remarkRemark hypothesisHypothesis claimClaim exampleExample Companion CurveW.
Wu and C. Chen ex_supplement

# A Companion Curve Tracing Method for Rank-deficient Polynomial Systems

Wenyuan Wu Chongqing Institute of Green and Intelligent Technology, Chinese
Academy of Sciences, Chongqing, China (, , https://www.arcnl.org).
<EMAIL_ADDRESS><EMAIL_ADDRESS>Changbo Chen11footnotemark: 1
Corresponding author.

###### Abstract

We propose a method for tracing implicit real algebraic curves defined by
polynomials with rank-deficient Jacobians. For a given curve $f^{-1}(0)$, it
first utilizes a regularization technique to compute at least one witness
point per connected component of the curve. We improve this step by
establishing a sufficient condition for testing the emptiness of $f^{-1}(0)$.
We also analyze the convergence rate and carry out an error analysis for
refining the witness points. The witness points are obtained by computing the
minimum distance of a random point to a smooth manifold embedding the curve
while at the same time penalizing the residual of $f$ at the local minima. To
trace the curve starting from these witness points, we prove that if one drags
the random point along a trajectory inside a tubular neighborhood of the
embedded manifold of the curve, the projection of the trajectory on the
manifold is unique and can be computed by numerical continuation. We then show
how to choose such a trajectory to approximate the curve by computing
eigenvectors of certain matrices. Effectiveness of the method is illustrated
by examples.

###### keywords:

rank-deficiency, real algebraic curve, curve tracing, numerical continuation,
penalty function, Tikhonov regularization

65H10, 14Q30, 90C23

##  1 Introduction

Given an implicit real algebraic curve in $\mathbb{R}^{n}$ defined by a finite
set of polynomials $f\subset\mathbb{R}[x_{1},\ldots,x_{n}]$, producing a
polygonal chain approximation of it is a classical problem. Existing numerical
continuation methods [Allgower2003] for solving this problem often require
that the Jacobian of $f$, denoted by $\mathcal{J}_{f}$, is of nullity one at
all (or almost all) points of the real zero set of $f$, that is $f^{-1}(0)$.
It remains a challenge to trace the curve defined by $f$ when
$\mathcal{J}_{f}$ is rank-deficient, that is of nullity greater than one,
which is the dimension of the curve. One typical example is when $f$ is a sum
of squares of polynomials or more generally when $f$ is nonnegtive.

For a smooth curve in $\mathbb{R}^{n}$ defined by
$f=\\{f_{1},\ldots,f_{n-1}\\}$, where $\mathcal{J}_{f}$ is of full rank at any
point of the curve, the main technical challenge would be to identify all
branches of $f^{-1}(0)$ and make sure that there is no jumping during curve
tracing. Identifying all branches of $f^{-1}(0)$ is the problem of computing
witness points for every connected component of $f^{-1}(0)$ [Rouillier2000,
Hauenstein2012, WR13]. Techniques for presenting or detecting curve jumping
also exist [Blum1997, Beltran2013, Martin2013, Yu2014, WRF2017].

For an almost smooth curve in $\mathbb{R}^{n}$ defined by
$f=\\{f_{1},\ldots,f_{n-1}\\}$, where $\mathcal{J}_{f}$ is of full rank at
nearly all points of the curve, one further difficulty is to trace across the
singular points and get the correct topology around the singular points. Many
work exist for handling such problems [Hong1996, Daouda2008, Cheng2010,
Gomes2014, DBLP:journals/jossac/ChenWF20, DBLP:journals/jossac/JinC20a].

