\ldots,x_{m}]^{u\times n}$.

Definition 3.5

Let $\bm{a}$ and $\bm{b}$ be column vectors with entries in $\mathbb{R}$ of dimension $m$ and $n$ respectively. We set $\psi(\bm{a},\bm{b})\mathrel{:=}(\bm{a}\otimes_{\mathrm{Kr}}\bm{b})^{\leq p}$, where $\otimes_{\mathrm{Kr}}$ denotes the Kronecker product, i.e., if $\bm{a}=\begin{pmatrix}a_{1}\\ \vdots\\ a_{m}\end{pmatrix}$, then $\psi(\bm{a},\bm{b})=\begin{pmatrix}a_{1}\bm{b}\\ \vdots\\ a_{m-2}\bm{b}\\ a_{m-1}\bm{b}^{\leq p-(m-2)n}\end{pmatrix}$.

For $Y\in\mathbb{R}^{u\times n\times m}$, we define $U(Y)$ to be the vector subspace of $\mathbb{R}^{p}$ generated by $\{\psi(\bm{a},\bm{b})\mid M(\bm{a}^{\top},Y)\bm{b}=\bm{0}\}$.

Lemma 3.6

For $T\in\mathscr{V}$, $\mathrm{rank}T=p$ if and only if $\dim U(\mu(\sigma(T)))=p$.

• Proof

Set $\mu(\sigma(T))=W=(W_{1};\cdots;W_{m})$ and $\ell=u-n$. Then by [SMS17, Theorem 6.5 (1)$\iff$(3)], we see that $\mathrm{rank}T=p$ if and only if there are $B=(\bm{b}_{1},\ldots,\bm{b}_{p})\in\mathbb{R}^{n\times p}$ and $p\times p$ diagonal matrices $D_{1}$, …, $D_{m}$ such that

[TABLE]

First suppose that there are $B=(\bm{b}_{1},\ldots,\bm{b}_{p})\in\mathbb{R}^{n\times p}$ and $D_{1}$, …, $D_{m}$ which satisfy $(\ast)$. If we set $\bm{a}_{j}=(d_{j1},\ldots,d_{jm})^{\top}$ for $1\leq j\leq p$, then

[TABLE]

for $1\leq j\leq p$ and

[TABLE]

is nonsingular. Therefore $\dim U(\mu(\sigma(T)))=p$.

**Conversely, assume that $\dim U(\mu(\sigma(T)))=p$. Then there are $\bm{a}_{1}$, …, $\bm{a}_{p}\in\mathbb{R}^{m}$ and $\bm{b}_{1}$, …, $\bm{b}_{p}\in\mathbb{R}^{n}$ such that $\psi(\bm{a}_{1},\bm{b}_{1})$, …, $\psi(\bm{a}_{p},\bm{b}_{p})$ are linearly independent. Set $\bm{a}_{j}=(d_{j1},\ldots,d_{jm})^{\top}$ for $1\leq j\leq p$, $D_{k}=\mathrm{Diag}(d_{1k},\ldots,d_{pk})$ for $1\leq k\leq m$ and $B=(\bm{b}_{1},\ldots,\bm{b}_{p})\in\mathbb{R}^{n\times p}$. Then it is easily verified that $B$ and $D_{1}$, …, $D_{m}$ satisfy $(\ast)$.  **

4 Determinantal varieties and Bezout’s theorem

From now on, we consider the case where $p=(m-1)(n-1)+1$. Then $u=m+n-2$.

Definition 4.1

We set

[TABLE]

for $1\leq k\leq m-1$ and

[TABLE]

where $\bm{e}_{1}=\begin{pmatrix}1\\ 0\\ \vdots\\ 0\end{pmatrix}\in\mathbb{R}^{m-1}$ and $A=(A_{1};\cdots;A_{m})\in\mathbb{R}^{u\times n\times m}$.

The next fact is the key lemma of this paper.

Lemma 4.2

Let $y$ be an indeterminate and $a_{1},\ldots,a_{m-1}\in\mathbb{C}$. Then the following conditions are equivalent, where $V_{a}(I)$ denotes the affine variety defined by an ideal $I$.

