数理解析研究所講究録
第 1544 巻 2007 年 13-21

13Real polynomials and flip relation
防衛大学校数学教育室 藤村雅代 (Masayo FUJIMURA)
Department of Mathematics,
National Defense Academy

masayo@nda.ac.jp
Abstract
be the space of all affine conjugacy classes of polynomial mapsLet
via the elementarytofromof degree . We define a map
points.
The image ofmultipliers
fixed
the
of
symmetric functions of. In this paper, we define the algebraic variety
is denoted byunderthat indicates essential property of M4, and introduce “flip relation” toconjugacy classes. First, we show that the image of the real moduli spacewith $discr\{G_{4}\}\geq 0$ , whereis the real section of
under
Next, we show connection between$G_{4}=0$ is the defining equation offlip relation and sign of the discriminant of G4.
$M_{n}$$M_{n}$$\Psi_{n}$$n$$\mathbb{C}^{n-1}$$M_{\mathfrak{n}}$$\Sigma(n)$$\Psi_{n}$$Q_{4}$$\Sigma(4)$$\Psi_{4}$$M_{4}(\mathbb{R})$$\mathcal{G}_{4}$1Introduction

be the space
Let $Poly_{n}$ be the space of all polynomial maps of degree , andof all affine conjugacy classes of these polynomials. We define a projection
via the elementary symmetric functions of the multipliers offromtothe fixed points. In [3], we show the projection is not surjective for every $n\geq 4$ .
and the complement
is denoted byunder
The image of. We remark that for $n=2$ or 3 thecalled the exceptional set, denoted bycoincides withexceptional set is empty and the real section of the space, the space of all conjugacy classes of polynomial
the real moduli spacemaps of degree with real coefficients (see [8]).
In this paper, we introduce “flip relation” and investigate relations between.
and the real moduli space M4
the real section of the space
and the algebraic variety that indicatesIn section 2, we define the map$Poly_{4}$. In section 3, we introduce flip relation to conjugacy
essential property ofclasses. First, we show $\Psi_{4}(M_{4}(\mathbb{R}))=\mathbb{R}^{3}\cap\Sigma(4)\cap\{G_{4}(s_{1}, s_{2}, s_{4})=0$ has areal root}, where $G_{4}(s_{1}, s_{2}, s_{4})=0$ is the defining equation of , as Theorem 1.
And next, we show connection between flip relation and sign of the discriminantas Theorem 2.
of the defining equation $G_{4}(s_{1}, s_{2}, s_{4})=0$ of$n$$M_{n}$$\Psi_{n}$$M_{n}$$\mathbb{C}^{n-1}$$M_{n}$$\mathbb{C}^{n-1}\backslash \Sigma(n)$$\Sigma(n)$$\Psi_{n}$$\mathcal{E}(n)$$\Sigma(n)$$M_{n}(\mathbb{R})$$n$$(\mathbb{R})$$\Sigma(4)$$\mathcal{G}_{4}$$\Psi_{4}$$g_{4}$$g_{4}$

2

The moduli space$M_{4}$

Let $Poly_{4}$ be the space of all polynomials of the form:$p(z)=a_{4}z^{4}+a_{3}z^{3}+a_{2}z^{3}+a_{1}z+a_{0}$$(a_{4}\neq 0)$.

