# Arithmetic properties of multiplier polynomials for certain polynomial maps

Yuya Murakami  Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-
ku, Fukuoka, 819-0395, JAPAN<EMAIL_ADDRESS>,  Kaoru Sano
NTT Institute for Fundamental Mathematics, NTT Communication Science
Laboratories, NTT Corporation, 2-4, Hikaridai, Seika-cho, Soraku-gun, Kyoto
619-0237, Japan<EMAIL_ADDRESS>and  Kohei Takehira  Graduate School of
Science, Tohoku University, 6-3, Aoba, Aramaki-aza, Aoba-ku, Sendai, 980-8578,
Japan<EMAIL_ADDRESS>
###### Abstract.

We investigate the arithmetic properties of the multiplier polynomials for
certain $1$-parameter families of polynomials. In particular, we answer the
unsolved Excercise 4.12 (c) of the book “The Arithmetic of Dynamical Systems”
by J. H. Silverman for broader classes of polynomials than quadratic
polynomials. As a corollary, we obtain the uniform upper bound of the naive
height of parabolic parameters of unicritical polynomials. Moreover,
candidates for the parabolic parameter are determined. We have also confirmed
that it is a parabolic parameter, except $c=-3/2$.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Fundamental facts on Number fields
    2. 2.2 Fundamental facts on Complete valuation fields
    3. 2.3 Fundamental facts on dynamics
  3. 3 Multiplier polynomials for $z^{d}+c$
    1. 3.1 Integrality
    2. 3.2 Valuations of roots
    3. 3.3 Monicness
    4. 3.4 Integrality of parameters with periodic points of fixed multiplier
    5. 3.5 A remark on logistic maps
  4. 4 Multiplier polynomials for $z^{d+1}+cz$
    1. 4.1 Replacement of polynomials
    2. 4.2 Integrality
    3. 4.3 Valuations of roots
    4. 4.4 Monicness
  5. 5 Height bound of the parabolic parameters
    1. 5.1 Height bound for $z^{d}+c$
    2. 5.2 Height bound for $z^{d+1}+cz$
  6. 6 Determination of rational parabolic parameters
  7. 7 Remarks on other polynomial families
    1. 7.1 The family $(z-c)z^{d}+c$
    2. 7.2 The family $z^{d+2}+cz^{2}$
  8. A Equalities on dynatomic modular curves
  9. B Lists of $\delta_{m}(x)$ and $\Delta_{n,m}$ for our polynomials
  10. C Exact description of Newton polygons for $z^{d+1}+cz$ and $z^{d+1}-cz+c$

##  1\. Introduction

For a polynomial $f$ of degree $d$ over an integral domain $\mathcal{O}$ and
positive integer $n$, its $n$-th dynatomic polynomial $\Phi_{n}^{\ast}(z)$ (or
$\Phi_{f,n}^{\ast}(z)$ for clarity) is

(1.1) $\Phi_{n}^{\ast}(z)=\prod_{k|n}(f^{\circ k}(z)-z)^{\mu(n/k)},$

where $f^{\circ k}$ is the $k$-th iterated composition $f\circ
f\circ\cdots\circ f$ (see [Sil07, Theorem 4.5]). It’s degree is
$d_{n}=\sum_{k|n}d^{k}\mu(\frac{n}{k})$. For a root
$\alpha\in\overline{\operatorname{Frac}(\mathcal{O})}$ of $\Phi_{n}^{\ast}$,
the multiplier $\omega_{n}(\alpha)$ is the derivative $(f^{\circ
n})^{\prime}(\alpha)$. By the chain rule, it is written as

(1.2) $\omega_{n}(\alpha)=\prod_{i=0}^{n-1}f^{\prime}(f^{\circ i}(\alpha)),$

where $f^{\circ 0}(z)$ is regarded as $z$. The $n$-th multiplier polynomial
$\delta_{n}(x)$ of $f$ is defined by

(1.3)
$\delta_{n}(x)^{n}=\prod_{\Phi_{n}^{\ast}(\alpha)=0}(x-\omega_{n}(\alpha)),$

where $\delta_{n}$ is indeed a polynomial (cf [VH92]). Let
$\Phi^{\mathrm{cyc}}_{i}(x)$ be the $i$-th cyclotomic polynomial. For positive
integers $m,n$ such that $m|n$ and $m<n$, the quantity $\Delta_{n,m}$ is
defined by

(1.4)
$\Delta_{n,m}=\operatorname{Res}_{x}(\Phi^{\mathrm{cyc}}_{n/m}(x),\delta_{m}(x)).$

In addition, we define the quantity $\Delta_{n,n}$ by

(1.5) $\delta_{n}(1)=\Delta_{n,n}\prod_{m|n,m\neq n}\Delta_{n,m}.$

In [MV95, Section 3, Table 1], the quantity $\Delta_{n,m}$ for $f(z)=z^{2}+c$
and $n,m\leq 6$ is listed. Following the list, it is conjectured that
$\Delta_{n,m}$ for $f=z^{2}+c$ is a monic integral polynomial in $4c$ in
[Sil07, Excercise 4.12 (c)**]. In this paper, we generalize this conjecture
for uni-critical polynomial maps of degree $d\geq 2$ and prove them.

######  Theorem 1.1.

Let $f_{c}(z)=z^{d}+c$ for an integer $d\geq 2$. Then for all integers $n,m$
with $m|n$ and $m<n$, there is a polynomial $\Psi_{n,m}(t)\in\mathbb{Z}[t]$
such that

(1.6) $\Delta_{n,m}(c)=\Psi_{n,m}(d^{d}c^{d-1}).$

Moreover, the polynomial
$(-1)^{\varphi(n)\left(\frac{d_{m}}{m}+d_{m}(d-1)\right)}\Psi_{n,m}(t)$ is
monic, where $d_{m}=\sum_{k|m}d^{k}\mu\left(\frac{m}{k}\right)$.