1. (1)

$(a_{1},\ldots,a_{m-1},-1)\in V_{a}(I_{n}(M(\bm{x},A)))$. 2. (2)

$y^{m-1}-a_{m-1}y^{m-2}-\cdots-a_{2}y-a_{1}$* is a factor of $y^{u}+1$.*

In order to prove this lemma, we need some preparation. First we make the following

Definition 4.3

Let $y$ be an indeterminate and $\{\mu_{t}\}_{t\geq 1}$ an infinite sequence of complex numbers. We set $\mathscr{I}(\{\mu_{t}\}_{t\geq 1})=\{f(y)\in\mathbb{C}[y]\mid f(y)=\sum_{k}c_{k}y^{k}$, $\sum_{k}c_{k}\mu_{k+t}=0$ for any $t\geq 1\}$.

It is easily verified that $\mathscr{I}(\{\mu_{t}\}_{t\geq 1})$ is an ideal of $\mathbb{C}[y]$.

Now let $a_{1}$, …, $a_{m-1}\in\mathbb{C}$. Set $\lambda_{t}=0$ for $1\leq t\leq m-2$, $\lambda_{m-1}=1$ and

[TABLE]

for $s\geq 1$, where the right hand side is an $s\times s$-determinant (some $a_{i}$’s may not appear for small $s$).

By the first row expansion, we see the following

Lemma 4.4

For $t\geq m$, we have $\lambda_{t}=\sum_{k=1}^{m-1}a_{m-k}\lambda_{t-k}$.

Set $h(y)=y^{m-1}-a_{m-1}y^{m-2}-\cdots-a_{2}y-a_{1}$. By the above lemma, we see that $h(y)\in\mathscr{I}(\{\lambda_{t}\}_{t\geq 1})$. Further, since $\lambda_{t}=0$ for $1\leq t\leq m-2$ and $\lambda_{m-1}=1$, there is no polynomial in $\mathscr{I}(\{\lambda_{t}\}_{t\geq 1})$ whose degree is less than $m-1$ except the zero polynomial, i.e., $\mathscr{I}(\{\lambda_{t}\}_{t\geq 1})$ is generated by $h(y)$.

Set

[TABLE]

and for integers $c_{1}$, …, $c_{n}$ with $1\leq c_{1}<\cdots<c_{n}\leq u$, we denote by $[c_{1},\ldots,c_{n}]_{N}$ the maximal minor of $N$ consisting of the $c_{1}$-th, …, $c_{n}$-th rows of $N$.

Now we state the following

Lemma 4.5

Under the notation above, the following conditions are equivalent.

1. (1)

$\mathrm{rank}N<n$. 2. (2)

$[i,m,m+1,\ldots,u]_{N}=0$* for $1\leq i\leq m-1$.* 3. (3)

$\lambda_{u+t}=0$* for $1\leq t\leq m-2$ and $\lambda_{u+m-1}=-1$.* 4. (4)

$\lambda_{u+t}=-\lambda_{t}$* for $t\geq 1$.* 5. (5)

$y^{u}+1\in\mathscr{I}(\{\lambda_{t}\}_{t\geq 1})$.

• Proof

Let

[TABLE]

be a $u\times u$ matrix. Then $\det({}^{t<}_{t<}U)=\lambda_{u+m-1-t}+\delta_{0,t}$ for $0\leq t\leq u-1$, where $\delta_{0,t}$ is the Kronecker’s delta.

(1)$\Longrightarrow$(3)****: Since ${}_{m-2<}U=N$ and $\mathrm{rank}N<n$ by assumption, we see that $\det({}^{t<}_{t<}U)=0$ for $0\leq t\leq m-2$. Thus, we see that $\lambda_{u+m-1-t}+\delta_{0,t}=0$ for $0\leq t\leq m-2$.

(3)$\Longrightarrow$(2)****: We see by the first row expansions of $\det({}_{t<}^{t<}U)$ and $[t+1,m,\ldots,u]$ and the assumption that

[TABLE]

for $0\leq t\leq m-2$.

(2)$\Longrightarrow$**(1)**** follows from the fact that the last $n-1$ rows of $N$ are linearly independent, (3)$\iff$(4) follows from the facts that $\lambda_{1}=\cdots=\lambda_{m-2}=0$ and $\lambda_{m-1}=1$ and Lemma 4.4 and (4)$\iff$(5) follows from the definition of $\mathscr{I}(\{\lambda_{t}\}_{t\geq 1})$.  **

Since $(a_{1},\ldots,a_{m-1},-1)\in V_{a}(I_{n}(M(\bm{x},A)))$ if and only if $\mathrm{rank}N<n$ and $y^{u}+1\in\mathscr{I}(\{\lambda_{t}\}_{t\geq 1})$ if and only if $h(y)$ divides $y^{u}+1$, we see Lemma 4.2 by Lemma 4.5.