In this paper, we are interested in the case that $\mathcal{J}_{f}$ is rank-
deficient at every point of $f^{-1}(0)$, such as when $f$ consists of a
polynomial in sum of squares. When $\mathcal{J}_{f}$ is rank-deficient, one of
the main obstacles is that it is hard to find a tracing direction since the
dimension of the nullspace of $\mathcal{J}_{f}$ is at least two. Our starting
point for solving this problem is a penalty function based method for
computing witness points for every connected component of $f^{-1}(0)$
[Wu2017]. We then extend it to a companion curve tracing method. The main idea
of the penalty function is to embed the curve in a high-dimensional smooth
manifold defined by a system $g$ with full rank Jacobians and compute the
minimum distance of a random point to the manifold while at the same time
penalizing the residual of $f$ at the local minima. To render the curve, one
natural idea is to move the random point (as a guiding point) along a
trajectory and hopefully the corresponding minima will be dragged
continuously. To implement this idea, there are two main challenges to
overcome. One is the potential occurence of discontinuity, which indeed may
happen as illustrated in Section 3 and proved in Section 5. Another is to make
sure the minima do move along the curve as one drags the guiding point along
the trajectory. We show that both challenges can be overcome and the idea of
moving the guiding point is indeed feasible if the point is moved along the
directions defined by eigenvectors of certain matrices and inside a tubular
neighborhood of the embedded manifold of the curve. The trajectory formed by
moving the random point is called a companion curve of $f^{-1}(0)$.

Interestingly, the penalty function method, although initially proposed in a
completely different context, turns out to be closely related to Tikhonov
regularization for solving rank-deficient nonlinear least squares problems
[Tikhonov95, Engl1996, Eriksson2005], which is explained in the preliminary
section.

The paper is structured as follows. In Section 2, we recall how to compute at
least one witness point for every connected component of an arbitrary real
algebraic variety, whose defining system allows to be rank-deficient, via the
so-called penalty function method [Wu2017]. In Section 3, we illustrate by a
simple example the challenge for tracing real algebraic curves defined by
rank-deficient systems, with the initial witness points provided by the
penalty function method, as well as the main idea of our companion curve
method for handling this problem. One weakness of the penalty function method
for computing witness points is that it always return a non-empty set of
points even the given real variety is empty. In Section 4, we provide a
sufficient criterion for testing emptiness of a variety by the penalty
function method. To successfully tracing a real algebraic curve, it is
important to make the approximate witness points close enough to the curve. In
Section 5, we propose a homotopy method for improving the precision of witness
points. With all these preparations and another tool from differential
geometry, namely Tubular Neighborhood Theorem, we propose a companion curve
method for curve tracing in Section LABEL:sec:pathtrack. The effectiveness of
the method is illustrated by several examples in Section LABEL:sec:ex.
Finally, in Section LABEL:sec:con, we draw the conclusion and propose several
ways to improve the current method.

##  2 Preliminaries

In this section, we recall some preliminary results that were introduced in
[Wu2017] for computing witness points of rank-deficient polynomial systems.

Let $x=(x_{1},\ldots,x_{n})$ and let
$f=\\{f_{1},...,f_{k}\\}\subset\mathbb{R}[x]$. Let $V_{\mathbb{R}}(f)$ be the
zero set of $f$ in $\mathbb{R}^{n}$. Let
$\mathfrak{a}=(\mathfrak{a}_{1},...,\mathfrak{a}_{n})\notin V_{\mathbb{R}}(f)$
be a point in $x$-space and consider the minimal distance from
$V_{\mathbb{R}}(f)$ to this point:

(1) $\displaystyle\min\;\sum_{i=1}^{n}(x_{i}-\mathfrak{a}_{i})^{2}$
$\displaystyle s.t.\hskip 28.45274ptf(x)=0.$

Clearly every semi-algebraically connected component of $V_{\mathbb{R}}(f)$
has at least one point attaining the local minimum. These points make the
following matrix lose full rank:

$A:=\left(\begin{array}[]{ccc}\partial f_{1}/\partial x_{1}&\cdots&\partial
f_{1}/\partial x_{n}\\\ \vdots&\ddots&\vdots\\\ \partial f_{k}/\partial
x_{1}&\cdots&\partial f_{k}/\partial x_{n}\\\
x_{1}-\mathfrak{a}_{1}&\cdots&x_{n}-\mathfrak{a}_{n}\\\ \end{array}\right).$

Note that the first $k$ rows of $A$ are exactly the the Jacobian matrix of $f$
w.r.t. $x$, denoted by $\mathcal{J}_{f}$. If $\mathcal{J}_{f}$ is rank-
deficient at every point of $V_{\mathbb{R}}(f)$, the matrix $A$ automatically
loses full rank and thus does not provide any extra helpful information on
computing the semi-algebraically connected components of $V_{\mathbb{R}}(f)$.

