# Deflation conjecture and local dimensions of Brent equations

Xin Li  Department of Mathematics, Zhejiang University of Technology, Hangzhou
310023, P. R. China<EMAIL_ADDRESS>(Xin Li) ,  Liping Zhang  School of
Mathematical Sciences, Qufu Normal University, Qufu 273165, P. R. China
<EMAIL_ADDRESS>(Liping Zhang) and  Yifen Ke  School of Mathematics and
Statistics, Fujian Normal University, 350117 Fuzhou, P.R. China
<EMAIL_ADDRESS>(Yifen Ke)

###### Abstract.

In this paper, a classical deflation process raised by Dayton, Li and Zeng is
realized for the Brent equations, which provides new bounds for local
dimensions of the solution set. Originally, this deflation process focuses on
isolated solutions. We generalize it to the case of irreducible components and
a related conjecture is given. We analysis its realization and apply it to the
Brent equations. The decrease of the nullities is easily observed. So the
deflation process can be served as a useful tool for determining the local
dimensions. In addition, our result implies that along with the decrease of
the tensor rank, the singular solutions will become more and more.

###### Key words and phrases:

Deflation, Singularity, Brent equations, Local dimension, Jacobian

###### 2020 Mathematics Subject Classification:

Primary 14Q15; Secondary 65H10

##  1\. Introduction

In [18, Sect. 8], Heule et al. raised the question: what is the dimension of
the solution set of Brent equations? Following our last paper [28], using the
deflation process raised in [7, 8], we continue discussing the local
dimensions of solutions of Brent equations.

Let $F(\textbf{x}):~{}\mathbb{C}^{n}\to\mathbb{C}^{m}$ be a polynomial system
with solution set $\textbf{V}(F)$. Let $J(\textbf{x})\in\mathbb{C}^{m\times
n}$ be the Jacobian of $F(\textbf{x})$. Given a solution
$\hat{\textbf{x}}\in\textbf{V}(F)$ and let
$\dim_{\hat{\textbf{x}}}\textbf{V}(F)$ denote its local dimension. The
Jacobian criterion of [11, Theorem 4.1.12] tells us that the nullity of
$J(\hat{\textbf{x}})$ is an upper bound of
$\dim_{\hat{\textbf{x}}}\textbf{V}(F)$:

$\dim_{\hat{\textbf{x}}}\textbf{V}(F)\leq
n-\operatorname{rank}~{}\left(J(\hat{\textbf{x}})\right).$ (1.1)

Usually, (1.1) may be not an equality, that is,

$\dim_{\hat{\textbf{x}}}\textbf{V}(F)<n-\operatorname{rank}~{}\left(J(\hat{\textbf{x}})\right).$
(1.2)

When (1.2) happens, in numerical algebraic geometry, the solution
$\hat{\textbf{x}}$ is said to be singular (or ultrasingular) [3, 32, 37]. It
is well-known that the locally quadratical convergence of Newton’s method is
only guaranteed when (1.1) is an equality (see e.g. [37]). Near the
neighbourhood of singular solutions, Newton’s method often loses its locally
quadratical convergence. In order to recover Newton’s method and turn (1.1)
into an equality, the “deflation” technique is introduced, which is also a
useful tool for the estimation of local dimensions.

The deflation technique was firstly introduced by Ojika [30, 31] for the
isolated solution. After that many improvements and generalizations were given
[7, 8, 16, 26]. Given a polynomial system and a solution. A deflation process
produces a new expanded polynomial system and a new expanded solution. For a
generic (smooth) point, the nullity of Jacobian of the new expanded solution
can be much closer to its local dimension. For isolated singular solutions, it
was shown in [7, 8, 26] that the nullity will decrease to zero after a finite
number of steps. For a positive dimensional irreducible component, Hauenstein
and Wampler introduced a new form of deflation called strong deflation [16],
and they refer to the deflation process raised in [7, 8, 26] as weak
deflation. Through strong deflation process, Hauenstein and Wampler proved
that their deflation sequence will decrease to the local dimension of the
generic (smooth) points of the irreducible component [16, Lemma 4.7]. For the
other (singular) points, the stabilized number of the (strong) deflation
sequence will be no greater than their local dimension [16, Sect.5.3].

For the (weak) deflation raised in [7, 8, 26], it focuses on the isolated
solutions. However, it is not hard to see that it is also suitable to the
positive dimensional solutions. In [37, Sect. 9], in order to recover the
quadratic convergence rate of Newton’s iteration at ultrasingular zeroes, Zeng
conjectured that for generic (smooth) points, the weak deflation process will
also terminate at the local dimension of positive dimensional solutions after
a finite number of steps. That is, similar properties of strong deflation in
[16] also hold for the weak deflation process raised in [7, 8] under the
positive dimensional generalization.

By Lemma 4.7 of [16], the strong deflation can turn (1.1) into an equality.
However, it will introduce much more new variables and polynomials than the
weak deflation. So the implementation of strong deflation is not so practical
for large polynomial systems, such as the Brent equations. In addition, the
local dimension determination methods raised in [1, 25, 36] are also not
suitable to large polynomial systems, too. Let $\langle m,n,p\rangle$ denote
the matrix multiplication tensor. We find that it is convenient to calculate
the first 4 numbers of the weak deflation sequences for $mnp\leq 125$. So our
results imply that the weak deflation is a useful tool for determining the
local dimensions for polynomial systems arising in the field of bilinear
complexity [24].

### Organization and main results

Firstly, after some preliminaries, more details of the conjecture in [37] are
given, see Section 2 and Conjecture 1 in Section 3. Some elementary results
are discussed, which are similar to those in [16]. Secondly, the constructions
of the Jacobians in the first three steps of the deflation process in [7, 8]
are analysed, which are also suitable for other steps, see Section 4. Thirdly,
numerical experiments are given for the solutions of the Brent equations, see
Section 5. The first 4 numbers of the deflation sequences of many known
solutions (see e.g. [14, 17]) are calculated, which provide bounds for their
local dimensions. Some interesting results are found. For example, the local
dimension of Strassen’s classical solution should be 23, see Section 5.1. For
the solutions provided in [17], after the first deflation step, we find that
most solutions are singular. This provides a new numerical evidence that
finding the tensor decomposition of low rank is hard [12, 19]. Some remarks
and open problems are given in Section 6.

Our numerical results are obtained by Matlab2022a. Some codes and data can be
found on [29]. The numerical experiments are taken on the server of ZJUT with
the configuration: Intel(R) Xeon(R) Silver 4210R CPU running at 2.40GHz plus
2.39 GHz and 256 GB memory.

##  2\. Preliminaries

Throughout this paper, $\mathbb{C}$ is the complex field. The vector space of
$m\times n$ complex matrices is denoted by $\mathbb{C}^{m\times n}$. Let
$\mathbb{C}^{n}$ (or $\mathbb{C}^{n\times 1}$) denote the $n$-dimensional
column vector space over $\mathbb{C}$. Given a matrix $A\in\mathbb{C}^{m\times
n}$. Let $\operatorname{rank}(A)$ denote the rank of $A$. The nullity of $A$
is

$n-\operatorname{rank}(A),$

which is the dimension of the null space of $A$ and denoted by
$\mathscr{N}(A)$. In the following, the deflation process always refers to the
deflation process raised in [7, 8] or its generalization.

Let $F:\mathbb{C}^{n}\to\mathbb{C}^{m}$ be a polynomial system. So there exist
$m$ polynomials $f_{i}:\mathbb{C}^{n}\to\mathbb{C}$ ($i=1,2,...m$) such that
$F=[f_{1},f_{2},...f_{m}]^{\textbf{T}}$. The solution set of $F$ over
$\mathbb{C}$ is called _an algebraic set_ and denoted by $\textbf{V}(F)$, that
is,

$\textbf{V}(F)=\\{\textbf{x}\in\mathbb{C}^{n}|f_{1}(\textbf{x})=f_{2}(\textbf{x})=\cdots=f_{m}(\textbf{x})=0\\}.$

Let $J(\textbf{x})$ denote the Jacobian matrix of $F(\textbf{x})$. So
$J(\textbf{x})\in\mathbb{C}^{m\times n}$. An algebraic set V is reducible if
there exist nonempty algebraic sets $V_{1},V_{2}\subsetneqq\textbf{V}$ such
that $\textbf{V}=V_{1}\cup V_{2}$. V is irreducible if no such decomposition
exists. Each algebraic set V can be decomposed into a unique finite union of
irreducible algebraic subsets:

$\textbf{V}=V_{1}\cup V_{2}\cup\cdots\cup V_{r},$

where $V_{i}$ is called the irreducible component of V. For an irreducible
component $V$, the set of manifold points in $V$ is connected and its
dimension as a complex manifold is defined to be the dimension of $V$, denoted
$\dim V$ [4, 32]. If $\dim V=k$, $V$ is said to be a $k$-dimensional
irreducible component. The local dimension of a point
$\textbf{x}\in\textbf{V}$ is the maximal dimension of the irreducible
components of V that contain x, which is denoted by
$\dim_{\textbf{x}}\textbf{V}$. For an irreducible component $V$, a property
$P$ holds generically on $V$ if there is a nonempty Zariski open set
$U\subseteq V$ such that $P$ holds on $V$. Each point of $U$ is called a
generic point of $V$ with respect to $P$.