14$p_{2}\in Poly_{4}$ are holomorphically conjugate, denoted by
Two mapswith $g\circ p_{1}\circ g^{-1}=p_{2}$ , where is the group of
if and only if there exists
all affine transformations.
of quartic polyThe space, $Poly_{4}/\sim$ ’ of holomorphic conjugacy classes
nomials is denoted by M4.
$z_{5}=\infty$ be the fixed points ofFor each $p(z)\in Poly_{4}$ , let , . . ,.
.
, i.e. $\mu_{j}=p’(z_{j})$ . Let$\mu_{5}=0$ the multipliers of,, and ,
be the elementary symmetric functions of these multipliers$p_{1}\sim p_{2}$$p_{1},$$g\in \mathfrak{U}$$\mathfrak{U}$$\langle p\rangle$$z_{1}$$p$$\cdot$$\mu_{4},$$\mu_{1}$$\sigma_{4,1},$$\sigma_{4,2},$$z_{4},$$\cdot$$\cdots,$$z_{j}$$\sigma_{4,5}$$\sigma_{4,1}=\mu_{1}+\mu_{2}+\mu_{3}+\mu_{4}$,,$\sigma_{4,2}=\mu_{1}\mu_{2}+\mu_{1}\mu_{3}+..\mu_{1}\mu_{4}+\mu_{2}\mu_{3}+\mu_{2}\mu_{4}+\mu_{3}\mu_{4}$,$\sigma_{4,3}=\mu_{1}\mu_{2}\mu_{3}+\mu_{1}\mu_{2}\mu_{4}+\mu_{1}\mu_{3}\mu_{4}+\mu_{2}\mu_{3}\mu_{4}$$\sigma_{4,4}=\mu_{1}\mu_{2}\mu_{3}\mu_{4}$$\sigma_{4,5}=0$.

,

These multipliers are invariant under taking an afBne conjugate.
The holomorphic index of a rational function at a fixed point
ed to be the complex number$f$$\zeta\in \mathbb{C}$is$defi\dot{n}$$\iota(f, \zeta)=\frac{1}{2\pi i}\oint\frac{dz}{z-f(z)}$,where we integrate in a small loop in the positive direction around .
The following results are well known as “Fatou’s index theorem”:$\zeta$$\bullet$$\bullet$If$\zeta$is a fixed point of multiplier$\mu\neq 1$, then$\iota(f, \zeta)=\frac{1}{1-\mu}$.

For any non-identical polynomial ,
$p$

,

(1)

$\sum_{\zeta\in \mathbb{C}}\iota(p, \zeta)=0$

where this summation is taken over all fixed points of .
$p$

be the elementary symmetric functions of the multipliersof $p(z)\in Poly_{4}$ . The following relation holds by Fatou’s index theorem.

Let$\sigma_{4,1},$$\cdots,$$\sigma_{4,4}$

Lemma (Theorem 1 in [4])Among$\sigma_{4,j’}s$, there is a linear relation$4-3\sigma_{4,1}+2\sigma_{4,2}-\sigma_{4,3}=0$.

Under the conjugacy of the action of , it can be assumed that any quartic
polynomial is (monic and “centered”, i.e. $p(z)=z^{4}+c_{2}z^{2}+c_{1}z+c_{0}$ . There
exceptare three monic and centered polynomiaJs in any conjugacy class$G(3)=$, and they are transformed each other under the action offor, where is the cube roots of unity.
$\mathfrak{U}$$\langle p\rangle$$\langle z^{4}+z\rangle$$\{1, \omega, \omega^{2}\}$

$w$

15
By this lemma, for a monic and centered quartic polynomial $p(z)=z^{4}+$$c_{2}z^{2}+c_{1}z+c_{0}$ , it is enoughto consider the three invariant values, determined by the transformation formula:of$\sigma_{4,1},$$\sigma_{4,2},$$\sigma_{4,4}$$\langle p\rangle$$\sigma_{4,1}=-8c_{1}+12$,$\sigma_{4,2}=4c_{2}^{3}-16c_{0}c_{2}+18c_{1}^{2}-60c_{1}+48$,$\sigma_{4,4}=16c_{0}c_{2}^{4}+(-4c_{1}^{2}+8c_{1})c_{2}^{3}-128c_{0}^{2}c_{2}^{2}+(144c0c_{1}^{2}-288c_{0}c_{1}$(2)+128 ) $c_{2}-27c_{1}^{4}+108c_{1}^{3}-144c_{1}^{2}+64c_{1}+256c_{0}^{3}$ .
$c_{0}$