Theorem 1.1 is an immediate consequence of the following theorem.

######  Theorem 1.2.

Let $f_{c}(z)=z^{d}+c$ for an integer $d\geq 2$. Then for all positive
integers $m$, the multiplier polynomial $\delta_{m}(x)$ is in the ring
$\mathbb{Z}[d^{d}c^{d-1},x]$ and
$(-1)^{\frac{d_{m}}{m}+d_{m}(d-1)}\delta_{m}(x)$ is monic in $d^{d}c^{d-1}$,
where $d_{m}=\sum_{k|m}d^{k}\mu\left(\frac{m}{k}\right)$.

Combining Theorem 1.1 and the bound of the Multibrot set, we get an upper
bound of the height of parabolic parameters $c$ as follows.

######  Theorem 1.3.

For $\gamma\in\mathbb{C}$, assume that $f_{\gamma}(z)$ has a parabolic
periodic point. Then $\gamma$ is in $\overline{\mathbb{Q}}$ and the inequality
$H(\gamma)\leq(2d^{d})^{1/(d-1)}$ holds.

The following conjecture is a central problem in the arithmetic of dynamics.

######  Conjecture 1.4 ([MS94]).

Let $N$ and $D$ be positive integers and $d\geq 2$ an integer. Let $K$ be a
number field. Then there is a constant $C(N,D,d,K)$ such that for any morphism
$f\colon\mathbb{P}^{N}\longrightarrow\mathbb{P}^{N}$ of degree $d$ defined
over $K$, the set

(1.7) $\left\\{x\in\mathbb{P}^{N}(L)\ \middle|\ [L:K]\leq D,x\text{ has a
finite }f\text{-orbit}\right\\}$

has almost $C(N,D,d,K)$ elements.

Assuming a generalization of the $abc$-conjecture called the
$abcd$-conjecture, Conjecture 1.4 is proved in [Loo21]. Even for quadratic
polynomial maps, Conjecture 1.4 is still widely open. The non-existence of
rational periodic rational points of the exact period $4$ and $5$ are shown in
[Mor98] and [FPS97], respectively. Assuming Birch–Swinerton-Dyer conjecture,
the same argument for the exact period $6$ is proved in [Sto08].

As a consequence of Theorem 1.1, we get a contribution to Conjecture 1.4. By
Theorem 1.3 and Northcott property of the height function, the set of the
parabolic rational parameter $c$ is finite, and the candidates are explicitly
given. Hence, we can determine the parabolic rational parameters through case-
by-case analysis.

######  Theorem 1.5.

For $\gamma\in\mathbb{Q}$, assume that $f_{\gamma}(z)=z^{2}+\gamma$ has a
parabolic periodic point in $\mathbb{C}$. Then $\gamma$ is either
$-7/4,-3/2,-5/4,-3/4$, or $1/4$.

We show the existence of parabolic periodic points for $\gamma$ other than
$-3/2$. But we don’t know if $z^{2}-3/2$ has a parabolic periodic point. In
[Oku22], readers can find the picture of Mandelbrot set for $z^{2}+\gamma$
with plots of parameters admitting parabolic periodic points in $\mathbb{C}$.

With some appropriate modifications, the same story seems to work for other
$1$-parameter families of polynomials. We prove a similar theorem for
$z^{d+1}+cz$ for $d\geq 1$.

######  Theorem 1.6.

Let $f_{c}(z)=z^{d+1}+cz$. For a positive integer $m$, the polynomial
$d^{\deg_{z}\Phi_{m}^{*}/(d+1)}\delta_{m}(x)$ is an element of
$\mathbb{Z}[dc,x]$ and monic in $dc$ up to multiplication by $\pm 1$.

######  Theorem 1.7.

Let $f_{c}(z)=z^{d+1}+cz$. For $\gamma\in\mathbb{C}$, assume that
$f_{\gamma}(z)$ has a parabolic periodic point in $\mathbb{C}$. Then $\gamma$
is in $\overline{\mathbb{Q}}$ and the inequality
$H(\gamma)\leq(2d+2)^{1/d}(d+1)$ holds.

######  Remark 1.8.

When we focus on the unicritical polynomial maps, the result [Ing19, Theorem
1] tells us that for any $d\geq 2$, $n\geq 1$, there exist constants $A$ and
$B$ such that for any $\lambda\in\overline{\mathbb{Q}}^{\times}$ and a
polynomial $f\in\overline{\mathbb{Q}}[z]$ admitting $n$-periodic points of
multiplier $\lambda$, the height $h(f)$ is bounded above by $Ah(\lambda)+B$.
While his original theorem holds for general polynomial maps of degree $d$ and
any multipliers values, the constants $A$ and $B$ depend on the period $n$.
From this perspective, Theorem 1.3 and Theorem 1.7 are although theorems for
specific polynomial maps and specific multipliers, they have the advantage
that the height bound is uniform for the period.

It is natural to expect that a similar argument works for any $1$-parameter
family of polynomials, but this is misleading. As the following theorem shows,
there are cases where the leading coefficient of $\Delta_{n,m}$ in the
parameter grows as $n$ grows. Thus, the uniform integrality for general
$1$-parameter families cannot be shown.