Considering $\textbf{V}(F)$ as a geometric object in $\mathbb{C}^{n}$, we have
the following definition which makes a slight change of [15, Def. 2.1].

######  Definition 2.1.

A point $\textbf{x}_{0}\in\textbf{V}(F)$ is called _geometric smooth_ , if
there exists a unique irreducible component $V\subseteq\textbf{V}(F)$
containing $\textbf{x}_{0}$ such that

$\dim T_{\textbf{x}_{0}}(V)=\dim V$

where $T_{\textbf{x}_{0}}(V)$ is the tangent space of $V$ at $\textbf{x}_{0}$.
Otherwise, $\textbf{x}_{0}$ is called a _geometric singular_ point of
$\textbf{V}(F)$. The set of all geometric singular points of $\textbf{V}(F)$
is denoted by $\operatorname{Sing}_{G}(\textbf{V}(F))$.

It is well-known that geometric smooth points are generic on an irreducible
component. On the other hand, we have the following definition.

######  Definition 2.2.

Let $F(\textbf{x})$ be a polynomial system whose solution set and Jacobian are
$\textbf{V}(F)$ and $J(\textbf{x})$, respectively. A point
$\textbf{x}_{0}\in\textbf{V}(F)$ is called _numerical smooth_ , if there
exists a unique irreducible component $V\subseteq\textbf{V}(F)$ containing
$\textbf{x}_{0}$ such that

$\mathscr{N}(J(\textbf{x}_{0}))=\dim V.$

Otherwise, $\textbf{x}_{0}$ is called a _numerical singular_ point of
$\textbf{V}(F)$. The set of all numerical singular points of $\textbf{V}(F)$
is denoted by $\operatorname{Sing}_{N}(\textbf{V}(F))$.

######  Remark 2.3.

It is well-known that if
$\textbf{x}_{0}\in\operatorname{Sing}_{N}(\textbf{V}(F))$, then

$\mathscr{N}(J(\textbf{x}_{0}))>\dim_{\textbf{x}_{0}}\textbf{V}(F).$ (2.1)

Comparing Definition 2.1 and 2.2, we know that numerical smooth points are
geometric smooth. However, numerical singular points may contain geometric
smooth points. For example, the irreducible components with multiplicity
greater than one [3, Example 8.10].

For the polynomial system $F=[f_{1},f_{2},...,f_{m}]^{\textsc{T}}$, let

$\langle
f_{1},f_{2},...,f_{m}\rangle\subseteq\mathbb{C}[x_{1},x_{2},...,x_{n}]$

denote the ideal generated by $f_{i}$. On the other hand, let
$\textbf{I}(\textbf{V}(F))$ denote the vanishing ideal on $\textbf{V}(F)$:

$\textbf{I}(\textbf{V}(F))=\\{f\in\mathbb{C}[x_{1},x_{2},...,x_{n}]|f(\textbf{x})=0~{}\text{for~{}all}~{}\textbf{x}\in\textbf{V}(F)\\}.$

By Proposition 2 of [6, Sect.6 Chapter 9] and Corollary 9 of [6, Sect.7
Chapter 9], we know that if

$\langle f_{1},f_{2},...,f_{m}\rangle=\textbf{I}(\textbf{V}(F)),$ (2.2)

then Definition 2.1 and 2.2 are equivalent, that is, the geometric smooth
(resp. singular) points and numerical smooth (resp. singular) points are the
same. However, for any polynomial system
$F=[f_{1},f_{2},...,f_{m}]^{\textsc{T}}$, it is well-known that

$\langle f_{1},f_{2},...,f_{m}\rangle\subseteq\textbf{I}(\textbf{V}(F)),$

and deciding if they are equal is usually hard. So, for those polynomial
systems we are interested, the geometric smooth points may be numerical
singular.

The definition of semiregularity and ultrasingularity of [37] are formalised
to irreducible components as follows. For the definition of intersection
multiplicity of an irreducible component, we refer to [3, Sec. 11.3].

######  Definition 2.4.

For a polynomial system $F$, given a $k$-dimensional irreducible component
$V\subseteq\textbf{V}(F)$. If all geometric smooth points are numerical
smooth, then we call $V$ is a _$k$ -dimensional semiregular irreducible
component_. That is, there is a nonempty Zariski open set $U\subseteq V$ such
that

$k=\dim_{\textbf{x}}\textbf{V}(F)=T_{\textbf{x}}V=\mathscr{N}(J(\textbf{x})),$

for all $\textbf{x}\in U$. Each point in $U$ is called a $k$-dimensional
semiregular solution.

On the other hand, if all geometric smooth points are numerical singular and
have the same intersection multiplicity, then we call $V$ is a _$k$
-dimensional ultrasingular irreducible component_. That is, there is a
nonempty Zariski open set $U\subseteq V$ and a constant positive number
$n_{0}$ such that

$k=\dim_{\textbf{x}}\textbf{V}(F)=T_{\textbf{x}}V<\mathscr{N}(J(\textbf{x}))=n_{0},$

for all $\textbf{x}\in U$. Each point in $U$ is called a $k$-dimensional
ultrasingular solution.

##  3\. Deflation conjecture for the irreducible component

The deflation process raised in [7, 8] focuses on isolated singular solutions.
In this section, we generalize it to the case of $k$-dimensional ultrasingular
irreducible components. More details of the conjecture in [37] for
ultrasingular solutions are given. Similar discussions are given in [16].

###  3.1. The first step of the deflation process

Let $V\subseteq\textbf{V}(F)$ be a $k$-dimensional ultrasingular irreducible
component, which has a nonzero Zariski open set

$U_{0}\subseteq V\quad\text{with}\quad k<\mathscr{N}(J(\textbf{x}))=n_{0}$
(3.1)

for all $\textbf{x}\in U_{0}$. Fix a $k$-dimensional ultrasingular solution
$\hat{\textbf{x}}_{1}\in U_{0}$. So we have

$k<\mathscr{N}\left(J(\hat{\textbf{x}}_{1})\right)=n_{0}.$ (3.2)

Then, choose $R_{1}\in\mathbb{C}^{n_{0}\times n}$ and a vector
$d_{1}\in\mathbb{C}^{n_{0}}$ randomly, such that the matrix

$\left[\begin{array}[]{c}J(\hat{\textbf{x}}_{1})\\\ R_{1}\\\
\end{array}\right],$ (3.3)

is of full (column) rank, and therefore the linear system

$\left[\begin{array}[]{c}J(\hat{\textbf{x}}_{1})\\\ R_{1}\\\
\end{array}\right]\textbf{x}=\left[\begin{array}[]{c}\textbf{0}\\\ d_{1}\\\
\end{array}\right]$ (3.4)

has a unique solution $\textbf{x}_{2}=\hat{\textbf{x}}_{2}\neq\textbf{0}$. By
the discussion in Section 3.1 of [16], we can see that besides
$\hat{\textbf{x}}_{1}$, for almost all $\hat{\textbf{x}}\in V$, the equation
(3.4) also has a unique solution. Let $\textbf{x}_{2}\in\mathbb{C}^{n}$ be new
variables and fixing $R_{1}$ for all points in $V$. Then the first step of
deflation of $F$ is defined by:

$F_{1}(\textbf{x}_{1},\textbf{x}_{2})=\left[\begin{array}[]{c}F(\textbf{x}_{1})\\\
\left[\begin{array}[]{c}J(\textbf{x}_{1})\\\ R_{1}\\\
\end{array}\right]\textbf{x}_{2}-\left[\begin{array}[]{c}\textbf{0}\\\
d_{1}\\\ \end{array}\right]\end{array}\right],~{}\text{for
all}~{}\textbf{x}_{1}\in V.$ (3.5)

So ($\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2}$) is a solution of the new
polynomial system $F_{1}(\textbf{x}_{1},\textbf{x}_{2})$.

When $m\geq n$, let $\pi_{n}:~{}\mathbb{C}^{m}\to\mathbb{C}^{n}$ denote the
natural projection map, which is defined by

$\pi_{n}:(x_{1},x_{2},...x_{m})\mapsto(x_{1},x_{2},...x_{n}).$ (3.6)

Let $\textbf{V}(F_{1})$ denote the solution set of $F_{1}$. For $U_{0}$
defined in (3.1), let

$W_{1}=\\{y\in\textbf{V}(F_{1})|\pi_{n}(y)\in
U_{0}\\}\subset\textbf{V}(F_{1}),$ (3.7)

and $\overline{W_{1}}$ be the Zariski closure of $W_{1}$. By the discussion of
Lemma 4.6 of [16], we also have the following lemma.