To overcome this algebraic rank deficiency problem, the paper [Wu2017]
introduces a penalty function based approach. Instead of solving the
optimization problem (1), one considers the following unconstrained
optimization problem:

(2) $\displaystyle\min\;\mu$
$\displaystyle=(\beta\cdot(f_{1}^{2}+\cdots+f_{k}^{2})+\sum_{i=1}^{n}(x_{i}-\mathfrak{a}_{i})^{2})/2.$

Note that as $\beta$ approaches infinity, $f_{i}$, $i=1,\ldots,k$ are forced
to be zero. Intuitively, this provides an approximate solution to the problem
(1) for large enough $\beta$. Indeed, the paper [Wu2017] proves the following
result justifying such intuitions.

######  Proposition  (Corollary $1$ in [Wu2017]).

Let $p$ be a local minimum of (1). There exists a local minimum $p^{\prime}$
of (2) for sufficiently large $\beta$, such that $\|p-p^{\prime}\|$ can be
arbitrarily small.

Moreover, one can get a rough estimation of the distance between $p$ and
$p^{\prime}$ via the notion of degree index.

######  Definition  (Definition $3$ in [Wu2017]).

For a given $v\neq 0\in\mathbb{R}^{n}$, let $f_{v}:=f(x=vt)$. Denote by
$\deg_{\min}(f_{v})$ the trailing degree of $f_{v}$. We call
$\deg_{ind}(f)=\max_{v}\deg_{\min}(f_{v})$ the degree index of $f$. Given a
point $p\in V_{\mathbb{R}}(f)$, the degree index of $f$ at $p$ is defined as
$\deg_{ind}(f(x+p))$.

######  Theorem 2.1 (Theorem $5$ in [Wu2017]).

For a random point $\mathfrak{a}\in\mathbb{R}^{n}$ and a sufficiently large
$\beta$, suppose that $p\in V_{\mathbb{R}}(f)$ attains the local minimal
distance to $\mathfrak{a}$. Then there is a solution $p^{\prime}$ of Equation
(3) such that $\|p^{\prime}-p\|=O(\sqrt[2\mathpzc{I}-1]{1/\beta}\,)$, where
$\mathpzc{I}=\max\\{\deg_{ind}(f_{i}(x+p)),i=1,...,k\\}$.

The local minima of (2) are exactly the points that vanish the gradient of
$\mu$, that is satisfying the following equation:

(3) $\left(\begin{array}[]{c}x_{1}\\\ \vdots\\\ x_{n}\\\
\end{array}\right)+\beta\cdot\mathcal{J}^{t}\cdot\left(\begin{array}[]{c}f_{1}\\\
\vdots\\\ f_{k}\\\
\end{array}\right)=\left(\begin{array}[]{c}\mathfrak{a}_{1}\\\ \vdots\\\
\mathfrak{a}_{n}\\\ \end{array}\right).\\\ $

where the $n\times k$ matrix $\mathcal{J}^{t}$ is the transpose of the
Jacobian of $f$.

The left hand side of Equation (3) defines a smooth mapping
$M:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$.

######  Lemma 2.2 (Lemma $2$ in [Wu2017]).

For almost all points
$\mathfrak{a}=(\mathfrak{a}_{1},...,\mathfrak{a}_{n})\notin
V_{\mathbb{R}}(f)$, $M^{-1}(\mathfrak{a})$ is a nonempty finite set and every
point of $M^{-1}(\mathfrak{a})$ is a regular point of $M$.

Proposition 2 and Lemma 2.2 together show that for almost all points
$\mathfrak{a}$ of $\mathbb{R}^{n}$, $M^{-1}(\mathfrak{a})$ contains points
meeting every semi-algebraically connected component of $V_{\mathbb{R}}(f)$.
We warn that $M^{-1}(\mathfrak{a})$ may contain extra points not belonging to
$V_{\mathbb{R}}(f)$. In particular, even $V_{\mathbb{R}}(f)$ is empty,
$M^{-1}(\mathfrak{a})$ is always nonempty. Numerically testing if a real
variety is empty in general is a difficult problem. We provide a partial
answer to this problem in Section 4.