There is a natural projection$\Psi_{4}$:M4$arrow$$\Sigma(4)$$w$$t\cup$$\langle p\rangle$$\mapsto$$(\sigma_{4,1}, \sigma_{4,2}, \sigma_{4,4})$,is the image of M4 under. We call, andmultiplier-coordinate space. The complement
is denoted bycalled the exceptional set.
In [5] and [2], we define the following parametrized algebraic variety, by
putting $c_{2}^{3}=c$ and $s_{j}=\sigma_{4,j}(j=1,2,4)$ . This variety indicates essential. or of $Poly_{4}$ .
property of the projection

where$\Sigma(4)$$\Sigma(4)=\{(\sigma_{4,1}, \sigma_{4,2}, \sigma_{4,4})\}$$\Psi_{4}$$\mathbb{C}^{3}\backslash \Sigma(4)$$\mathcal{E}(4)$$\Psi_{4}$Deflnition 1
An algebraic variety, is defined by$\mathcal{G}_{4}$in$\mathbb{C}^{3}\cong\{(s_{1}, s_{2}, s_{4})\}$

with a parameter$c\in \mathbb{C}$$G_{4}(s_{1}, s_{2}, s_{4})=262144(s_{1}-4)^{2}c^{2}+1024(27s_{1}^{4}+(-144s_{2}-576)s_{1}^{2}$$+(384s_{2}+1280)s_{1}\cdot+128s_{2}^{2}-256s_{2}-512s_{4}-768)c$$+(9s_{1}^{2}+24s_{1}-32s_{2}-48)^{3}=0$.

The number of conjugacy classes corresponding to a point
equals to the number of possible parameter values on. The following
result is obtained by counting the number of solution of the defining equation$(s_{1}, s_{2}, s_{4})\in \mathbb{C}^{3}$$c$$g_{4}$$c$$G_{4}(s_{1}, s_{2}, s_{4})=0$.

The exceptional set$\mathcal{E}(4)$is parametrized as follows (see [5]):$(s_{1}, s_{2}, s_{4})=(4,$ $s,$ $\frac{(s-4)^{2}}{4})$3,$s\in \mathbb{C}\backslash \{6\}$.

Real polynomials and flip relation

For the case of quadratic rational maps or cubic polynomial maps, the real
section of the moduli space coincides with the real moduli space, the space ofconjugacy classes of maps with real coefficients (see [7] and [8]). However, thisphenomena do not succeed to the case $n=4$ . We have the following results.

Theorem 1 ([6])$\Psi_{4}(M_{4}(\mathbb{R}))=\mathbb{R}^{3}\cap\Sigma(4)\cap${$G_{4}(s_{1},$ $s_{2},$$s_{4})=0$

has a real root}.

16
The assertion of Theorem 1 is directly obtained by the following three lemmas 1, 2 and 3.
be the space of all complex valued quartic polynomial maps
Letwith real coefficients.
$Poly_{4}(\mathbb{R})$

Lemma 1 If

$p\in Poly_{4}(\mathbb{R})$then$\Psi_{4}(\langle p\rangle)$

is in$\mathbb{R}^{3}$.

Proof. This lemma is clear from the multipliers of satisfy one of the following
three cases: ‘four real values’, ‘two real and a pair of complex conjugates’, or‘two pair of complex conjugates’.
$p$$\blacksquare$

Both figures indicate the surface where discriminant of.
vanishes, on the space

Figure 1:$\mathbb{R}^{3}=\{(s_{1}, s_{2}, s_{4})\}$$G_{4}(s_{1}, s_{2}, s_{4})=0$

,
, let
with
For a point$G_{4}(s_{1},s_{2},
s_{4})=0$
has two oeal rvots if and only if). Then, the equation

Lemma 2$s_{2},$ $s_{4}$$\langle p\rangle\in M_{4}(\mathbb{R})$$(s_{1}, s_{2,8_{4})}\in \mathbb{R}^{3}\cap\Sigma(4)$$s_{1}\neq 4$$(p\rangle$$\in\Psi_{4}^{-1}(s_{1}$.