######  Lemma 3.1.

$\overline{W_{1}}$ is an irreducible component of $\textbf{V}(F_{1})$ which is
generically isomorphic to $V$. In particular,

$\dim\overline{W_{1}}=\dim V=k.$

Let $J_{1}(\textbf{x}_{1},\textbf{x}_{2})$ denote the Jacobian of
$F_{1}(\textbf{x}_{1},\textbf{x}_{2})$. Then by Proposition 4 and Remark 5 of
[27], with probability one, there is an open neighbourhood $\tilde{U}_{1}$ of
$(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2})$ such that the nullity

$\mathscr{N}(J_{1}(\textbf{x}_{1},\textbf{x}_{2}))=\mathscr{N}(J_{1}(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2}))=n_{1},$
(3.8)

for all $(\textbf{x}_{1},\textbf{x}_{2})\in\tilde{U}_{1}$. Then we let

$U_{1}=\overline{W}_{1}\cap\tilde{U}_{1}.$ (3.9)

###  3.2. The second and subsequent steps

In (3.8), if $n_{1}=k$, then ($\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2}$) is
a $k$-dimensional semiregular zero of $F_{1}(\textbf{x}_{1},\textbf{x}_{2})$,
so the singularity of $F(\textbf{x})$ at $\hat{\textbf{x}}_{1}$ and its
neighbourhood is “deflated”. However,
($\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2}$) may still be an ultrasingular
zero of $F_{1}(\textbf{x}_{1},\textbf{x}_{2})$, that is, $n_{1}>k$. In this
case, we can repeat the same deflation process above for $F_{1}$ and obtain a
new polynomial system $F_{2}$. More precisely, we can find and fix
$R_{2}\in\mathbb{C}^{n_{1}\times 2n}$ and $d_{2}\in\mathbb{C}^{n_{1}}$ such
that

$F_{2}(\textbf{x}_{1},\textbf{x}_{2},\textbf{x}_{3},\textbf{x}_{4})=\left[\begin{array}[]{c}F_{1}(\textbf{x}_{1},\textbf{x}_{2})\\\
\left[\begin{array}[]{c}J_{1}(\textbf{x}_{1},\textbf{x}_{2})\\\ R_{2}\\\
\end{array}\right]\left[\begin{array}[]{c}\textbf{x}_{3}\\\ \textbf{x}_{4}\\\
\end{array}\right]-\left[\begin{array}[]{c}\textbf{0}\\\ d_{2}\\\
\end{array}\right]\end{array}\right]~{}\text{for all}~{}\textbf{x}_{1}\in V,$
(3.10)

and the solution ($\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2}$) extends to a
solution
($\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2},\hat{\textbf{x}}_{3},\hat{\textbf{x}}_{4}$)
of $F_{2}$, uniquely. Let $\textbf{V}(F_{2})$ denote the solution set of
$F_{2}$. For $U_{0}$ and $\pi_{n}$ defined in (3.1) and (3.6) respectively,
let

$W_{2}=\\{y\in\textbf{V}(F_{2})|\pi_{n}(y)\in
U_{0}\\}\subset\textbf{V}(F_{2}),$ (3.11)

and $\overline{W_{2}}$ be the Zariski closure of $W_{2}$. Then following the
same reason as Lemma 3.1, we also have $\dim\overline{W_{2}}=k$. Just as
(3.9), there is also a neighbourhood $U_{2}\subseteq\overline{W_{2}}$ of
$(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2},\hat{\textbf{x}}_{3},\hat{\textbf{x}}_{4})$
such that the nullity

$\mathscr{N}(J_{2}(\textbf{x}_{1},\textbf{x}_{2},\textbf{x}_{3},\textbf{x}_{4}))=\mathscr{N}(J_{2}(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2},\hat{\textbf{x}}_{3},\hat{\textbf{x}}_{4}))=n_{2},$
(3.12)

for all $(\textbf{x}_{1},\textbf{x}_{2},\textbf{x}_{3},\textbf{x}_{4})\in
U_{2}$.

Generally, assume ($\hat{\textbf{x}}_{1},...,\hat{\textbf{x}}_{2^{i}}$) is an
ultrasingular zero of $F_{i}(\textbf{x}_{1},...,\textbf{x}_{2^{i}})$ after $i$
steps of deflation. Let

$W_{i}=\\{y\in\textbf{V}(F_{i})|\pi_{n}(y)\in
U_{0}\\}\subset\textbf{V}(F_{i}),$ (3.13)

and $\overline{W_{i}}$ be the Zariski closure of $W_{i}$. Then
$\dim\overline{W_{i}}=k$ and the Jacobian
$J_{i}(\textbf{x}_{1},...,\textbf{x}_{2^{i}})$ maintains a nullity $n_{i}>k$,
in a neighborhood $U_{i}\subset\overline{W_{i}}$ of
$(\hat{\textbf{x}}_{1},...,\hat{\textbf{x}}_{2^{i}})$. That is,

$\mathscr{N}\left(J_{i}(\textbf{x}_{1},...,\textbf{x}_{2^{i}})\right)=\mathscr{N}\left(J_{i}(\hat{\textbf{x}}_{1},...,\hat{\textbf{x}}_{2^{i}})\right)=n_{i},$
(3.14)

for all $(\textbf{x}_{1},...,\textbf{x}_{2^{i}})\in U_{i}$. Then the next
deflation step expands the system to

$F_{i+1}(\textbf{x}_{1},\textbf{x}_{2},...,\textbf{x}_{2^{i+1}})=\left[\begin{array}[]{c}F_{i}(\textbf{x}_{1},...,\textbf{x}_{2^{i}})\\\
\left[\begin{array}[]{c}J_{i}(\textbf{x}_{1},...,\textbf{x}_{2^{i}})\\\
R_{i+1}\\\ \end{array}\right]\left[\begin{array}[]{c}\textbf{x}_{2^{i}+1}\\\
\vdots\\\ \textbf{x}_{2^{i+1}}\\\
\end{array}\right]-\left[\begin{array}[]{c}\textbf{0}\\\ d_{i+1}\\\
\end{array}\right]\end{array}\right],$ (3.15)

for all $\textbf{x}_{1}\in V$, where $R_{i+1}\in\mathbb{C}^{n_{i}\times
2^{i}n}$ and $d_{i+1}\in\mathbb{C}^{n_{i}}$ are (randomly) chosen and fixed
such that almost all points of $\overline{W_{i}}$ extend to a solution of
$F_{i+1}$, uniquely.

######  Theorem 3.2.

In the deflation process above, for $k$, $n_{0}$ and $n_{i}$, we have

$k\leq n_{i}\leq\cdots\leq n_{1}\leq n_{0},$ (3.16)

for each $i\geq 1$. Moreover, suppose that
$J_{i}(\hat{\textbf{x}}_{1},...,\hat{\textbf{x}}_{2^{i}})\in\mathbb{C}^{a_{i}\times
2^{i}n}$ for some $a_{i}$. Let
$J_{i}(\hat{\textbf{x}}_{1})\in\mathbb{C}^{a_{i}\times n}$ be the submatrix of
$J_{i}(\hat{\textbf{x}}_{1},...,\hat{\textbf{x}}_{2^{i}})$ which is obtained
by taking the first $n$ columns of
$J_{i}(\hat{\textbf{x}}_{1},...,\hat{\textbf{x}}_{2^{i}})$. Then

$n_{i}=n-rank~{}J_{i}(\hat{\textbf{x}}_{1}).$ (3.17)

###### Proof.

By Lemma 3.1, we can see that without changing its dimension, after each step
of deflation, the $k$-dimensional ultrasingular irreducible component is
embedded into a new $k$-dimensional irreducible component. So we have $k\leq
n_{i}$ for each $i$.

For $n_{i}\leq\cdots\leq n_{1}\leq n_{0}$, it suffices to show that $n_{1}\leq
n_{0}$ by the recursive process of the deflation. From (3.5), we know that

$J_{1}(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2})=\left[\begin{array}[]{cc}J(\hat{\textbf{x}}_{1})&\textbf{0}\\\
J\left((J(\hat{\textbf{x}}_{1})\hat{\textbf{x}}_{2}),\hat{\textbf{x}}_{1}\right)&J(\hat{\textbf{x}}_{1})\\\
\textbf{0}&R_{1}\\\ \end{array}\right]\in\mathbb{C}^{(2m+n_{0})\times 2n},$
(3.18)

where
$J\left((J(\hat{\textbf{x}}_{1})\hat{\textbf{x}}_{2}),\hat{\textbf{x}}_{1}\right)$
is the partial Jacobian of $J(\textbf{x}_{1})\textbf{x}_{2}$ with respect to
$\textbf{x}_{1}$ which takes the value at
$(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2})$. By the choice of $R_{1}$, we
have

$rank~{}J_{1}(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2})\geq n_{0}+n.$

So $n_{1}=2n-rank~{}J_{1}(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2})\leq
n_{0}.$

For (3.17), when $i=0$, the proof is followed by
$J_{0}(\hat{\textbf{x}}_{1})=J(\hat{\textbf{x}}_{1})$. For $i\geq 1$, from
(3.15) and the choice of $R_{i}$ in each step, we only show the case when
$i=1$. The others are similar too.