Sometimes, it is useful to use the following two equivalent formulations to
(2).

Let $z=(z_{1},..,z_{k})$ be $k$ slack variables and
$g=\\{f_{1}+z_{1},f_{2}+z_{2},...,f_{k}+z_{k}\\}$. Note that we have
$g\subset\mathbb{R}[x,z]$ and $V_{\mathbb{R}}(g)\subseteq\mathbb{R}^{n+k}$.

$\displaystyle\min\;\mu$
$\displaystyle=(\beta\cdot(z_{1}^{2}+\cdots+z_{k}^{2})+\sum_{i=1}^{n}(x_{i}-\mathfrak{a}_{i})^{2})/2$
$\displaystyle s.t.\hskip 28.45274ptg(x,z)=0.$

Let $z_{i}=w_{i}/\sqrt{\beta}$, $i=1,\ldots,k$, and substitute them into (2).
Let $h=\\{f_{1}+w_{1}/\sqrt{\beta},\ldots,f_{k}+w_{k}/\sqrt{\beta}\\}$.

$\displaystyle\min\;$
$\displaystyle(w_{1}^{2}+\cdots+w_{k}^{2}+\sum_{i=1}^{n}(x_{i}-\mathfrak{a}_{i})^{2})/2$
$\displaystyle s.t.\hskip 28.45274pth=0.$

One nice thing about (2) and (2) is that the real varieties defined by $g$ and
$h$ are smooth submanifolds of $\mathbb{R}^{n+k}$.

######  Remark 2.3.

If we replace $\beta=1/t$, we obtain another equivalent formulation:

(6) $\displaystyle\min\;\mu$
$\displaystyle=((f_{1}^{2}+\cdots+f_{k}^{2})+t\sum_{i=1}^{n}(x_{i}-\mathfrak{a}_{i})^{2})/2.$

Such a formulation is exactly the Tikhonov regularization for nonlinear least
squares problems [Tikhonov95, Engl1996, Eriksson2005], where
$t\sum_{i=1}^{n}(x_{i}-\mathfrak{a}_{i})^{2})/2$ is the regularization term
for the minimization problem: $\min\|f\|^{2}$. Here, one must be cautious
about the choice of $\mathfrak{a}$, since Lemma 2.2 does not exclude the
possibility that the solution of problem (6) is not regular for certain
$\mathfrak{a}$, which indeed poses a challenge for curve tracing as explained
in next section.

##  3 An introductory example

In this section, we illustrate by an example the challenge of a pure numerical
method for tracing algebraic curves defined by rank-deficient polynomials as
well as the main idea of our companion curve method.

Let
$f:={x_{{1}}}^{6}-2\,{x_{{1}}}^{3}x_{{2}}+{x_{{2}}}^{2}=(x_{1}^{3}-x_{2})^{2}$.
Recall from Section 2 that, one can obtain approximate witness points of
$V_{\mathbb{R}}(f)$ by solving Equation (3). Choosing $\beta=10^{4}$ and
$\mathfrak{a}=(0,-1)$, the equation becomes
$\\{60000\,{x_{{1}}}^{11}-180000\,{x_{{1}}}^{8}x_{{2}}+180000\,{x_{{1}}}^{5}{x_{{2}}}^{2}-60000\,{x_{{1}}}^{2}{x_{{2}}}^{3}+x_{{1}},-20000\,{x_{{1}}}^{9}+60000\,{x_{{1}}}^{6}x_{{2}}-60000\,{x_{{1}}}^{3}{x_{{2}}}^{2}+20000\,{x_{{2}}}^{3}+x_{{2}}+1\\}$.
By the homotopy continuation method, say the one implemented in Hom4PS-2.0
[Lee2008], one obtains three approximate witness points: $(-0.3639,-0.0840)$,
$(-0.8296,-0.5982)$, $(0,-0.0364)$. Geometrically, these points are actually
the projection onto $(x_{1},x_{2})$-space of the following three local minima
of the optimization problem (2):

$(-0.3639,-0.0840,-0.1280),(-0.8296,-0.5982,-0.0739),(0,-0.0364,-0.1324).$

These three points attain the local minimum distance from the point $(0,-1,0)$
to the manifold $f+w/100=0$, as illustrated by Figure 1.