######  Theorem 1.9.

Let $\Delta_{n,1}$ be the first multiplier polynomial for
$f_{c}(z)=z^{d+2}+cz^{2}$. Then the leading coefficient of $\Delta_{n,1}$ in
$c$ is $\pm d^{d\varphi(n)}\Phi^{\mathrm{cyc}}_{n}(2)$. In particular, for all
$n\geq 7$, the leading coefficient has a prime divisor $q$ such that $q\equiv
1\bmod n$.

Morton–Vivaldi shows the following theorem on the relation between
$\Phi_{n}^{*}(z)$ and $\Delta_{n,m}$.

######  Theorem 1.10 ([MV95, Theorem 2.2]).

Let $f(z)$ be a monic polynomial over an integral domain $\mathcal{O}$. Then,
we have

(1.8)
$\operatorname{Res}_{z}\left(\Phi_{n}^{*}(z),\Phi_{m}^{*}(z)\right)=\pm\Delta_{n,m}^{m}$

for all positive integers $n$ and $m$ such that $n>m$ and $m|n$.

######  Remark 1.11.

In [Oku22], Theorem 1.10 is generalized to rational functions on the
projective line.

In the proof of Theorem 1.10, Morton–Vivaldi considered the expression

(1.9)
$\operatorname{Res}_{z}\left(\Phi_{m}^{*}(z),\Phi_{k}^{*}(z)\right)=(\text{unit})\cdot\prod_{\alpha\in
Z(\Phi_{k}^{*})}\Phi^{\mathrm{cyc}}_{m/k}\left(\left(f^{\circ
k}\right)^{\prime}(\alpha)\right)$

in [MV95, p. 575, line 14], where $\Phi^{\mathrm{cyc}}_{n}$ is the $n$-th
cyclotomic polynomial. In connection, we give the following equalities on
dynatomic modular curves.

######  Proposition 1.12.

Let $f(z)$ be a monic polynomial over an integral domain $\mathcal{O}$ and $k$
and $m$ be positive integers with $k\mid m$. Let
$z_{0}\in\overline{\operatorname{Frac}(R)}$ be a periodic point of $f$ of
exact period $k$ and $\lambda\coloneqq\omega_{k}(z_{0})$. We denote the
$k$-part and the prime to $k$-part of $m$ by

(1.10) $\widetilde{m}\coloneqq\prod_{p\mid k}p^{v_{p}(m)},\quad
m^{\prime}\coloneqq\prod_{p\nmid k}p^{v_{p}(m)}=\frac{m}{\widetilde{m}}$

respectively, where $v_{p}$ is the $p$-adic valuation with respect to a prime
number $p$. Then, we have

(1.11)
$\prod_{d\mid\widetilde{m}}\Phi_{f,dm^{\prime}}^{*}(z_{0})=\Phi_{f^{\circ\widetilde{m}},m^{\prime}}^{*}(z_{0})=\Phi^{\mathrm{cyc}}_{n^{\prime}}\left(\lambda^{\widetilde{m}/k}\right).$

######  Remark 1.13.

When the previous version of this paper was completed, we shared it with
Yûsuke Okuyama. Then, he let us know that the calculation of multiplier
polynomials is dealt with in Valentin Huguin’s thesis [Hug21]. We found that
Theorem 1.2 is the same statement as [Hug21, Proposition 1.51] except for the
explicit description of the sign. Although we looked forward to published
papers in which [Hug21, Proposition 1.51] is written, they are not found.

The proof of ours and that of [Hug21, Proposition 1.51] are independent, and
Theorem 1.6 suggests that our techniques can be used for broader classes of
polynomial maps than unicritical polynomials. In addition, the uniform height
bound and the determination of rational parabolic parameters are new results.

### Organization of this paper

Section 2 is devoted to preparing notation and some fundamental facts on
valuations and complex dynamics.

We prove the theorems on unicritical polynomials $z^{d}+c$ in Section 3. The
fact that $\delta_{m}(x)$ is in the ring $\mathbb{Z}[d^{d}c^{d-1},x]$ is
proved in Section 3.1. In Section 3.2, we calculate the $c^{-1}$-adic
valuation of roots of $\Phi_{n}^{\ast}$ and their multipliers through the
discussion of Newton polygon. Using these calculations, we show the monicness
of $\delta_{m}(x)$ in $d^{d}c^{d-1}$ in Section 3.3. In Section 3.4, we prove
more general results than Theorem 1.1 as a corollary of the results
established in the previous sections. Our results on uni-critical polynomials
imply the results on the bifurcation parameters of the logistic maps. It is
written in Section 3.5. We prove the theorems on non-unicritical polynomials
$z^{d+1}+az$ in Section 4. In Section 4.1, we reduce the assertion to
Proposition 4.2 on the map $\tilde{f}(z)=z^{d+1}-cz^{d}+c$ by dividing by the
automorphisms of $z^{d+1}+cz$. Proposition 4.2 asserts that
$d^{m((d+1)^{k-1}-1)/d}R_{k,m}(x)$ is in the ring $\mathbb{Z}[dc,x]$ and is
monic in $dc$ up to multiplication by $\pm 1$, where $R_{k,m}(x)$ is the
resultant of some polynomials. In Section 4.2, we show that
$d^{m((d+1)^{k-1}-1)/d}R_{k,m}(x)$ is in the ring $\mathbb{Z}[dc,x]$. In
Section 4.3, we calculate the valuations of quantities involving the $n$-th
periodic points of $\tilde{f}$. In Section 4.4, we prove the statement on the
monicness of Proposition 4.2

In Section 5.1 and Section 5.2, we give the upper bound of the heights of the
parabolic parameters of the polynomial maps $z^{d}+c$ and $z^{d+1}+az$,
respectively. Using this bound, the rational parabolic parameters are
determined in Section 6.