When $i=1$, for $J_{1}(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2})$, from
(3.18), we have

$J_{1}(\hat{\textbf{x}}_{1})=\left[\begin{array}[]{c}J(\hat{\textbf{x}}_{1})\\\
J\left((J(\hat{\textbf{x}}_{1})\hat{\textbf{x}}_{2}),\hat{\textbf{x}}_{1}\right)\\\
\textbf{0}\\\ \end{array}\right].$ (3.19)

On the other hand, by choice of $R_{1}$, we can see that the submatrix
obtained by taking columns $n+1$ to $2n$:

$\left[\begin{array}[]{c}\textbf{0}\\\ J(\hat{\textbf{x}}_{1})\\\ R_{1}\\\
\end{array}\right]$

is of full column rank $n$. So

$n_{1}=2n-rank~{}J_{1}(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2})=n-rank~{}J_{1}(\hat{\textbf{x}}_{1}).$

∎

######  Remark 3.3.

For the points in $V\setminus U_{0}$ which are geometric singular, we can also
apply the deflation process defined in (3.15) to them. Similar to (3.16), with
the same reason, we still have

$n_{i}\leq\cdots\leq n_{1}\leq n_{0}.$

However, for points in $V\setminus U_{0}$, we can see that

$n_{i}<k$

can be happen for some $i$, see Example 3.6 and 3.7. So deflation process
provides lower bound of local dimension for them.

From (3.19), we can see that after each deflation step, the addition of new
nonzero submatrices to the bottom of the first $n$ columns is the reason that
makes the nullities decrease.

###  3.3. The deflation conjecture

By Theorem 3.2, we can see that the nullity $n_{i}$ can not be always strictly
decreased. It should be stabilized after a finite number of steps.

######  Definition 3.4.

Suppose that $\hat{\textbf{x}}$ is a solution of $F(\textbf{x})$. With
notations above, for the deflation process (3.15), let $i_{0}\geq 0$ be the
smallest integer such that

$n_{i+1}=n_{i}\qquad\text{for~{}all}\quad i\geq i_{0}.$

Then

$(n_{0},n_{1},...,n_{i_{0}})$ (3.20)

is called the _deflation sequence_ of $\hat{\textbf{x}}$.

######  Definition 3.5.

Let $\hat{\textbf{x}}\in\textbf{V}(F)$ with local dimension
$\dim_{\hat{\textbf{x}}}\textbf{V}(F)=k$. Let $(n_{0},n_{1},...,n_{i_{0}})$
denote its deflation sequence. If

$n_{i_{0}}=k,$

then we say that the deflation sequence $(n_{0},n_{1},...,n_{i_{0}})$ is
_regular_ and $\hat{\textbf{x}}$ has _a regular deflation process_.

######  Example 3.6.

Let $F(x,y)=y^{2}-x^{3}$ with $z_{1}=(0,0)$ and $z_{2}=(1,1)$ in
$\textbf{V}(F)$. By direct computer verification, the first 4 numbers of their
deflation sequences are

$(n_{0},n_{1},n_{2},n_{3})=(2,1,1,0)~{}\text{and}~{}(1,1,1,1),~{}\text{respectively}.$

######  Example 3.7.

Let $F(x,y,z)=x^{2}-y^{2}z$ be the Whitney umbrella. Consider the three points
of $\textbf{V}(F)$: (2, 2, 1), (0, 0, 1) and (0, 0, 0). The first 4 numbers of
their deflation sequences are:

$(n_{0},n_{1},n_{2},n_{3})=(2,2,2,2),~{}(3,2,1,1)~{}\text{and}~{}(3,2,2,1),~{}\text{respectively}.$

It is proved in [7, 26] that all isolated singular solutions have regular
deflation processes. For $k$-dimensional ultrasingular irreducible components,
Zeng’s conjecture in [37, Sect. 9] can be formalised as follows.

######  Conjecture 1 (_Deflation Conjecture_).

Let $V\subseteq\textbf{V}(F)$ be a $k$-dimensional ultrasingular irreducible
component. So there is a nonempty Zariski open set $U\subseteq V$ which
consists of geometric smooth points and a constant number $n_{0}$ such that
$\mathscr{N}\left(J(\textbf{x})\right)=n_{0}>k$ for all $\textbf{x}\in U$. We
have:

  1. (1)

The recursive deflation process (3.15) terminates in finitely many steps such
that the deflation sequences of all $\textbf{x}\in U$ are equal and regular.

  2. (2)

For the point $\hat{\textbf{x}}\in V\setminus U$ with deflation sequence
$(n_{0},n_{1},...,n_{i_{0}})$, $n_{i_{0}}$ is a lower bound of its local
dimension. That is,

$\dim_{\hat{\textbf{x}}}\textbf{V}(F)\geq n_{i_{0}}.$

######  Remark 3.8.

In [16], using the results in [2], Hauenstein and Wampler showed that
Conjecture 1 holds for their strong deflation. Leykin et al.’s deflation
process can also be given a similar conjecture [26], and similarly for other
deflation processes in the literatures.

######  Remark 3.9.

If Conjecture 1 is true, the number in the deflation sequence provides upper
bounds of the local dimensions of geometric smooth points. The last number of
the deflation sequence is a lower bound of the local dimensions of the
geometric singular points. This leads to the problem of determining the
singularity of a solution (see also Section 6.2).

######  Remark 3.10.

Given a set of solutions of a polynomial system. We usually don’t know the
relations between them, such as if they lie on a same irreducible component.
So if we apply the deflation process to them, the $R_{i}$ and $d_{i}$ defined
in (3.15) are chosen separately for each solution. Thus, in practical
numerical experiment, if we don’t know the relation between the solutions, the
deflation process is the same as the deflation process for isolated solutions
in [7, 8]. For example, the solutions of Brent equations studied in Section 5.

##  4\. On realization of the deflation process

In this section, the details of the construction of the Jacobians of the first
3 steps of the deflation process raised in [7, 8] are given. Although they may
not be new, we have not seen them in the literatures. We can see that the
Jacobians are not hard to be realized numerically. So it is a useful tool for
the determination of local dimensions of the algebraic sets.

Firstly, we introduce some notations. The original variable
$\textbf{x}=[x_{1},\cdots,x_{n}]^{\textbf{T}}$ will be denoted by
$\textbf{x}_{1}$ in accordance with the notation for the auxiliary (vector)
variables $\textbf{x}_{2}$, $\textbf{x}_{3}$,… etc. For any fixed or variable
vector $\textbf{y}=[y_{1},...,y_{n}]^{\textbf{T}}$, the directional
differentiation operator along y is defined as

$\nabla_{\textbf{y}}=y_{1}\frac{\partial}{\partial
x_{1}}+y_{2}\frac{\partial}{\partial
x_{2}}+\cdots+y_{n}\frac{\partial}{\partial x_{n}}.$

######  Lemma 4.1.

The composite of $\nabla_{\textbf{y}}$ and $\nabla_{\textbf{z}}$ is
commutative, that is,

$\nabla_{\textbf{y}}\nabla_{\textbf{z}}=\nabla_{\textbf{z}}\nabla_{\textbf{y}}$

###### Proof.

Since $\frac{\partial^{2}}{\partial x_{i}\partial
x_{j}}=\frac{\partial^{2}}{\partial x_{j}\partial x_{i}}$, we have

$\begin{split}\nabla_{\textbf{y}}\nabla_{\textbf{z}}&=\nabla_{\textbf{y}}\left(z_{1}\frac{\partial}{\partial
x_{1}}+z_{2}\frac{\partial}{\partial
x_{2}}+\cdots+z_{n}\frac{\partial}{\partial x_{n}}\right)\\\
&=\sum_{i=1}^{n}\sum_{j=1}^{n}y_{i}z_{j}\frac{\partial^{2}}{\partial
x_{i}\partial
x_{j}}=\sum_{i=1}^{n}\sum_{j=1}^{n}z_{i}y_{j}\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}}\\\ &=\nabla_{\textbf{z}}\nabla_{\textbf{y}}.\end{split}$

∎

For any variable $\textbf{u}=(u_{1},...,u_{n})^{\textbf{T}}$, the gradient
operator is defined by

$\triangle_{\textbf{u}}=\left[\frac{\partial}{\partial
u_{1}},\frac{\partial}{\partial u_{2}},\cdots,\frac{\partial}{\partial
u_{n}}\right],$

whose “dot product” with a vector $\textbf{v}=[v_{1},...,v_{n}]^{\textbf{T}}$
is defined as

$\textbf{v}\cdot\triangle_{\textbf{u}}=v_{1}\frac{\partial}{\partial
u_{1}}+v_{2}\frac{\partial}{\partial
u_{2}}+\cdots+v_{n}\frac{\partial}{\partial u_{n}}.$

So we have
$\nabla_{\textbf{y}}=\textbf{y}\cdot\triangle_{\textbf{x}}=\textbf{y}\cdot\triangle_{\textbf{x}_{1}}$.