Now we recall the following facts about determinantal varieties (see e.g. [Har92, p.151 and p.243]).

Fact 4.6

*Let $X$ be a $u\times n$ matrix of indeterminates. Then the projective variety in $\mathbb{P}_{\mathbb{C}}\mathbb{C}^{u\times n}$ defined by $I_{n}(X)$ has degree ${u\choose n-1}$ and codimension $u-n+1$. *

Note that ${u\choose n-1}={u\choose m-1}$ and $u-n+1=m-1$ since $u=m+n-2$. Note also that there are ${u\choose m-1}$ monic factors of $y^{u}+1$ of degree $m-1$ in $\mathbb{C}[y]$.

In view of this fact, we make the following

Definition 4.7

For $B=(B_{1};\cdots;B_{m})\in\mathbb{C}^{u\times n\times m}$, we set $\varphi_{B}\colon\mathbb{C}^{m}\to\mathbb{C}^{u\times n}$, $(\alpha_{1},\ldots,\alpha_{m})\mapsto\alpha_{1}B_{1}+\cdots+\alpha_{m}B_{m}$.

Then by Lemma 4.2, Fact 4.6 and Bezout’s theorem, we see the following

Corollary 4.8

Let $\mathbb{P}_{\mathbb{C}}({\mathrm{Im}}\varphi_{A})$ be the linear subspace of $\mathbb{P}_{\mathbb{C}}\mathbb{C}^{u\times n}$ defined by ${\mathrm{Im}}\varphi_{A}$. Then $\mathbb{P}_{\mathbb{C}}({\mathrm{Im}}\varphi_{A})$ and $V_{p}(I_{n}(X))$ intersect transversely at ${u\choose m-1}$ distinct points, where $V_{p}(I)$ denotes the projective variety defined by the homogeneous ideal $I$.

By the implicit function theorem, we see the following fact.

Corollary 4.9

There is a Euclidean open neighborhood $\mathscr{U}$ of $A$ in $\mathbb{R}^{u\times n\times m}$ such that if $B\in\mathscr{U}$, then $\varphi_{B}$ is injective and the number of real points of $V_{p}(I_{n}(M(\bm{x},B)))\subset\mathbb{P}_{\mathbb{C}}\mathbb{C}^{m}$ is the number of real monic polynomials of degree $m-1$ which divide $y^{u}+1$, where we say a point of a complex projective space is real if all possible ratios of its homogeneous coordintes are real numbers.

We denote the number of real monic polynomials of degree $m-1$ which divides $y^{u}+1$ by $\alpha(m,n)$. Then we see the following

Lemma 4.10

[TABLE]

By replacing $\mathscr{U}$ to a smaller neighborhood if necessary, we may assume that $\mathrm{rank}(M(\bm{a},B)_{\leq n-1})=n-1$ for any $B\in\mathscr{U}$ and $\bm{a}\in\mathbb{R}^{m}\setminus\{\bm{0}\}$ (cf. [SMS17, Corollary 4.20]). Then we have the following fact.

Lemma 4.11

Suppose $B\in\mathscr{U}$. Then $\#\{[\psi(\bm{a},\bm{b})]\in\mathbb{P}_{\mathbb{R}}\mathbb{R}^{p}\mid M(\bm{a},B)\bm{b}=\bm{0},\bm{a}\in\mathbb{R}^{m}\setminus\{\bm{0}\},\bm{b}\in\mathbb{R}^{n}\setminus\{\bm{0}\}\}=\alpha(m,n)$, where $[\bm{x}]$ denotes the point of $\mathbb{P}_{\mathbb{R}}\mathbb{R}^{p}$ defined by $\bm{x}\in\mathbb{R}^{p}\setminus\{\bm{0}\}$.

5 Plural typical ranks of some formats of 3-tensors

In this section, we show that in certain formats of 3-tensors, there are plural typical ranks. We use the notation of the previous section.

First we recall the following fact.

Fact 5.1** ([SMS17, Proposition 2.4, Lemma 3.5, Theorems 7.3 and 8.1])**

If $m-1$ and $n-1$ are not bit-disjoint, then ${\mathrm{trank}}(n,p,m)=\{p,p+1\}$, where two positive integers are bit-disjoint if there are no 1’s in the same place of their binary notation.