Proof. At first, we renark that $G_{4}(s_{1}, s_{2}, s_{4})=0$ has real coefficients, since
.
in
point
If $G_{4}(s_{1}, s_{2}, s_{4})=0$ has two real roots, we can choose real values as coeffi.
cients of . Therefore )Conversely, we suppose $G_{4}(S_{1},82,8_{4})=0$ has a pair of complex conjugateroots, and let $p=z^{4}+c_{2}z^{2}+c_{1}z+c_{0}$ be a corresponding monic and centeredrepresentative. Then, from the argument of case . in the proof of Theorem 2. Therefore
described in below, the coefficient is never real and1$G_{4}(s_{1}, s_{2}, s_{4})=0$ have two real roots, counted including multiplicity.
$(s_{1}, s_{2}, s_{4})$$p$$\mathbb{R}^{3}$$\langle p$$\in M_{4}(\mathbb{R})$$c$$c_{2}$$\langle p\rangle\not\in M_{4}(\mathbb{R})$

Lemma 3 For each point $(4, s_{2}, s_{4})\in(\{4\}xR^{2})\cap\Sigma(4)$ , the linear equation$G_{4}(s_{1}, s_{2}, s_{4})=0$ has a real root.
That is, there is a unique conjugacy class incorresponding to thi8 point.
$M_{4}(\mathbb{R})$17
Proof. Let $p(z)=z^{4}+c_{2}z^{2}+c_{1}z+c_{0}$ be a representative such that$(4, s_{2}, s_{4})$$\Psi_{4}(\langle p\rangle)=$.

From the transformation formula (2), $s_{1}=4=-8c_{1}+12$ , we have $c_{1}=1$ .
Substituting $s_{1}=4$ to $G_{4}(s_{1}, s_{2}, s_{4})=0$ , we have$131072(s_{2}^{2}-8s_{2}-4s_{4}+16)c-32768(s_{2}-6)^{3}=0$ .

Therefore, the unique root is always real valued. Hence, we can choose realwith $c_{2}^{3}=c$ . Then,value
is also real valued from the transformation
formula. Thus we have the assertion.
$c$$c_{2}$$c_{0}$$\iota$We remark that, the intersection of the plane
and the locus
where the discriminant ofis vanished coincides with the set.
This set corresponds to$\{(4, s_{2}, s_{4})\}$$\{(4, s, \frac{(\epsilon-4)^{2}}{4})\}$$G_{4}$$\mathcal{E}(4)\cup\{(4,6,1)\}$.

Now, we introduce the following “flip relation” between two classes, to prove

Theorem 2.

Deflnition 2
Two conjugacy classes
and (are under flip relation, denoted by (, if there exist two representatives,
such that.
$q\rangle$$\langle p\rangle$$p\rangle$$\in M_{4}$$\mapsto\langle q\rangle$$p\in\langle p\rangle$$q\in(q\rangle$$p(\overline{z})=\overline{q(z)}$The flip relation is not an equivalence relation, because ($p\rangle$$\not\simeq\langle p\rangle$

in general.

The following theorem contrasts with Theorem 1.
Theorem 2 ([6]) Suppose that two classes
andwith. Then, the following two are equivalent.
$\langle p\rangle$$\langle p\rangle\neq\langle q\rangle$$\langle q\rangle$

satish$\Psi_{4}(\langle p\rangle)=\Psi_{4}(\langle q\rangle)\in \mathbb{R}^{3}$(1)$\langle p\rangle\mapsto\langle q\rangle$.

(2) The equation $G_{4}(s_{1}, s_{2}, s_{4})=0$ has no real root.
$G_{4}(s_{1}, s_{2}, s_{4})=0$ is a quadratic equation.
Remark 1
In the case of
Therefore the condition (2) in Theorem 2-equivalents to the condition “Theequation $G_{4}(s_{1}, s_{2}, s_{4})=0$ has negative discriminant”. As we see later, for every
point, the corresponding equation $G_{4}(s_{1}, s_{2}, s_{4})=0$
in
has real coefficients. Hence, if one of the roots is real, the other is also real.
$s_{1}\neq 4,$$(s_{1}, s_{2}, s_{4})$$\Psi_{4}(M_{4}(\mathbb{R}))$