Let $J(\textbf{x})$ be the Jacobian of the polynomial system
$F(\textbf{x})=[f_{1}(\textbf{x}),f_{2}(\textbf{x}),...f_{m}(\textbf{x})]^{\textbf{T}}$.
Then $J(\textbf{x})\textbf{y}$ can be denoted by

$J(\textbf{x})\textbf{y}=\nabla_{\textbf{y}}F=\left[\begin{array}[]{c}\nabla_{\textbf{y}}f_{1}\\\
\nabla_{\textbf{y}}f_{2}\\\ \vdots\\\ \nabla_{\textbf{y}}f_{m}\\\
\end{array}\right].$

Similarly, set

$\nabla_{\textbf{u}}\nabla_{\textbf{v}}F=\left[\begin{array}[]{c}\nabla_{\textbf{u}}\nabla_{\textbf{v}}f_{1}\\\
\nabla_{\textbf{u}}\nabla_{\textbf{v}}f_{2}\\\ \vdots\\\
\nabla_{\textbf{u}}\nabla_{\textbf{v}}f_{m}\\\ \end{array}\right].$

Let
$G(\textbf{x}_{1},\textbf{x}_{2},...,\textbf{x}_{k}):\mathbb{C}^{kn}\to\mathbb{C}^{m}$
be a polynomial system and $G=[g_{1},g_{2},...,g_{m}]^{\textbf{T}}$, its
_partial Jacobian_ with respect to some
$\textbf{x}_{i}=[x_{i1},...,x_{in}]^{\textbf{T}}$ is defined by

$J(G,\textbf{x}_{i}):=\left[\begin{array}[]{c}\triangle_{\textbf{x}_{i}}g_{1}\\\
\triangle_{\textbf{x}_{i}}g_{2}\\\ \vdots\\\
\triangle_{\textbf{x}_{i}}g_{m}\\\
\end{array}\right]=\left[\begin{array}[]{ccc}\frac{\partial g_{1}}{\partial
x_{i1}}&\cdots&\frac{\partial g_{1}}{\partial x_{in}}\\\ \vdots&&\vdots\\\
\frac{\partial g_{m}}{\partial x_{i1}}&\cdots&\frac{\partial g_{m}}{\partial
x_{in}}\\\ \end{array}\right].$ (4.1)

Let $J(J(\textbf{x})\textbf{v},\textbf{x})$ be the partial Jacobian of
$J(\textbf{x})\textbf{v}$ with respect to x, and similarly for
$J(J(J(\textbf{x})\textbf{w},\textbf{x})\textbf{v},\textbf{x})$. Then we have
the following lemma.

######  Lemma 4.2.

$J(J(\textbf{x})\textbf{v},\textbf{x})\textbf{u}=\nabla_{\textbf{u}}\nabla_{\textbf{v}}F=\left[\begin{array}[]{c}\nabla_{\textbf{u}}\nabla_{\textbf{v}}f_{1}\\\
\nabla_{\textbf{u}}\nabla_{\textbf{v}}f_{2}\\\ \vdots\\\
\nabla_{\textbf{u}}\nabla_{\textbf{v}}f_{m}\\\
\end{array}\right]=\nabla_{\textbf{v}}\nabla_{\textbf{u}}F.$

$J(J(J(\textbf{x})\textbf{w},\textbf{x})\textbf{v},\textbf{x})\textbf{u}=\nabla_{\textbf{u}}\nabla_{\textbf{v}}\nabla_{\textbf{w}}F=\left[\begin{array}[]{c}\nabla_{\textbf{u}}\nabla_{\textbf{v}}\nabla_{\textbf{w}}f_{1}\\\
\nabla_{\textbf{u}}\nabla_{\textbf{v}}\nabla_{\textbf{w}}f_{2}\\\ \vdots\\\
\nabla_{\textbf{u}}\nabla_{\textbf{v}}\nabla_{\textbf{w}}f_{m}\\\
\end{array}\right]=\nabla_{\textbf{a}}\nabla_{\textbf{b}}\nabla_{\textbf{c}}F,$

where a, b, c is any permutation of u, v, w.

###### Proof.

The proof is followed by Lemma 4.1. Since for $i=1,2,...,m$, we have

$\nabla_{\textbf{u}}\nabla_{\textbf{v}}f_{i}=\nabla_{\textbf{v}}\nabla_{\textbf{u}}f_{i}$

and

$\nabla_{\textbf{u}}\nabla_{\textbf{v}}\nabla_{\textbf{w}}f_{i}=\nabla_{\textbf{a}}\nabla_{\textbf{b}}\nabla_{\textbf{c}}f_{i},$

where a, b, c is any permutation of u, v, w. ∎

For the partial Jacobians of $\nabla_{\textbf{u}}F(\textbf{x})$,
$\nabla_{\textbf{u}}\nabla_{\textbf{v}}F(\textbf{x})$ and
$\nabla_{\textbf{u}}\nabla_{\textbf{v}}\nabla_{\textbf{w}}F(\textbf{x})$, we
have the following formulas.

######  Lemma 4.3.

With notation in (4.1), we have

  1. (1)

$J(\nabla_{\textbf{u}}F(\textbf{x}),\textbf{v})=\textbf{0}$,

  2. (2)

$J(\nabla_{\textbf{u}}F(\textbf{x}),\textbf{u})=J(\textbf{x})$,

  3. (3)

$J(\nabla_{\textbf{v}}\nabla_{\textbf{u}}F(\textbf{x}),\textbf{u})=J(\nabla_{\textbf{u}}\nabla_{\textbf{v}}F(\textbf{x}),\textbf{u})=J(\nabla_{\textbf{v}}F(\textbf{x}),\textbf{x})$,

  4. (4)

$J(\nabla_{\textbf{v}}\nabla_{\textbf{w}}\nabla_{\textbf{u}}F(\textbf{x}),\textbf{u})=J(\nabla_{\textbf{v}}\nabla_{\textbf{u}}\nabla_{\textbf{w}}F(\textbf{x}),\textbf{u})=J(\nabla_{\textbf{u}}\nabla_{\textbf{v}}\nabla_{\textbf{w}}F(\textbf{x}),\textbf{u})=J(\nabla_{\textbf{v}}\nabla_{\textbf{w}}F(\textbf{x}),\textbf{x})$.

###### Proof.

The proof is followed by direct verification and Lemma 4.2. ∎

With preparation above, in the following we describe the first 3 steps of the
deflation process explicitly. From this, we can obtain the first 4 numbers
($n_{0},n_{1},n_{2},n_{3}$) of the deflation sequence.

Step 0. Given a polynomial system
$F(\textbf{x})=[f_{1}(\textbf{x}),f_{2}(\textbf{x}),...f_{m}(\textbf{x})]^{\textbf{T}}$,
whose Jacobian is $J(\textbf{x})$. Given a solution
$\hat{\textbf{x}}\in\textbf{V}(F)$. Then

$n_{0}=n-rank~{}J(\hat{\textbf{x}})$

is the nullity which is the first number of the deflation sequence of
$\hat{\textbf{x}}$.

Step 1. Let $n_{0}$ and $\hat{\textbf{x}}_{1}=\hat{\textbf{x}}$ be the number
and solution in Step 0, respectively. We choose a random matrix
$R_{1}\in\mathbb{C}^{n_{0}\times n}$ and a random vector
$d_{1}\in\mathbb{C}^{n_{0}}$. By (3.5), the first deflation of $F(\textbf{x})$
is

$F_{1}(\textbf{x}_{1},\textbf{x}_{2})=\left[\begin{array}[]{c}F(\textbf{x}_{1})\\\
J(\textbf{x}_{1})\textbf{x}_{2}\\\
R_{1}\textbf{x}_{2}-d_{1}\end{array}\right],$ (4.2)

whose Jacobian is

$J_{1}(\textbf{x}_{1},\textbf{x}_{2})=\left[\begin{array}[]{cc}J(\textbf{x}_{1})&\textbf{0}\\\
J(\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1}),\textbf{x}_{1})&J(\textbf{x}_{1})\\\
\textbf{0}&R_{1}\\\ \end{array}\right],$ (4.3)

where
$J(\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1}),\textbf{x}_{1})=J(J(\textbf{x}_{1})\textbf{x}_{2},\textbf{x}_{1})$.

Fixing $\hat{\textbf{x}}_{1}$, let $\hat{\textbf{x}}_{2}$ be the solution of
the linear system $F_{1}(\hat{\textbf{x}}_{1},\textbf{x}_{2})=\textbf{0}$ (see
also (3.4)). Then we have

$n_{1}=2n-rank~{}J_{1}(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2}),$

which is the second number of the deflation sequence.

Step 2. Let $n_{1}$ and $(\hat{\textbf{x}}_{1},~{}\hat{\textbf{x}}_{2})$ be
the number and solution in Step 1, respectively. We choose a random matrix
$R_{2}\in\mathbb{C}^{n_{1}\times 2n}$ and a random vector
$d_{2}\in\mathbb{C}^{n_{1}}$. If we continue taking the deflation process on
$F_{1}(\textbf{x}_{1},\textbf{x}_{2})$, then by (3.15) we have

$F_{2}(\textbf{x}_{1},\textbf{x}_{2},\textbf{x}_{3},\textbf{x}_{4})=\left[\begin{array}[]{c}F_{1}(\textbf{x}_{1},\textbf{x}_{2})\\\
J_{1}(\textbf{x}_{1},\textbf{x}_{2})~{}\left[\begin{array}[]{c}\textbf{x}_{3}\\\
\textbf{x}_{4}\\\ \end{array}\right]\\\
R_{2}\left[\begin{array}[]{c}\textbf{x}_{3}\\\ \textbf{x}_{4}\\\
\end{array}\right]-d_{2}\end{array}\right]=\left[\begin{array}[]{c}g_{1}\\\
g_{2}\\\ \vdots\\\ g_{7}\\\ \end{array}\right],$ (4.4)

and from (4.2) and (4.3) we have

$\begin{split}&g_{1}=F(\textbf{x}_{1}),~{}g_{2}=J(\textbf{x}_{1})\textbf{x}_{2}=\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1}),~{}g_{3}=R_{1}\textbf{x}_{2}-d_{1},~{}g_{4}=J(\textbf{x}_{1})\textbf{x}_{3}=\nabla_{\textbf{x}_{3}}F(\textbf{x}_{1}),\\\
&g_{5}=J(\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1}),\textbf{x}_{1})\textbf{x}_{3}+J(\textbf{x}_{1})\textbf{x}_{4}=\nabla_{\textbf{x}_{3}}\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1})+\nabla_{\textbf{x}_{4}}F(\textbf{x}_{1}),\\\
&g_{6}=R_{1}\textbf{x}_{4},~{}g_{7}=R_{2}\left[\begin{array}[]{c}\textbf{x}_{3}\\\
\textbf{x}_{4}\\\ \end{array}\right]-d_{2}.\end{split}$ (4.5)