Figure 1: (Color online) Left: the random point $\mathfrak{a}=(0,-1,0)$ (in
red $\bullet$), the three local minima (in blue $+$), the smooth surface
defined by $f+w/100=0$ and its intersection with $w=0$ (gold curve). Right:
the random point $\mathfrak{a}=(0,-1)$ (in red $\bullet$), the three
approximate witness points (in blue $+$), and the curve $f=0$ (gold curve).

###  3.1 The challenge

Next we would like to generate more points of $V_{\mathbb{R}}(f)$ by curve
tracing with the three initial points. One natural idea is to consider the
following linear homotopy:

(7) $H_{\tau}(x,\tau)=x+\beta_{0}\mathcal{J}^{t}\cdot f-(\tau
p_{1}+(1-\tau)p_{0})\equiv 0,$

where $\beta_{0}=10000$ and one moves $\mathfrak{a}$ from $p_{0}=(0,-1)$ to
$p_{1}=(-0.5,-1.5)$ in a line. As long as the number of solutions of
$H(x,\mathfrak{a})=x+\beta_{0}\mathcal{J}^{t}\cdot f-\mathfrak{a}$ in $x$
remains unchanged and the graphs of the solutions of $H(x,\mathfrak{a})$ as
functions of $\mathfrak{a}$ remain disjoint and smooth, one should encounter
no much difficulty during curve tracing. However, as illustrated by the left
subfigure of Fig. 2, the path of $\mathfrak{a}$ crosses the discriminant locus
[LazardRouillier2007] of $H(x,\mathfrak{a})$, which can be obtained by a
Gröbner basis computation [LazardRouillier2007] by Lemma 5.1 in Section 5. As
illustrated by the right subfigure of Fig. 2, when $\mathfrak{a}$ approaches
the discriminant locus, the solution curve starting with $(0,-0.0364)$ (blue
$+$) and the solution curve starting with $(-0.3639,-0.0840)$ (purple
$\bullet$) get close to each other gradually and finally are no longer
solutions of $H(x,\mathfrak{a})$. Interestingly, if we apply Newton iteration
now starting with these non-solution points, they finally converge to the
third solution curve traced starting with the local minimum
$(-0.8296,-0.5982)$ (gold $\star$). Therefore, curve jumping happened when the
parameter $\mathfrak{a}$ crossed the discriminant locus.

Figure 2: (Color online) Left: the discriminant locus of $H(x,\mathfrak{a})$
(two “V” curves in dark green) and the segment $\overline{p_{0}p_{1}}$ (in
red). Right: the traced curve starting with the initial point
$(-0.8296,-0.5982)$ (gold $\star$), the traced curve starting with the initial
point $(0,-0.0364)$ (blue $+$), and the traced curve starting with the initial
point $(-0.3639,-0.0840)$ (purple $\bullet$).

###  3.2 The companion curve solution

Note that in Fig. 1, there are three points on the smooth surface attaining
the local minimum distance to the given random point $(0,-1,0)$. Intuitively,
there should only be one local minimum point if the given random point is
close enough to the surface and the surface is connected. This is indeed true,
which will be proved in Section LABEL:sec:pathtrack, thanks to the Tubular
Neighborhood Theorem in differential geometry.