We prove Proposition 1.12 in Appendix A. We give lists of computation of
$\delta_{m}$ and $\Delta_{n,m}$ for some polynomials in Appendix B. In
Appendix C, we give the complete description of Newton polygon of some
polynomials, parts of which are proved in Section 4.3 and used in Section 4.4.

## Acknowledgement

The authors would like to thank Takashi Taniguchi, Kazunari Sugiyama, and
Yasuhiro Ishitsuka, who organized “Number Theory Summer School 2023” and gave
the authors a chance to discuss the topic of this paper. The authors also
would like to thank Rob Benedetto, Ben Hutz, Patrick Ingram, Yûsuke Okuyama,
Joe Silverman, and Takayuki Watanabe for giving some comments on the recent
research of parabolic parameters. The first author would like to thank
Masanobu Kaneko for giving many comments. The third author would like to thank
Nobuo Tsuzuki for his helpful comments. The first author is supported by JSPS
KAKENHI Grant Number JP23KJ1675. The second author is supported by JSPS
KAKENHI Grant Number JP20K14300. The third author is financially supported by
JSPS KAKENHI Grant Number JP 22J20227 and the WISE Program for AI Electronics,
Tohoku University.

##  2\. Preliminaries

Let $K$ be a field and $F,G$ be polynomials over $K$. Denote $Z(F)$ the
multiset of its roots in $\overline{K}$ counted with multiplicity. Write
$F(z)=\sum_{i=0}^{n}a_{i}z^{i}$ and $G(z)=\sum_{j=0}^{m}b_{j}z^{j}$ with
$a_{n}\neq 0$ and $b_{m}\neq 0$, Then, the resultant of $F$ and $G$ with
respect to $z$ is the quantity

(2.1) $\displaystyle\operatorname{Res}_{z}(F,G)$
$\displaystyle=a_{n}^{m}b_{m}^{n}\prod_{\alpha\in Z(F)}\prod_{\beta\in
Z(G)}(\alpha-\beta)$ (2.2) $\displaystyle=a_{n}^{m}\prod_{\alpha\in
Z(F)}G(\alpha).$

######  Lemma 2.1.

Let $F$ and $G$ as above.

  1. (i)

The quantity $\operatorname{Res}_{z}(F,G)$ is equal to the following
determinant of $(m+n)\times(m+n)$ matrix

(2.3) $\begin{vmatrix}a_{n}&a_{n-1}&\cdots&\cdots&a_{0}&&&\\\
&a_{n}&a_{n-1}&\cdots&\cdots&a_{0}&&\\\ &&\ddots&&&&\ddots&\\\
&&&a_{n}&a_{n-1}&\cdots&\cdots&a_{0}\\\
b_{m}&b_{m-1}&\cdots&\cdots&b_{0}&&&\\\
&b_{m}&b_{m-1}&\cdots&\cdots&b_{0}&&\\\ &&\ddots&&&&\ddots&\\\
&&&b_{m}&b_{m-1}&\cdots&\cdots&b_{0}\end{vmatrix},$

where the first $m$ row vectors are coefficients of $F$, and the latter $n$
low vectors are coefficients of $G$.

  2. (ii)

Let $\phi\in K[z]$ be a non-constant monic polynomial. Then we have

(2.4)
$\operatorname{Res}_{z}(F\circ\phi,G\circ\phi)=\operatorname{Res}(F,G)^{\deg\phi}.$

###  2.1. Fundamental facts on Number fields

For a number field $K$, let $M_{K}$ be the set of standard absolute values on
$K$ whose restriction to $\mathbb{Q}$ is a $p$-adic absolute value or the
Archimedean absolute value. We write $M_{K}^{\infty}$ for the archimedean
absolute values of $K$ lying above $|\cdot|_{\infty}$ and $M_{K}^{0}$ for the
non-archimedean absolute values of $K$ lying above the $p$-adic absolute
values of $\mathbb{Q}$. For $v\in M_{K}$, let $K_{v}$ denote the completion of
$K$ at $v$. Let $n_{v}$ be the local degree $[K_{v}:\mathbb{Q}_{p}]$, where
$p$ is the prime such that $v$ lying above the $p$-adic absolute value. For
$a\in K$, we set $\|a\|_{v}\coloneq|a|_{v}^{n_{v}}$. The (absolute
logarithmic) height function $h\colon K\longrightarrow\mathbb{R}$ is defined
by

(2.5) $h(a)\coloneq\frac{1}{[K:\mathbb{Q}]}\sum_{v\in
M_{K}}\log\max\\{1,\|a\|_{v}\\}.$

for $a\in K$. The following theorems yield the well-definedness of $h$ as a
function on $\mathbb{P}^{1}(\overline{\mathbb{Q}})$.

######  Theorem 2.2 (cf [Lan94, Section II.1]).

Let $L/K/\mathbb{Q}$ be a tower of number fields and $v\in M_{K}$ be an
absolute value on K. Then

(2.6) $\frac{1}{[L:K]}\sum_{\begin{subarray}{c}w\in M_{L}\\\
w|v\end{subarray}}n_{w}=n_{v}.$

######  Theorem 2.3 (cf [Lan94, Section V.1]).

Let $K$ be a number field. Then, for all $a\in K^{\times}$, we have

(2.7) $\prod_{v\in M_{K}}\|a\|_{v}=1.$

The following theorem is a specific version of a well-known theorem called the
Northcott property, which is used to prove finiteness theorems in Diophantine
geometry.