Let $J_{2}(\textbf{x}_{1},\textbf{x}_{2},\textbf{x}_{3},\textbf{x}_{4})$ be
the Jacobian of
$F_{2}(\textbf{x}_{1},\textbf{x}_{2},\textbf{x}_{3},\textbf{x}_{4})$. It
consists of $7\times 4=28$ partial Jacobian blocks, which can be seen as
follows

$J_{2}(\textbf{x}_{1},\textbf{x}_{2},\textbf{x}_{3},\textbf{x}_{4})=\left[\begin{array}[]{ccc}J(g_{1},\textbf{x}_{1})&\cdots&J(g_{1},\textbf{x}_{4})\\\
J(g_{2},\textbf{x}_{1})&\cdots&J(g_{2},\textbf{x}_{4})\\\ \vdots&&\vdots\\\
J(g_{7},\textbf{x}_{1})&\cdots&J(g_{7},\textbf{x}_{4})\\\ \end{array}\right].$
(4.6)

In (4.6), we only describe the partial Jacobians $J(g_{5},\textbf{x}_{2})$, by
Lemma 4.3, which can be calculated as follows

$\begin{split}J(g_{5},\textbf{x}_{2})&=J\left(\nabla_{\textbf{x}_{3}}\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1}),\textbf{x}_{2}\right)+J\left(\nabla_{\textbf{x}_{4}}F(\textbf{x}_{1}),\textbf{x}_{2}\right)\\\
&=J\left(\nabla_{\textbf{x}_{2}}\nabla_{\textbf{x}_{3}}F(\textbf{x}_{1}),\textbf{x}_{2}\right)+\textbf{0}\\\
&=J\left(\nabla_{\textbf{x}_{3}}F(\textbf{x}_{1}),\textbf{x}_{1}\right).\end{split}$
(4.7)

The other partial Jacobians are not hard to be obtained.

Fixing $\hat{\textbf{x}}_{1}$, $\hat{\textbf{x}}_{2}$, let
$\hat{\textbf{x}}_{3}$, $\hat{\textbf{x}}_{4}$ be the solution of the linear
system
$F_{2}(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2},\textbf{x}_{3},\textbf{x}_{4})=\textbf{0}$.
Then we have

$n_{2}=4n-rank~{}J_{2}(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2},\hat{\textbf{x}}_{3},\hat{\textbf{x}}_{4}),$

which is the third number of the deflation sequence.

Step 3. Let $n_{2}$ and
$(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2},\hat{\textbf{x}}_{3},\hat{\textbf{x}}_{4})$
be the number and solution in Step 2, respectively. We choose a random matrix
$R_{3}\in\mathbb{C}^{n_{2}\times 4n}$ and a random vector
$d_{3}\in\mathbb{C}^{n_{2}}$. If we continue taking the deflation process on
$F_{2}(\textbf{x}_{1},\textbf{x}_{2},\textbf{x}_{3},\textbf{x}_{4})$, then by
(3.15) we have

$F_{3}(\textbf{x}_{1},\textbf{x}_{2},...,\textbf{x}_{8})=\left[\begin{array}[]{c}F_{2}(\textbf{x}_{1},\textbf{x}_{2},\textbf{x}_{3},\textbf{x}_{4})\\\
J_{2}(\textbf{x}_{1},\textbf{x}_{2},\textbf{x}_{3},\textbf{x}_{4})~{}\left[\begin{array}[]{c}\textbf{x}_{5}\\\
\textbf{x}_{6}\\\ \textbf{x}_{7}\\\ \textbf{x}_{8}\\\ \end{array}\right]\\\
R_{3}\left[\begin{array}[]{c}\textbf{x}_{5}\\\ \textbf{x}_{6}\\\
\textbf{x}_{7}\\\ \textbf{x}_{8}\\\
\end{array}\right]-d_{3}\end{array}\right]=\left[\begin{array}[]{c}h_{1}\\\
h_{2}\\\ \vdots\\\ h_{15}\\\ \end{array}\right],$ (4.8)

and from (4.4), (4.5) and (4.6) we have

$\begin{split}&h_{i}=g_{i},\quad\text{for}~{}i=1,2,...7,\\\
&h_{8}=\nabla_{\textbf{x}_{5}}F(\textbf{x}_{1}),\\\
&h_{9}=\nabla_{\textbf{x}_{5}}\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1})+\nabla_{\textbf{x}_{6}}F(\textbf{x}_{1}),\\\
&h_{10}=R_{1}\textbf{x}_{6},\\\
&h_{11}=\nabla_{\textbf{x}_{5}}\nabla_{\textbf{x}_{3}}F(\textbf{x}_{1})+\nabla_{\textbf{x}_{7}}F(\textbf{x}_{1}),\\\
&h_{12}=\nabla_{\textbf{x}_{5}}\nabla_{\textbf{x}_{3}}\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1})+\nabla_{\textbf{x}_{5}}\nabla_{\textbf{x}_{4}}F(\textbf{x}_{1})+\nabla_{\textbf{x}_{6}}\nabla_{\textbf{x}_{3}}F(\textbf{x}_{1})\\\
&\qquad+\nabla_{\textbf{x}_{7}}\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1})+\nabla_{\textbf{x}_{8}}F(\textbf{x}_{1}),\\\
&h_{13}=R_{1}\textbf{x}_{8},\\\
&h_{14}=R_{2}\left[\begin{array}[]{c}\textbf{x}_{7}\\\ \textbf{x}_{8}\\\
\end{array}\right]\\\ &h_{15}=R_{3}\left[\begin{array}[]{c}\textbf{x}_{5}\\\
\textbf{x}_{6}\\\ \textbf{x}_{7}\\\ \textbf{x}_{8}\\\
\end{array}\right]-d_{3}\end{split}$ (4.9)

Let $J_{3}(\textbf{x}_{1},\textbf{x}_{2},...,\textbf{x}_{8})$ be the Jacobian
of $F_{3}(\textbf{x}_{1},\textbf{x}_{2},...,\textbf{x}_{8})$. It consists of
$15\times 8=120$ partial Jacobian blocks, which can be seen as follows

$J_{3}(\textbf{x}_{1},\textbf{x}_{2},...,\textbf{x}_{8})=\left[\begin{array}[]{ccc}J(h_{1},\textbf{x}_{1})&\cdots&J(h_{1},\textbf{x}_{8})\\\
J(h_{2},\textbf{x}_{1})&\cdots&J(h_{2},\textbf{x}_{8})\\\ \vdots&&\vdots\\\
J(h_{15},\textbf{x}_{1})&\cdots&J(h_{15},\textbf{x}_{8})\\\
\end{array}\right].$ (4.10)

In (4.10), we only describe the partial Jacobians $J(h_{12},\textbf{x}_{2})$
and $J(h_{12},\textbf{x}_{3})$. By Lemma 4.3 we have

$\begin{split}J(h_{12},\textbf{x}_{2})&=J\left(\nabla_{\textbf{x}_{5}}\nabla_{\textbf{x}_{3}}\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1}),\textbf{x}_{2}\right)+J\left(\nabla_{\textbf{x}_{7}}\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1}),\textbf{x}_{2}\right)\\\
&=J\left(\nabla_{\textbf{x}_{5}}\nabla_{\textbf{x}_{3}}F(\textbf{x}_{1}),\textbf{x}_{1}\right)+J\left(\nabla_{\textbf{x}_{7}}F(\textbf{x}_{1}),\textbf{x}_{1}\right),\end{split}$
(4.11)

and

$\begin{split}J(h_{12},\textbf{x}_{3})&=J\left(\nabla_{\textbf{x}_{5}}\nabla_{\textbf{x}_{3}}\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1}),\textbf{x}_{3}\right)+J\left(\nabla_{\textbf{x}_{6}}\nabla_{\textbf{x}_{3}}F(\textbf{x}_{1}),\textbf{x}_{3}\right)\\\
&=J\left(\nabla_{\textbf{x}_{5}}\nabla_{\textbf{x}_{2}}F(\textbf{x}_{1}),\textbf{x}_{1}\right)+J\left(\nabla_{\textbf{x}_{6}}F(\textbf{x}_{1}),\textbf{x}_{1}\right)\end{split}$
(4.12)

Fixing $\hat{\textbf{x}}_{1}$, $\hat{\textbf{x}}_{2}$, $\hat{\textbf{x}}_{3}$,
$\hat{\textbf{x}}_{4}$, let $\hat{\textbf{x}}_{5}$, $\hat{\textbf{x}}_{6}$,
$\hat{\textbf{x}}_{7}$, $\hat{\textbf{x}}_{8}$ be the solution of the linear
system

$F_{3}(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2},\hat{\textbf{x}}_{3},\hat{\textbf{x}}_{4},\textbf{x}_{5},\textbf{x}_{6},\textbf{x}_{7},\textbf{x}_{8})=\textbf{0}.$

Then we have

$n_{3}=8n-rank~{}J_{3}(\hat{\textbf{x}}_{1},\hat{\textbf{x}}_{2},...,\hat{\textbf{x}}_{8}),$

which is the fourth number of the deflation sequence.