Now suppose that $\mathfrak{a}=p_{0}$ is the initial random point and
$(x,w)=(q_{0},r_{0})$ is an approximate local minimum of the optimization
problem (2). Suppose that $\|r_{0}\|$ is small enough such that $x=q_{0}$ can
be seen as an approximate witness point of $V_{\mathbb{R}}(f)$. We first move
the point $\mathfrak{a}$ to another point $p_{1}$ on the segment
$\overline{p_{0}q_{0}}$ such that $(p_{1},0)$ is close to the surface and hope
that there is only one point $(q_{1},r_{1})$ on the surface with local minimum
distance to $(p_{1},0)$ nearby. We then pick a well chosen direction, say
$\vec{v}$, by some eigenvectors computation and move $\mathfrak{a}$ from
$p_{1}$ to $p_{1}^{\prime}$. Accordingly, $x$ is moved in the same direction
from $q_{1}$ to $q_{1}^{\prime}$ (may be further refined to
$q_{1}^{\prime\prime}$ by Newton iteration if necessary). It is possible that
$(p_{1}^{\prime},0)$ is now outside the tubular neighborhood of the surface
and one may repeat the previous step to drag $p_{1}^{\prime}$ to $p_{2}$ and
produce $q_{2}$. And then move $q_{2}$ to $q_{2}^{\prime}$, $p_{2}$ to
$p_{2}^{\prime}$, so on so forth. The polygonal chain formed by the sequence
of points $p_{1},p_{2},\ldots$ is called the companion curve of
$V_{\mathbb{R}}(f)$, as illustrated in Fig. 3.

Figure 3: (Color online) Left: illustrating the idea of companion curve
tracing method. Right: an approximation of $V_{\mathbb{R}}(f)$ (polygonal
chain in gold $\bullet$) and its companion curve (polygonal chain in blue
$+$). The initial random point $(0,-1)$ (in red $\bullet$) at the bottom was
moved to a point (in blue $\Box$) close to $V_{\mathbb{R}}(f)$.

##  4 Emptiness of real variety

In this section, we propose a criterion for testing the emptiness of a given
real variety.

Lemma 2.2 implies that all the solutions of Equation (3) can be obtained by
applying homotopy continuation methods. Among these solutions, we look for
solutions with small residuals i.e. $\|z\|\ll 1$. It is possible that such
points do not exist, which then provides strong evidence that
$V_{\mathbb{R}}(f)$ is empty. Intuitively, this is because if
$V_{\mathbb{R}}(f)$ is not empty, increasing the penalty factor $\beta$ will
force $\|z\|$ close to zero. Thus, the minimal value of $\mu$ will be slightly
larger than the distance from $\mathfrak{a}$ to $V_{\mathbb{R}}(f)$.

To study the relationship between $\mu_{\min}$ and the emptiness of
$V_{\mathbb{R}}(f)$, we homogenize the system $f$ by adding a variable $x_{0}$
satisfying a new equation $\bar{f}_{k+1}=\sum_{i=0}^{n}x_{i}^{2}-1=0$ to
obtain a homogenized system $\bar{f}$ except for the new inhomogeneous
equation. The corresponding unconstrained optimization problem is

(8) $\displaystyle\min\;\bar{\mu}$
$\displaystyle=(\beta\cdot(\bar{f}_{1}^{2}+\cdots+\bar{f}_{k}^{2}+\bar{f}_{k+1}^{2})+\sum_{i=0}^{n}(x_{i}-\mathfrak{a}_{i})^{2})/2.$

where $\mathfrak{a}$ is chosen randomly in the unit ball $B(0;1)$ of
$\mathbb{R}^{n+1}$.

######  Proposition 1.

Let $\bar{\mu}_{\min}$ be the global minimal value of $\bar{\mu}$ in the
optimization problem (8). If $\bar{\mu}_{\min}>2$, then
$V_{\mathbb{R}}(f)=\emptyset$.

######  Proof 4.1.

We prove it by contradiction. Suppose $V_{\mathbb{R}}(f)\neq\emptyset$. Then
$V_{\mathbb{R}}(\bar{f})\neq\emptyset$ and let $p\in V_{\mathbb{R}}(\bar{f})$.
We know that $p,\mathfrak{a}\in B(0;1)$ which implies $\|p-\mathfrak{a}\|\leq
2$. Thus, $\bar{\mu}_{\min}\leq\beta\cdot 0+\|p-\mathfrak{a}\|^{2}/2\leq 2$.
It contradicts the assumption $\bar{\mu}_{\min}>2$. $\square$