The flip relation correlates between periodic cycles of and by symmetric
relation for real axis. The property of a periodic cycle of , attracting, repellingetc., implies the same one of . (See Figure 2.) We will show this fact as thefollowing proposition.
$p$

$q$

$p$

$q$

18

Figure 2: The left figure indicates the dynamics of $p_{1}(z)=z^{4}+iz^{2}+iz$ (upper),and $p_{2}(z)=z^{4}-0.39685iz^{2}+iz-0.66933$ (lower).
has a parabolic fixed point at the$(12-8i, 30-64i, 125-48i)$ . The map
origin with multiplier . The right figure indicates the dynamics of $q_{1}(z)=$$z^{4}-iz^{2}-iz$ (upper), and $q_{2}(z)=z^{4}+0.39685iz^{2}-iz-0.66933$ (lower).
$\Psi_{4}(\langle q_{1}\rangle)=\Psi_{4}(\langle q_{2}\rangle)=(12+8i, 30+64i, 125+48i)$ . The map
has a parabolic
fixed point at the origin with multiplier $-i$ . These maps satisfy, and dynamics of these maps are symmetry for the real axis.
and$\Psi_{4}(\Phi_{1}\rangle)=\Psi_{4}((p_{2}\rangle)=$$p_{1}$$i$$q_{1}$$\langle p_{1}\rangle\mapsto\langle q_{1}\rangle$$\langle p_{2}\rangle\angle_{\nabla}\langle q_{2}\rangle$

19. And suppose two, andProposition 1 Suppose $p \oint q,$ $i.e$ .
is a periodic point of ofrepresentatives and are chosen as (3). If
is a periodic point of of peri $odn$ with multiplierwith multiplier , then
. Therefore, the dynamics of and are symmetr3t for the real axis.
$\langle p\rangle\neq\langle q\rangle$

$p$

$n$

$q$

$\mu$

$\langle p\rangle\angle_{\nabla}\langle q\rangle$ca$peri\cdot od$$P$$z_{0}$$q$$p$$\overline{\mu}$$q$.
holds by
Proof. The equation
is a periodic point of of period , we obtain
Since
fact of a relation between two multipliers is given by$\overline{p(z)}=q(\overline{z})$$\overline{p^{n}(z)}=q^{n}(\overline{z})$

$n$

$p$

$z_{0}$$\overline{\mu}=\overline{\{p^{n}(z_{0})\}’}=\{\overline{p^{n}(z_{0})}\}’=\{q^{n}(\overline{z_{0}})\}’$$\overline{z_{0}}=q^{n}(\overline{z_{0}})$

. The

.
I

or , are invariant under the
Remark that the coefficients of the , i.e.
action of cube roots of unity.
Rom now on, we assume that and are monic and centered representative
quartic polynomial maps. If necessary, we consider or are the map that is.
taken conjugate by, we can choose two representatives with following forms:
If
$z$

$p$

$\overline{c_{1}}$$c_{1}$

$q$

$p$

$q$

$zrightarrow\omega z$$\langle p\rangle\mapsto\langle q\rangle$$p=z^{4}+c_{2}z^{2}+c_{1}z+c_{0}$Lemma 4.

,

Under the condition that, then$\langle p\rangle\mapsto\langle q\rangle$$(s_{1}, s_{2}, s_{4})\in \mathbb{R}^{3}$$\Psi_{4}(\langle q\rangle)=(s_{1}, s_{2}, s_{4})$(3)$q=z^{4}+\overline{c_{2}}z^{2}+\overline{c_{1}}z+\overline{c_{0}}$

.

with

$\langle p\rangle\neq\langle q\rangle$, if$\Psi_{4}(\langle p\rangle)=$, the following is directly given by the
By the condition (Proof.
transformation formula (2) (cf. Figure 2),$p\rangle$$\mapsto\langle q\rangle$

and$\langle p\rangle\in\Psi_{4}^{-1}(s_{1}, s_{2}, s_{4})$$\langle q$)$\in\Psi_{4}^{-1}(\overline{s_{1}},\overline{s_{2}},\overline{s_{4}})$.