For the polynomial system $F=[f_{1},f_{2},...f_{m}]^{\textsc{T}}$, the degree
of $F$ is defined by

$Deg~{}F=\max\\{Deg~{}f_{i}|i=1,2,...m\\},$

where $Deg~{}f_{i}$ are the degrees of the polynomials $f_{i}$. So from above
analysis, we have the following proposition.

######  Proposition 4.4.

If $Deg~{}F=3$, in the deflation process (3.15), the Jacobians
$J_{i}(\textbf{x}_{1},...,\textbf{x}_{2^{i}})$ are builded up by $R_{i}$ and
the following partial Jacobians

$J(\textbf{x}),~{}J(\nabla_{\textbf{u}}F(\textbf{x}),\textbf{x}),~{}J(\nabla_{\textbf{u}}\nabla_{\textbf{v}}F(\textbf{x}),\textbf{x}),$

where $\textbf{u},~{}\textbf{v}$ are (partial) solutions of the extended
system.

###### Proof.

Since $Deg~{}F=3$, we can see that
$J(\nabla_{\textbf{u}}\nabla_{\textbf{v}}\nabla_{\textbf{w}}F(\textbf{x}),\textbf{x})=\textbf{0}$,
for any solutions u, v, w of the extended system. ∎

The degree of the polynomial systems obtained by tensor decompositions of
tensors of order 3 are all of degree 3, such as the Brent equations in next
section.

##  5\. Numerical experiments on Brent equations

Firstly, we recall the definition of Brent equations (see also [28]). Let
$V_{1}=\mathbb{C}^{m\times n}$, $V_{2}=\mathbb{C}^{n\times p}$ and
$V_{3}=\mathbb{C}^{p\times m}$ denote the set of complex $m\times n$, $n\times
p$ and $p\times m$ matrices, respectively. The matrix multiplication tensor is
defined by (see e.g. [5, 24])

$\langle m,n,p\rangle=\sum_{i=1}^{m}\sum_{j=1}^{n}\sum_{k=1}^{p}e_{ij}\otimes
e_{jk}\otimes e_{ki}\in\mathbb{C}^{m\times n}\otimes\mathbb{C}^{n\times
p}\otimes\mathbb{C}^{p\times m},$ (5.1)

where $e_{ij}$ denotes the matrix with a 1 in the $i$th row and $j$th column,
and other entries are 0. Finding a tensor decomposition of $\langle
m,n,p\rangle$ with rank $r$ is equivalent to find solutions of a system of
polynomial equations called the _Brent equations_ , which is given as follows:

$\sum_{i=1}^{r}\alpha_{i_{1},i_{2}}^{(i)}\beta_{j_{1},j_{2}}^{(i)}\gamma_{k_{1},k_{2}}^{(i)}=\delta_{i_{2},j_{1}}\delta_{j_{2},k_{1}}\delta_{k_{2},i_{1}},$
(5.2)

where $k_{2},i_{1}\in\\{1,2,...,m\\}$, $i_{2},j_{1}\in\\{1,2,...,n\\}$,
$j_{2},k_{1}\in\\{1,2,...,p\\}$, $\delta_{ij}$ is the Kronecker symbol,
$\alpha_{i_{1},i_{2}}^{(i)}$, $\beta_{j_{1},j_{2}}^{(i)}$ and
$\gamma_{k_{1},k_{2}}^{(i)}$ are unknown variables.

In the following, the polynomial system in (5.2) is called _the Brent
equations of type $[m,n,p|r]$_, which is denoted by $B(m,n,p|r)$. We can see
that the polynomial system $B(m,n,p|r)$ has $(mn+np+pm)r$ variables and
$(mnp)^{2}$ equations. So $B(m,n,p|r)$ can be considered as a polynomial
mapping:

$B(m,n,p|r)\colon\mathbb{C}^{(mn+np+pm)r}\to\mathbb{C}^{(mnp)^{2}}.$

Usually, $B(m,n,p|r)$ is an overdetermined polynomial system. The solution set
of $B(m,n,p|r)$ over $\mathbb{C}$ is denoted by $\textbf{V}(m,n,p|r)$.

For the convenience of coding, an order can be given for the variables in
(5.2). For example, for $i=1,2,...r$ let
$A_{i}=\left(\alpha_{i_{1},i_{2}}^{(i)}\right)\in\mathbb{C}^{m\times n}$,
$B_{i}=\left(\beta_{j_{1},j_{2}}^{(i)}\right)\in\mathbb{C}^{n\times p}$ and
$C_{i}=\left(\gamma_{k_{1},k_{2}}^{(i)}\right)\in\mathbb{C}^{n\times p}$. For
any matrix in $\mathbb{C}^{a\times b}$, define the columnwise vectorization
map $\mathcal{C}\colon\mathbb{C}^{a\times b}\to\mathbb{C}^{ab\times 1}$ by
concatenating the columns of the matrix in $\mathbb{C}^{a\times b}$ as a
column vector of $\mathbb{C}^{ab\times 1}$ (or $\mathbb{C}^{ab}$). For
example, let $a=\left[\begin{array}[]{cc}a_{11}&a_{12}\\\ a_{21}&a_{22}\\\
\end{array}\right]\in\mathbb{C}^{2\times 2}$, then
$\mathcal{C}(a)=[a_{11},a_{21},a_{12},a_{22}]^{\textbf{T}}\in\mathbb{C}^{4}$.
For $i=1,2,...r$, let

$\textbf{x}_{i}=\left[\begin{array}[]{c}\mathcal{C}(A_{i})\\\
\mathcal{C}(B_{i})\\\ \mathcal{C}(C_{i})\\\
\end{array}\right]\in\mathbb{C}^{mn+np+pm}.$

Then we can order the variables in (5.2) by

$\textbf{x}=\left[\begin{array}[]{c}\textbf{x}_{1}\\\ \vdots\\\
\textbf{x}_{r}\\\ \end{array}\right]\in\mathbb{C}^{(mn+np+pm)r}.$ (5.3)

In the following of this section, we report our experimental results. By
Theorem 3.2 and Remark 3.3, the deflation sequence tells us the bound of the
local dimension. We can see that the as $r$ in (5.2) decreases, singular
solutions become more and more.

###  5.1. Deflation test on $\textbf{V}(2,2,2|7)$: Strassen’s classical
solution

For $B(2,2,2|7)$, it is an underdetermined system which has $m=64$ equations
and $n=84$ variables. So from equation (3) of [36], we know that for any
$p\in\textbf{V}(2,2,2|7)$, its local dimension has a lower bound

$20=84-64=n-m\leq\dim_{p}\textbf{V}(2,2,2|7).$ (5.4)

In particular, for the classical solution of $B(2,2,2|7)$ given by Strassen
[34] (see also [5, 24]), after doing deflation 3 times, the first 4 numbers of
deflation sequence are

$(23,23,23,23).$

So its local dimension should be 23. Moreover, $\dim\textbf{V}(2,2,2|7)$
should be 23.

######  Remark 5.1.

It was proved in [10] that $\textbf{V}(2,2,2|7)$ has projective dimension 9.

###  5.2. Deflation test on $\textbf{V}(3,3,3|23)$

In the website [17], using SAT solvers Heule et al. provided 17376 solutions
in $\textbf{V}(3,3,3|23)$ which are inequivalent under the de Groote group
action [5, 9]. We take deflation test on these solutions. We found that most
of these solutions are ultrasingular. Let $(n_{0},n_{1},n_{2},n_{3})$ be the
first 4 numbers of the deflation sequence. Overall, from the experimental
result, we can see that $76\leq n_{0}\leq 95$ (see also [28, Sect. 3]). After
doing deflation 3 times, we can see that $70\leq n_{3}\leq 86$. The data can
be found on [29]. The results are summarized below.

####  5.2.1. The first step of deflation

Among the 17376 solutions, after doing the first step of deflation, there are
16511 solutions whose nullities of Jacobians are strictly decreased, that is,
$n_{0}>n_{1}$. Let $d_{1}=n_{0}-n_{1}$. Then the numerical experiment tells us
that $0\leq d_{1}\leq 23$. The distribution is as follows.

$d_{1}=n_{0}-n_{1}$ | $d_{1}=0$ | $1\leq d_{1}\leq 3$ | $4\leq d_{1}\leq 10$ | $11\leq d_{1}\leq 23$ | Total number
---|---|---|---|---|---
Number | 865 | 5495 | 9375 | 1641 | 17376

####  5.2.2. The second step of deflation

After doing the second step of deflation, there are 16206 solutions whose
nullities of Jacobians are strictly decreased, that is, $n_{1}>n_{2}$. Let
$d_{2}=n_{1}-n_{2}$. Then the numerical experiment tells us that $0\leq
d_{2}\leq 10$. The distribution is as follows.