Example 5.2

${\mathrm{trank}}(n,p,m)=\{p,p+1\}$ in the following cases.

1. (1)

Both $m$ and $n$ are even. 2. (2)

$m=5$ and $n\equiv 5,6,7,8\pmod{8}$. 3. (3)

$m=6$ and $n\equiv 2,4,5,6,7,8\pmod{8}$. 4. (4)

$m=7$ and $n\equiv 3,4,5,6,7,8\pmod{8}$. 5. (5)

$m=8$ and $n\equiv 2,3,4,5,6,7,8\pmod{8}$. 6. (6)

$m=9$ and $n\equiv 9,10,11,12,13,14,15,16\pmod{16}$.

Set $A^{\prime\prime}=(A_{2};\cdots;A_{m-2};A_{m};-A_{m-1};-A_{1})$. Then there is a permutation matrix $P\in{\mathrm{GL}}(u,\mathbb{R})$ such that ${}_{p<}\mathrm{fl}_{1}(PA^{\prime\prime})=-E_{u}$. Set $A^{\prime}=PA^{\prime\prime}$ and $W_{0}=\mathrm{fl}_{1}(A^{\prime})_{\leq p}$. Further set $\rho\colon\mathbb{R}^{u\times n\times m}\to\mathbb{R}^{u\times n\times m}$, $B=(B_{1};\cdots;B_{m})\mapsto P(B_{2};\cdots;B_{m-2};B_{m};-B_{m-1};-B_{1})$ and ${\mathcal{U}}=\mu^{-1}(\rho(\mathscr{U}))$, where $\mu$ is the map defined in Definition 3.2. Then ${\mathcal{U}}$ is an open neighborhood of $W_{0}$. Further, we see the following fact by Lemma 4.11.

Lemma 5.3

If $T\in\sigma^{-1}({\mathcal{U}})$, then $\#\{[\psi(\bm{a},\bm{b})]\in\mathbb{P}_{\mathbb{R}}\mathbb{R}^{p}\mid M(\bm{a},\mu(\sigma(T)))\bm{b}=\bm{0},\bm{a}\in\mathbb{R}^{m}\setminus\{\bm{0}\},\bm{b}\in\mathbb{R}^{n}\setminus\{\bm{0}\}\}=\alpha(m,n)$.

Since $\sigma^{-1}({\mathcal{U}})$ contains $\tau(W_{0})$, $\sigma^{-1}({\mathcal{U}})$ is not an empty set. Therefore, we see by Lemma 3.6 that if $\alpha(m,n)<p$, then there exists a non-empty Euclidean open subset $\sigma^{-1}({\mathcal{U}})$ of $\mathbb{R}^{n\times p\times m}$ consisting of tensors of rank larger than $p$. Further, since we see by [SMS17, Theorem 8.1] that typical ranks of $n\times p\times m$ tensors are less than or equal to $p+1$, we see the following fact.

Lemma 5.4

If $\alpha(m,n)<p$, then ${\mathrm{trank}}(n,p,m)=\{p,p+1\}$.

Now we state the following

Theorem 5.5

Suppose $3\leq m\leq n$ and $p=(m-1)(n-1)+1$. Then ${\mathrm{trank}}(n,p,m)=\{p,p+1\}$ in the following cases.

1. (1)

$m=3$* or $m=4$.* 2. (2)

$m=5$* and $n\leq 26$ or $n=28$.* 3. (3)

$m=6$* and $n\leq 34$.* 4. (4)

$m=7$* and $n\leq 16$.* 5. (5)

$m=8$* and $n\leq 16$.*

• Proof

**By Lemma 4.10 and computation, we see that $\alpha(m,n)<p$ in the following cases: (1) $m=3$ or $m=4$, (2) $m=5$ and $n\leq 26$ or $n=28$, (3) $m=6$ and $n\leq 34$, (4) $m=7$ and $n\leq 12$, (5) $m=8$ and $n\leq 14$ and (6) $m=9$ and $n=10$. Thus, we see the result by Example 5.2 and Lemma 5.4.  **

Remark 5.6

Set $m=3$ and $n=5$. Then $m-1$ and $n-1$ are bit-disjoint. However, by Theorem 5.5, we see that ${\mathrm{trank}}(n,p,m)=\{p,p+1\}$. Thus the converse of Fact 5.1 does not valid.