######  Example 4.2.

Let $f={x}^{4}+{y}^{4}-3\,xy+3/2$. We can verify that

$f={\frac{159}{784}}\,{x}^{4}+{\frac{15}{64}}\,{y}^{4}+{\frac{3}{50}}+\left(\frac{6}{5}-\frac{5}{4}\,xy\right)^{2}+\left({\frac{25}{28}}\,{x}^{2}-{\frac{7}{8}}\,{y}^{2}\right)^{2}>0.$

Homogenizing $f$ yields
$\bar{f}=\\{{x}^{4}+{y}^{4}-3\,{h}^{2}xy+3/2\,{h}^{4},{h}^{2}+{x}^{2}+{y}^{2}-1\\}$.
Choose $\mathfrak{a}=(0.2,0.5,0.3)$ and $\beta=10000$ and solve the
corresponding system of Equation (3) by Hom4Ps2. It gives $23$ real roots
among $111$ complex ones. The minimal value of $\bar{\mu}$ is $28.6$ at
$(x=0.565,y=0.565,h=0.596)$. By Proposition 1, it indicates that
$V_{\mathbb{R}}(f)=\emptyset$.

To apply this lemma we may choose sufficiently large $\beta$ to make
$\bar{\mu}_{\min}>4$ possible. However, for some positive polynomials it will
never happen no matter how large $\beta$ is.

######  Example 4.3.

Consider $f=(xy-1)^{2}+y^{2}$ and it is positive but arbitrarily close to
zero. Fixing $\mathfrak{a}=(0,0,0)$, by Equation (8), we have

$\bar{\mu}=\beta\left({h}^{4}-2\,{h}^{2}xy+{h}^{2}{y}^{2}+{x}^{2}{y}^{2}\right)^{2}+\beta\left({h}^{2}+{x}^{2}+{y}^{2}-1\right)^{2}+{h}^{2}+{x}^{2}+{y}^{2}.$

Numerical computation shows that $\bar{\mu}_{\min}=0.999975$ at the point
$(x=2.9\times 10^{-8},y=0.9999,h=8.5\times 10^{-9})$ when $\beta=10^{4}$.
Actually, $\bar{\mu}_{\min}$ must be no greater than $1$ since
$\bar{\mu}(0,1,0)=1$ for any $\beta$. It means that we cannot tell if the real
variety of $f$ is empty or not by Proposition 1.

Proposition 1 only gives a sufficient condition for
$V_{\mathbb{R}}(f)=\emptyset$. If $\bar{\mu}_{\min}<2$, deciding the emptiness
is an open question. In the rest of this paper, we always assume that
$V_{\mathbb{R}}(f)\neq\emptyset$.

##  5 Refinement

By Proposition 2, theoretically we can use Equation (3) to update the
approximate root $x^{\prime}$ by increasing $\beta$. Suppose we have all the
real solutions of Equation (3) for $\beta=\beta_{0}$ denoted by
$R_{\beta_{0}}$. When we increase $\beta$ from $\beta_{0}$ to $\beta_{1}$,
there are two ways to obtain $R_{\beta_{1}}$. One way is to solve Equation (3)
in the complex field and then keep the real solutions. Alternatively, we may
trace the real curves of the following homotopy starting from all points of
$R_{\beta_{0}}$ by moving $\beta$ from $\beta_{0}$ to $\beta_{1}$
continuously.

(9) $H(x,\beta)=(x-\mathfrak{a})+\beta\cdot\mathcal{J}^{t}\cdot f\equiv 0.$

But we have to be aware of singular Jacobian for a successful tracing.

######  Lemma 5.1.

For any polynomial system $f\subset\mathbb{R}[x]$, let
$F=\\{{x}-\mathfrak{a}+\beta\cdot\mathcal{J}^{t}\cdot{f}\\}$ and
$G=\\{F,\det(\frac{\partial F}{\partial
x})\\}\subset\mathbb{R}[x,\mathfrak{a},\beta]$. Then there is a nonzero
polynomial $\phi(\mathfrak{a},\beta)\in\langle G\rangle$.


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