######  Theorem 2.4.

Let $K$ be a number field. Then for any integer $d\geq 1$ and any constant
$B>0$, the set

(2.8) $\\{a\in\overline{K}\ |\ [K(a):K]\leq d\text{ and }h(a)<B\\}$

is a finite set.

###  2.2. Fundamental facts on Complete valuation fields

The facts written in this section will be used in the latter sections for
$K_{0}((c^{-1}))$, and the $c^{-1}$-adic absolute value, where $K_{0}$ is some
field and $K_{0}((t))$ is the field of Laurent series on $t$. $K_{0}((t))$ is
the fraction field of the ring of formal power series.

Throughout this section, let $K$ be a complete non-archimedean valuation field
for an additive valuation $v$.

######  Theorem 2.5 (cf [Neu99, Chapter II, (4.8) Theorem]).

For any algebraic extension $L/K$, a unique extension $w$ of $v$ to $L$
exists. Moreover, when $L/K$ has a finite degree $n$, it is given by the
formula

(2.9) $w(a)=\frac{1}{n}v\left(N_{L/K}(a)\right)$

for $a\in L$, where $N_{L/K}\colon L\longrightarrow K$ is the norm map.

######  Definition 2.6.

Let $F(x)=\sum_{i=0}^{n}a_{i}x_{i}\in K[x]$ be a polynomial satisfying
$a_{n}\neq 0$. To each term $a_{i}x^{i}$, we associate a point
$(i,v(a_{i}))\in\mathbb{R}^{2}$, where we regard $(i,\infty)$ when $a_{i}=0$.
The Newton polygon of $F$ is the lower convex envelope of the set of the
points

(2.10) $\\{(i,v(a_{i}))\ |\ 0\leq i\leq n\\}.$

When $a_{i}=0$ for all $1\leq i\leq k-1$ and $a_{k}\neq 0$, we regard that the
Newton polygon contains the line segment of slope $-\infty$ and of horizontal
length $k$.

######  Theorem 2.7 (cf [Neu99, Chapter II, (6.3) Proposition]).

Let $f(x)=\sum_{i=0}^{n}\in K[x]$ be a polynomial with $a_{n}\neq 0$. Let $L$
be the splitting field of $f$ over $K$ and $w$ be the unique extension of $v$.
If a line segment of slope $-m$ and of horizontal length $r$ occurs in the
Newton polygon of $f$, then $f(x)$ has precisely $r$ roots
$\alpha_{1},\ldots,\alpha_{r}\in L$ with the valuation

(2.11) $w(\alpha_{1})=\cdots=w(\alpha_{r})=m.$

For $F=\sum_{i=m}^{\infty}a_{i}t^{i}\in K_{0}((t))$ with $a_{m}\neq 0$, we set
$v_{t}(F)=m$. If $F$ is a polynomial in $c$, then we have

(2.12) $\deg_{c}(F)=-v_{c^{-1}}(F).$

We need the following construction to prove Theorem 1.6 in Section 4.

Let $K_{0}$ be a field, and $v$ be an additive valuation on $K_{0}$. Then we
can construct an additive valuation $\tilde{v}$ on $K_{0}(t)$ by

(2.13) $\tilde{v}\left(\sum_{i=0}^{n}a_{i}x^{i}\right)=\inf_{0\leq i\leq
n}v(a_{i}),$

where $a_{i}\in K_{0}$.

For example, by applying the above construction for
$\overline{\mathbb{Q}((c^{-1}))}\subset\overline{\mathbb{Q}((c^{-1}))}(y)\subset\overline{\mathbb{Q}((c^{-1}))}(x,y)$,
we obtain an additive valuation $v$ on $\overline{\mathbb{Q}((c^{-1}))}(x,y)$
which satisfies

  * •

$v=-\deg_{c}$ on $\mathbb{Q}[c,x,y]$,

  * •

$v(x+f)=\min\\{0,v(f)\\}$ for all $f\in\overline{\mathbb{Q}((c^{-1}))}(y)$,
and

  * •

$v(y+\alpha)=\min\\{0,v_{c^{-1}}(\alpha)\\}$ for all
$\alpha\in\overline{\mathbb{Q}((c^{-1}))}$.

###  2.3. Fundamental facts on dynamics

Let $\alpha\in\mathbb{C}$ is a $n$-periodic point of $f(z)\in\mathbb{C}[z]$.
If $\omega_{n}(\alpha)$ is a root of unity, we say that $\alpha$ is a
parabolic periodic point. If $|\omega_{n}(\alpha)|<1$, we say that $\alpha$ is
an attracting periodic point. For such a periodic point, the following theorem
is well-known.

######  Theorem 2.8 (cf [Mil06, Theorem 8.6, 10.15]).

Let $f$ be a rational map on $\mathbb{P}^{1}$ of degree $d\geq 2$. Let
$\alpha\in\mathbb{P}^{1}(\mathbb{C})$ be a parabolic or attracting
$n$-periodic point of $f$. Then there is a critical point $\beta$ of $f$
satisfying

(2.14) $\lim_{m\to\infty}f^{\circ(nm+i)}(\beta)=\alpha$

for some $0\leq i\leq n-1$.

For a polynomial map $f_{c}(z)=z^{d}+c$ with $d\geq 2$, we say that
$\gamma\in\mathbb{C}$ is a parabolic parameter if $f_{\gamma}$ has a parabolic
periodic point in $\mathbb{C}$. We set

(2.15) $\mathbb{M}_{d}\coloneq\\{\gamma\in\mathbb{C}\colon(f_{\gamma}^{\circ
n}(0))_{n=0}^{\infty}\text{ is bounded}\\}.$

Sometimes $\mathbb{M}_{d}$ is called Multibrot set. By Theorem 2.8, parabolic
parameters are in $\mathbb{M}_{d}$. Moreover, they are known to lie at the
nodes of $\mathbb{M_{d}}$.