Therefore, the assertion is clearly obtained. (See Figure 3.)

1. From Lemma 4, we
Proof of Theorem 2. First, we suppose that {. Hence, $G_{4}(s_{1}, s_{2}, s_{4})=0$ has real coefficients.
have
has a
, each representative corresponding to the pointAsfollows:
real coefficient of . Therefore, two representatives and are$p\rangle$$\mapsto\langle q\rangle$$(s_{1}, s_{2}, s_{4})\in \mathbb{R}^{3}$$(s_{1}, s_{2}, s_{4})$$s_{1}\in \mathbb{R}$

$p$

$z$

$p=z^{4}+c_{2}z^{2}+c_{1}z+c_{0}$,$give\bm{n}^{-}as$$q$$q=z^{4}+\overline{c_{2}}z^{2}+c_{1}z+\overline{c_{0}}$$(c_{1}\in \mathbb{R})$.

Now, we need to consider the following three cases.
$a$. The equation

has multiple roots:
In this case, there is a unique conjugacy class corresponding to the point
and. While, from the assumption ($\#\Psi_{4}^{-1}(s_{1},s_{2},
s_{4})=2$. This result contradicts with, we have$G_{4}(s_{1}, s_{2}, s_{4})=0$$(s_{1}, s_{2}, s_{4})$$p\rangle$$(s_{1}, s_{2}, s_{4})$$\#\Psi_{4}^{-1}(s_{1}, s_{2}, s_{4})=1$.

$\neq\langle q\rangle$$\Psi_{4}(\langle p\rangle)=\Psi_{4}((q\rangle)=$20

Figure 3: The left figure indicates the dynamics of $p(z)=z^{4}+(1.04113-$
0.28573 ) $z^{2}+0.82627z+(0.41265-0.10426i)$ . The map has two attracting
fixed points at $0.33745+0.71902i$ and 0. $21049-0.61365i$ with multiplier 0. $8i$
$and-0.8i$ respectively. The right figure indicates the dynamics of $q(z)=z^{4}+$$(1.04113+0.28573i)z^{2}+0.82627z+(0.41265+0.10426i)$ . We remark that,$\Psi_{4}(\langle p\rangle)=\Psi_{4}(\langle q\rangle)=(5.38983, 7.80949, 4.58847)\in \mathbb{R}^{3}$ , and.
$\dot{j}$$P$$\langle p\rangle\mapsto\langle q\rangle$$b$. The equation

has two real roots:
be two real roots of $G_{4}(s_{1}, s_{2}, s_{4})=0$ . In this case we canandLet. Then, two values
are also real values,
choose two real values$s_{2}=4c_{2}^{3}-16c_{0}c_{2}+18c_{1}^{2}-60c_{1}+48$. Hence the relation
since
this
with
the assumption. (Moreover,obtained,andresultcontradicts
is)is clear from
$G_{4}(s_{1}, s_{2}, s_{4})=0$$\tilde{c}_{2}^{3}$$c_{2}^{3}$$c_{0},\tilde{c}_{0}$$c_{2},\tilde{c}_{2}$$\langle p\rangle\not\simeq\langle q\rangle$$p \oint q$$c$$c_{2}^{3}\neq\tilde{c}_{2}^{3}.$. The equation
Letvalues$c_{2}^{3},\tilde{c}_{2}^{3}$

has a pair of complex conjugate roots:
be a pair of complex conjugate roots. We can choose two. Then,is given by $s_{2}=$byas$c_{2},\tilde{c}_{2}$$G_{4}(s_{1}, s_{2}, s_{4})=0$$c_{2}=\overline{\tilde{c}_{2}}$$c_{0}=\overline{\tilde{c}_{0}}$$c_{2}^{3}=\overline{\tilde{c}_{2}^{3}}$$4c_{2}^{3}-16c_{0}c_{2}+18c_{1}^{2}-60c_{1}+48$.