$d_{2}=n_{1}-n_{2}$ | $d_{2}=0$ | $1\leq d_{2}\leq 3$ | $4\leq d_{2}\leq 10$ | Total number
---|---|---|---|---
Number | 1168 | 14732 | 1476 | 17376

####  5.2.3. The third step of deflation

In this step, there are only 978 solutions whose nullities of Jacobians are
strictly decreased, that is, $n_{2}>n_{3}$. Let $d_{3}=n_{2}-n_{3}$. Then the
numerical experiment tells us that $0\leq d_{3}\leq 5$. By Theorem 3.2, we
know that $n_{2}\geq n_{3}$. However, due to the test is numerical, some
errors appear for 41 solutions, that is, $n_{2}-n_{3}<0$. On the other hand,
if we restart the deflation process separately for these solutions, we can
find that $n_{2}\geq n_{3}$. The distribution can be seen as follows.

$d_{3}=n_{2}-n_{3}$ | $d_{3}=0$ | $d_{3}=1$ | $2\leq d_{3}\leq 5$ | Error ($d_{3}<0$) | Total number
---|---|---|---|---|---
Number | 16357 | 795 | 183 | 41 | 17376

The first 4 numbers of deflation sequence of Laderman’s solution [23] are
(76,76,76,76). The first 4 numbers of deflation sequence of Smirnov’s solution
[33] in $\textbf{V}(3,3,3|23)$ are (87,82,79,79).

###  5.3. Deflation test on $\textbf{V}(4,4,4|49)$

In [13, 14], Fawzi et al. provided 14236 solutions in $\textbf{V}(4,4,4|49)$,
which are inequivalent under the isotopy group action. We take deflation on
these points and find that the ultrasingular solutions are relatively small
among all the 14236 solutions.

Note that $B(4,4,4|49)$ has 2352 variables and 4096 equations. For each
solution, it takes about 10 minutes for the computation of
$(n_{0},n_{1},n_{2},n_{3})$. It will spend a lot of time if we take the first
3 steps of deflation for all 14236 solutions. On the other hand, the time for
the computation of $(n_{0},n_{1})$ is relatively acceptable. So we firstly
take the first step of deflation on Fawzi et al.’s solutions and obtain
$(n_{0},n_{1})$ of the 14236 solutions, the data can be found on [29]. The
distribution is summarized as follows.

| $n_{1}=n_{0}=197$ | $n_{0}>n_{1}$ | Total number
---|---|---|---
Number | 13530 | 706 | 14236

Secondly, from above table, for the 13530 solutions whose $n_{0}=n_{1}=197$,
we randomly choose 100 solutions and take the second and third deflation steps
to obtain $(n_{2},n_{3})$. For all the 100 solutions, we find that
$n_{2}=n_{3}=197$, so the first 4 numbers of their deflation sequences are all
equal to: (197,197,197,197). So this implies that for all 13530 solutions we
should have $n_{0}=n_{1}=n_{2}=n_{3}=197$.

For the remaining 706 solutions, their first 4 numbers of deflation sequences
($n_{0},~{}n_{1},~{}n_{2},~{}n_{3}$) are computed. We found that:

  1. (1)

$151\leq n_{0}\leq 208$;

  2. (2)

$145\leq n_{3}\leq 182$;

  3. (3)

for most solutions we have $n_{2}=n_{3}$.

###  5.4. Deflation test on other well-known solutions

Firstly, the first 4 numbers of deflation sequence Smirnov’s solution [33]
that lies in $\textbf{V}(3,3,6|40)$ are: (131, 131, 131, 131). So it seems
that this solution is semiregular. Recently, Kauers and Moosbauer found many
new solutions of $B(m,n,p|r)$ and put them on Github [20, 21]. We take
deflation process for some solutions that are suitable to the complex field.
The first 4 numbers of deflation sequence are summarized in Table 1.

$(m,n,p|r)$ | $(n_{0},n_{1},n_{2},n_{3})$
---|---
$(3,4,5|47)$ | (152,149,148,148 )
$(4,4,5|62)$ | (201,194,193,193 )
$(4,5,5|76)$ | (251,251,251,251 )
$(5,5,5|97)$ | (395,344,339,339 )
Table 1. Deflation sequences for some solutions in [20, 21]

In Table 1, $(m,n,p|r)$ means the solution lie in the set
$\textbf{V}(m,n,p|r)$, which can be found in the Github database of the
article [20, 21]. In particular, the computation of $(395,344,339,339)$ for
$(5,5,5|97)$ spends the time about 4.8 hours on our server.

###  5.5. Deflation test on natural algorithms

When $r=mnp$, the definition of $\langle m,n,p\rangle$ in (5.1) provides a
solution of $B(m,n,p|mnp)$ which is denoted by $N(m,n,p)$. Before Strassen’s
algorithm [34], we usually use the algorithm induced in (5.1) for the
computation of matrix multiplication. So we call $N(m,n,p)$ the _natural
algorithm_.

Intuitively, $N(m,n,p)$ should be semiregular. So we should have
$n_{0}=n_{1}=n_{2}=n_{3}$ for their deflation sequences. When $2\leq m,n,p\leq
4$, our experimental results are summarized in Table 2. In Table 2, it is
interesting to see that there is a big drop between $n_{1}$ and $n_{2}$. It
tells us that the natural algorithms are geometric singular.

| $(n_{0},n_{1},n_{2},n_{3})$
---|---
$N(2,2,2)$ | (40,40,32,22 )
$N(2,2,3)$ | (72,72,48,34)
$N(2,3,3)$ | (126,126,54,50)
$N(3,3,3)$ | (216,216,72,72)
$N(3,3,4)$ | (324,324,96,96)
$N(3,4,4)$ | (480,480,126,126)
$N(4,4,4)$ | (704,704,164,164)
Table 2. Deflation sequences for natural algorithms

##  6\. final remarks and open problems

###  6.1. The deflation sequence as an invariant

In [28, Quest. 3.15], we observed that the rank of Jacobian of $B(m,n,p|r)$
may be served as an invariant to decide the equivalent classes under the de
Groote group action [5, 9]. Further discussions of de Groote group action can
be seen from [18, 24, 35], etc. In this paper, from the numerical experiment,
we also find that the deflation sequences are unchanged when the solutions are
transformed by the de Groote group action. So we have the following question.

######  Question 6.1.

For two points in $\textbf{V}(m,n,p|r)$, if their deflation sequences are
different, do they also lie in different group orbits under the de Groote
group action?

For the deflation sequence $(n_{0},n_{1},...)$, $n_{0}$ is the nullity of
Jacobian. So the deflation sequence contains more information and is more
useful to distinguish the equivalent classes. For the solutions in [17], in
our numerical experiment, it is easy to find two deflation sequences
$(n_{0},n_{1},...)$ and $(n_{0}^{\prime},n_{1}^{\prime},...)$ such that
$n_{0}=n_{0}^{\prime}$ but $n_{1}\neq n_{1}^{\prime}$.

However, for the solutions provied in [14], almost all solutions have the
deflation sequence:(197,197,197,197). So in this case, distinguishing
equivalent classes by deflation sequences is not so effective. On the other
hand, this implies that the tensor rank of $\langle 4,4,4\rangle$ should be
less than 49.

###  6.2. Deciding the singularity of a solution

Given a polynomial system and a solution. Which type of the solution belongs
to: numerical smooth/singular or geometric smooth/singular? The deflation
sequence only tells us partial information about the type.

###  6.3. The real deflation process

By now, the deflation processes in [7, 8, 16, 26] are given for the complex
algebraic set. In practical, the polynomial system we are interested is often
of real coefficients and has real solutions. For example, the Brent equations
are of this type. Is the results in [7, 8, 16, 26] also suitable to the real
algebraic set? For example, it is not hard to see that Conjecture 1 also has a
real counterpart.

###  6.4. On local real dimension and local complex dimension

From the relation between fast matrix algorithms and solutions of the Brent
equations [5, 18], we can see that integer, rational and real solutions are
more useful than complex solutions. For real solutions of the Brent equation,
we want to know the difference between the local real dimension and local
complex dimension. Related discussions are given in the paper [15].

###  6.5. On the numerical determination of local dimension

On how to determine the local dimension, Kuo and Li provided a numerical
method in [22]. In addition, in the neighbourhood of the solution, the
convergence behavior under the rank-$r$ projection iteration in [37] can also
help us identify the local dimension of the solution.

## Acknowledgments

Xin Li is supported by National Natural Science Foundation of China (Grant
No.11801506). Liping Zhang is supported by the Zhejiang Provincial Natural
Science Foundation of China (Grant No. LY21A010015) and Grant 11601484, the
National Natural Science Foundation, People’s Republic of China.

The discussion with Professor Zhonggang Zeng of Northeastern Illinois
University inspired us a lot. We are grateful to him for so many valuable
comments and suggestions. We are grateful to Yuan Feng, Xiaodong Ding,
Liaoyuan Zeng, Changfeng Ma and Jinyan Fan for their kind supports and
suggestions. Many thanks to Professor Petr Tichavský for many helpful
discussions when we prepare this paper. Xin Li show his appreciation to
Yaqiong Zhang for her support and patience.

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