##  3\. Multiplier polynomials for $z^{d}+c$

In this section, we fix an integer $d\geq 2$ and consider a polynomial map
$f_{c}(z)\coloneqq z^{d}+c$ with a parameter $c$. This section aims to prove
Theorem 1.2. We prove that $\delta_{m}(x)$ $(m\geq 1)$ is in
$\mathbb{Z}[d^{d}c^{d-1},x]$ in Proposition 3.3. The monicness of
$\delta_{m}(x)$ in $d^{d}c^{d-1}$ is proved in Proposition 3.5.

###  3.1. Integrality

In this subsection, we show the part of the statement in Theorem 1.2 that
$\delta_{m}(x)$ is an element of $\mathbb{Z}[d^{d}c^{d-1},x]$.

######  Lemma 3.1.

For a positive integer $k$, let $F_{k}(z):=f_{c}^{\circ k}(z)-z$. Then, the
following statements hold.

  1. (i)

The polynomial $d^{d^{k}/(d-1)}F_{k}(z/d^{1/(d-1)})$ is in the ring
$\mathbb{Z}[d^{d/(d-1)}c,z]$ and is monic in $z$.

  2. (ii)

For $\alpha\in Z(F_{k})$, the element $d\alpha^{d-1}$ is integral over
$\mathbb{Z}[d^{d/(d-1)}c]$.

###### Proof.

To prove the statements of (i), it is enough to show that
$d^{d^{k}/(d-1)}f_{c}^{\circ k}(z/{d^{1/(d-1)}})$ is in the ring
$\mathbb{Z}[d^{d/(d-1)}c,z]$ and is monic in $z$. We prove the statements by
induction on $k$. For $k=1$, since we have
$d^{d/(d-1)}f_{c}(z/d^{1/(d-1)})=z^{d}+d^{d/(d-1)}c$, the statement holds.
Assume the statement for $k$. Then

(3.1)
$d^{\frac{d^{k+1}}{d-1}}f_{c}^{\circ(k+1)}\left(\frac{z}{d^{\frac{1}{d-1}}}\right)=\left(d^{\frac{d^{k}}{d-1}}f_{c}^{\circ
k}\left(\frac{z}{d^{\frac{1}{d-1}}}\right)\right)^{d}+d^{\frac{d^{k+1}-d}{d-1}}\cdot
d^{\frac{d}{d-1}}c$

is in the ring $\mathbb{Z}[d^{d/(d-1)}c,z]$ and is monic in $z$ by the
assumption.

(ii). For $\alpha\in Z(F_{k})$ let $\beta\coloneq d^{1/(d-1)}\alpha$. Since we
have $d^{d^{k}/(d-1)}F_{k}(\beta/d^{1/(d-1)})=d^{d^{k}/(d-1)}F_{k}(\alpha)=0$,
the element $\beta$ is integral over $\mathbb{Z}[d^{d/(d-1)}c]$ by (i). Thus
$d\alpha^{d-1}=\beta^{d-1}$ is also integral over $\mathbb{Z}[d^{d/(d-1)}c]$.
∎

######  Lemma 3.2.

Let $k\geq 1$ and $m\geq 1$. For a polynomial
$G\in\mathbb{Z}[x,x_{0},x_{1},\ldots,x_{m-1}]$, define
$R_{k}(G)\coloneq\operatorname{Res}_{z}(f_{c}^{\circ
k}(z)-z,G(x,z,f_{c}(z),\ldots,f_{c}^{\circ(m-1)}(z))\in\mathbb{Z}[c,x]$. Then,
the following statements hold.

  1. (i)

All coefficients of $R_{k}(G)\in\mathbb{Z}[c][x]$ in the indeterminate $x$ are
integral over $\mathbb{Z}[d^{d/(d-1)}c]$.

  2. (ii)

Regard $G$ as a polynomial in indeterminates $x_{0},x_{1},\ldots,x_{m-1}$ with
coefficients in $\mathbb{Z}[x]$. If the total degree of each term of $G$ is
divisible by $d-1$, the polynomial $R_{k}(G)$ is in the ring
$\mathbb{Z}[d^{d}c^{d-1},x]$.

###### Proof.

(i) By Lemma 3.1 (ii), $f_{c}^{\prime}(\alpha)$ is integral over
$\mathbb{Z}[d^{d/(d-1)}c]$ for all $\alpha\in Z(F_{k})$. Since all
coefficients of $R_{k,m}(G)$ in $x$ are given by multiplication and addition
of these values, they are integral over $\mathbb{Z}[d^{d/(d-1)}c]$.

(ii) Let $\zeta=\zeta_{d-1}$ be a $d-1$-th root of unity and
$\nu\colon\mathbb{P}^{1}\longrightarrow\mathbb{P}^{1}$ be the multiplication
by $\zeta$. Then, since the diagram

(3.2)
${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$$\scriptstyle{\nu}$$\scriptstyle{f_{c}}$$\scriptstyle{f_{\zeta
c}}$$\scriptstyle{\nu}$

commutes, $\nu$ induces a bijection between $Z(f_{c}^{\circ k}-z)$ and
$Z(f_{\zeta c}^{\circ{k}}-z)$. Thus we have