Therefore the equation $G_{4}(s_{1}, s_{2}, s_{4})=0$ has negative discriminant.
Conversely, suppose the following quadratic equation,$262144(s_{1}-4)^{2}c^{2}+1024(27s_{1}^{4}+(-144s_{2}-576)s_{1}^{2}+(384s_{2}$$+1280)s_{1}+128s_{2}^{2}-256s_{2}-512s_{4}-768)c+(9s_{1}^{2}$$+24s_{1}-32s_{2}-48)^{3}=0$,

has negative discriminant, i.e.
$1073741824(54s_{1}^{5}+(-81s_{2}-27s_{4}-135)s_{1}^{4}+(36s_{2}^{2}-144s_{2}-1008)s_{1}^{3}$$+(-4s_{2}^{3}+360s_{2}^{2}+(144s_{4}+2976)s_{2}+576s_{4}+4192)s_{1}^{2}+(-160s_{2}^{3}$$-2176s_{2}^{2}+(-384s_{4}-6400)s_{2}-$1280s4–$5376$ ) $s_{1}+16s_{2}^{4}+448s_{2}^{3}$.

$+(-128s_{4}+2176)s_{2}^{2}+(256s_{4}+3840)s_{2}+256s_{4}^{2}+768s_{4}+2304)<0$21
Then, for a monic and centered representative, theterm of degree one, i.e.
. Because the equation
is given by$G_{4}(s_{1}, s_{2}, s_{4})=0$ has negative discriminant with real coefficients, there exists
two roots
satisfying. We can choose. Then,holds by $s_{2}=4c_{2}^{3}-16c_{0}c_{2}+18c_{1}^{2}-60c_{1}+48$ . Now we obtain the following two
polynomial maps belonging to:$p\in\langle p\rangle=\Psi_{4}^{-1}(s_{1}, s_{2}, s_{4})$$c_{1}=-\frac{s-1}{8}\underline{2}$$c_{1)}$$c=c_{2}^{3},\tilde{c}_{2}^{3}$$c_{2}^{3}=\overline{\tilde{c}_{2}^{3}}$$c_{2}=\overline{\tilde{c}_{2}}$$c_{0}=\overline{\tilde{c}_{0}}$$(s_{1}, s_{2}, s_{4})$$p=z^{4}+c_{2}z^{2}+c_{1}z+c_{0}$Thus,$\langle p\rangle\mapsto\langle q\rangle$,.

$q=z^{4}+\tilde{c}_{2}z^{2}+c_{1}z+\tilde{c}_{0}=z^{4}+\overline{c_{2}}+c_{1}z+\overline{c_{0}}$holds. 1

References[1] A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. $Ec$ . Nonm. Sup., 18 (1985), 287-343.
[2] M. FMjimura, Complex Dynamical Systems of the Quartic Polynomials,
RIMS Koukyuroku, 1415 (2005), 176-184.

[3] M. FMjimura, Data on Multipliers as the Moduli Space of the Polynomials.
unpublished.
[4] M. Fujimura and K. Nishizawa, Moduli spaces and symmetry loci of polynomial maps, In Proc. of the 1997 Intemat. Symposium Symb. Alg. Compt.,(1997), 342-348.

[5] M. Fujimura and K. Nishizawa, Some Dynamical Loci of Quartic Polynomials, J. of Japan Soc. Symb. Alg. Compt., 11 (2005), 57-68.
[6] M. Fujimura and K. Nishizawa, The real multiplier-coordinate space ofthe quartic polynomials, accepted to Proc. of the Intemat. Conference

Nonlinear Anal. Convex Anal. 2005.

[7] J. Milnor, Geometry and dynamIcs of quadratic rational maps, $E\varphi e\dot{n}mental$
Mathematics, 2 (1993), 37-83.

[8] J. Milnor, Remarks on iterated cubic maps, Experiment. Math., 1 (1992),

5-24.
[9] K. Nishizawa, Parametrization by fixed-points multipliers of the polynomials with degree , RIMS Kokyuroku, 1199 (2001), 127-131.
$n$