(3.3) $\displaystyle R_{k}(G)|_{c\mapsto\zeta c}$
$\displaystyle=\prod_{\alpha^{\prime}\in Z(f_{\zeta c}^{\circ
k}-z)}G(x,\alpha^{\prime},f_{\zeta c}(\alpha^{\prime}),\ldots,f_{\zeta
c}^{\circ(m-1)}(\alpha^{\prime}))$ (3.4) $\displaystyle=\prod_{\alpha\in
Z(f_{c}^{\circ k}-z)}G(x,\zeta\alpha,\zeta f_{c}(\alpha),\ldots,\zeta
f_{c}^{\circ(m-1)}(\alpha))$ (3.5) $\displaystyle=\prod_{\alpha\in
Z(f_{c}^{\circ
k}-z)}G(x,\alpha,f_{c}(\alpha),\ldots,f_{c}^{\circ(m-1)}(\alpha))$
$\displaystyle\text{by the assumption of }G$ (3.6) $\displaystyle=R_{k}(G).$

Hence all the coefficients of $R_{k}(G)\in\mathbb{Z}[c][x]$ as a polynomial of
$x$ are elements of $\mathbb{Z}[c^{d-1}]$ and integral over
$\mathbb{Z}[d^{d/(d-1)}c]$ by Lemma 3.1 (ii). Since $\mathbb{Z}[d^{d/(d-1)}c]$
is normal, the polynomial $R_{k}(G)$ is in
$\mathbb{Z}[c^{d-1}][x]\cap\mathbb{Z}[d^{d/(d-1)}c][x]=\mathbb{Z}[d^{d}c^{d-1},x]$.
∎

######  Proposition 3.3.

For any positive integer $m\geq 1$, the multiplier polynomial $\delta_{m}(x)$
is in $\mathbb{Z}[d^{d}c^{d-1},x]$.

###### Proof.

Applying Lemma 3.1 (i) to $G=x-\prod_{i=0}^{m-1}(f_{c})^{\prime}(x_{i})$, we
see that $\operatorname{Res}_{z}(f_{c}^{\circ k}(z)-z,x-(f_{c}^{\circ
m})^{\prime}(z))$ is in $\mathbb{Z}[d^{d}c^{d-1},x]$. It is known that
$\delta_{m}(x)^{m}=\prod_{k|m}\operatorname{Res}_{z}(f_{c}^{\circ
k}(z)-z,x-(f_{c}^{\circ m})^{\prime}(z))^{\mu(m/k)}$ is in $\mathbb{Z}[c,x]$
by [MV95]. Since each factor $\operatorname{Res}_{z}(f_{c}^{\circ
k}(z)-z,x-(f_{c}^{\circ m})^{\prime}(z))$ is a polynomial in $c^{d-1}$ and
$x$, $\delta_{m}(x)^{m}$ is as well. By Gauss’ lemma, $\delta_{m}(x)^{m}$ is a
polynomial in $d^{d}c^{d-1}$ and $x$. Here, we note that
$\mathbb{Z}[c^{d-1},x]\subset\mathbb{Q}(d^{d}c^{d-1},x)=\operatorname{Frac}(\mathbb{Z}[d^{d}c^{d-1},x])$.
Since the ring $\mathbb{Z}[d^{d}c^{d-1},x]$ is normal, $\delta_{m}(x)$ is in
this ring. ∎

###  3.2. Valuations of roots

In this subsection, we study valuations of roots of $f_{c}^{\circ k}(z)-z$ and
$\delta_{m}(x)$. Our primary tool is Newton polygons. The content of this
section might be well-known to experts, but it is written for the reader’s
convenience.

Let $v\colon\mathbb{Q}((c^{-1}))\longrightarrow\mathbb{Z}$ be the
$c^{-1}$-adic valuation. We also denote $v$ the unique extension of $v$ to the
algebraic closure $\overline{\mathbb{Q}((c^{-1}))}$.

######  Lemma 3.4.

The following statements hold for all positive integers $k$ and $m$, the
following statements hold.

  1. (i)

The Newton polygon of $f_{c}^{\circ k}(z)\in\mathbb{Q}((c^{-1}))[z]$ consists
of one line segment of slope $1/d$ connecting points $(0,-d^{k-1})$ and
$(d^{k},0)$ as shown in Figure 2.

  2. (ii)

The Newton polygon of $f_{c}^{\circ k}(z)-z\in\mathbb{Q}((c^{-1}))[z]$
consists of one line segment of slope $1/d$ connecting points $(0,-d^{k-1})$
and $(d^{k},0)$ as shown in Figure 2.

  3. (iii)

The Newton polygon of $\operatorname{Res}_{z}(f_{c}^{\circ
k}(z)-z,x-(f_{c}^{\circ m})^{\prime}(z))\in\mathbb{Q}((c^{-1}))[x]$ consists
of one line segment of slope $m(d-1)/d$ connecting points $(0,-m(d-1)d^{k-1})$
and $(d^{k},0)$ as shown in Figure 2.

  4. (iv)

The Newton polygon of $\delta_{m}(x)\in\mathbb{Q}((c^{-1}))[x]$ consists of
one line segment of slope $m(d-1)/d$ connecting points
$(0,-\sum_{k|m}m(d-1)d^{k-1}\mu(m/k))$ and $(\sum_{k|m}d^{k}\mu(m/k),0)$ as
shown in Figure 2.

Figure 1. The Newton polygons of $f_{c}^{\circ k}(z)$ and $f_{c}^{\circ
k}(z)-z$.

Figure 2. The Newton polygons of $\operatorname{Res}_{z}(f_{c}^{\circ
k}(z)-z,x-(f_{c}^{\circ m})^{\prime}(z))$ and $\delta_{m}(x)$.

###### Proof.

(i) We prove that the Newton polygon of $f_{c}^{\circ k}$ consists of one line
segment of slope $1/d$ by induction on $k$. This is trivial for $k=1$. Assume
the statement for $k$. Then since $Z((f_{c}^{\circ k}(z))^{d})$ is a $d$ copy
of $Z(f_{c}^{\circ k})$, the Newton polygon of $(f_{c}^{\circ k}(z))^{d}$
consists of one line segment of slope $1/d$. The point $(0,v(c))=(0,-1)$ is
above the Newton polygon of $(f_{c}^{\circ k})^{d}$, and thus the Newton
polygon of $f_{c}^{\circ(k+1)}$ is the same as that of $(f_{c}^{\circ
k})^{d}$.

(ii) Since the point $(1,v(1))=(1,0)$ is above the Newton polygon by (i), the
Newton polygon of $F_{k}=f_{c}^{\circ k}-z$ is the same as that of
$f_{c}^{\circ{k}}$.

(iii) For all $\alpha\in Z(f_{c}^{\circ k}(z)-z)$, we have

(3.7)
$\displaystyle\phantom{{}={}}v\left((f_{c}^{\circ{m}})^{\prime}(\alpha)\right)=v\left(\prod_{i=0}^{m-1}d\cdot
f_{c}^{\circ i}(\alpha)^{d-1}\right)$ (3.8)
$\displaystyle=\sum_{i=0}^{m-1}(d-1)\cdot v(f_{c}^{\circ
i}(\alpha))=-m(d-1)/d,$

where note that for all $i\geq 0$, the equality $v(f_{c}^{\circ
i}(\alpha))=-1/d$ holds by (ii) since $f_{c}^{\circ i}(\alpha)$ for $0\leq
i\leq m-1$ are elements in $Z(f_{c}^{\circ k}(z)-z)$ as well. The equation
(3.8) implies the statement.

(iv) Since we have

(3.9)
$\delta_{m}(x)^{m}=\operatorname{Res}_{z}\left(\Phi_{m}^{*}(z),x-(f_{c}^{\circ
m})^{\prime}(z)\right)=\prod_{k|m}\operatorname{Res}_{z}\left(f_{c}^{\circ
k}(z)-z,x-(f_{c}^{\circ{m}})^{\prime}(z)\right)^{\mu(m/k)},$

the valuation of all the roots of $\delta_{m}(x)$ are equal to $-m(d-1)/d$ by
(iii).

∎

###  3.3. Monicness

In this subsection, we show the statement in Theorem 1.2 that $\delta_{m}(x)$
is monic in $d^{d}c^{d-1}$ up to multiplication by $\pm 1$.

######  Proposition 3.5.

For all positive integers $k,m$, the following statements hold.

  1. (i)

The leading term of $\operatorname{Res}_{z}\left(f_{c}^{\circ
k}(z)-z,x-(f^{\circ m})^{\prime}(z)\right)\in\mathbb{Z}[x,d^{d}c^{d-1}]$ in
$c$ is $(d^{d}c^{d-1})^{md^{k-1}}$. In particular, it is monic in
$d^{d}c^{d-1}$ up to multiplication by $(-1)^{(m+1)d^{k}}$.

  2. (ii)

The multiplier polynomial $\delta_{m}(x)\in\mathbb{Z}[x,d^{d}c^{d-1}]$ is
monic in $d^{d}c^{d-1}$ up to multiplication by
$(-1)^{\frac{d_{m}}{m}+d_{m}(d-1)}$.

###### Proof.

(i) By Lemma 3.4(iv), the highest degree of
$\operatorname{Res}_{z}(f_{c}^{\circ k}(z)-z,x-(f^{\circ m})^{\prime}(z))$ in
$c$ is realized only in the constant term for $x$. The constant term for $x$
is calculated as follows:

(3.10) $\displaystyle\operatorname{Res}_{z}(f_{c}^{\circ
k}(z)-z,-(f_{c}^{\circ m})^{\prime}(z))$ $\displaystyle=\prod_{\alpha\in
Z(f_{c}^{\circ k}(z)-z)}\prod_{i=0}^{m-1}\left(-d(f_{c}^{\circ
i}(\alpha))^{d-1}\right)$ (3.11) $\displaystyle=(-1)^{md^{k}}\cdot
d^{md^{k}}\left(\prod_{\alpha\in Z(f_{c}^{\circ
k}(z)-z)}\alpha^{d-1}\right)^{m}$ (3.12)
$\displaystyle=(-1)^{(m+1)d^{k}}d^{md^{k}}f_{c}^{\circ k}(0)^{(d-1)m}$ (3.13)
$\displaystyle=(-1)^{(m+1)d^{k}}(d^{d}c^{d-1})^{md^{k-1}}+(\text{terms of
lower degrees in }c).$

Thus, the statement holds.

(ii) Let $\alpha_{1},\alpha_{2},\ldots,\alpha_{r}$ represent the
$f_{c}$-orbits of $Z(\Phi_{m}^{\ast})$, where $r=d_{m}/m$ and
$d_{m}=\deg_{z}\Phi_{m}^{\ast}(z)$. By Lemma 3.4 (iv), the highest degree of
$\delta_{m}(x)$ in $c$ is realized only in the constant term for $x$. Hence,
the following calculation of the constant term in $x$ implies the assertion.

(3.14) $\displaystyle\delta_{m}(0)$
$\displaystyle=\prod_{i=1}^{r}(-(f_{c}^{\circ
m})^{\prime}(\alpha_{i}))=(-1)^{r}\prod_{i=1}^{r}\prod_{j=1}^{m-1}\left(d(f_{c}^{\circ
j}(\alpha_{i}))^{d-1}\right)$ (3.15)
$\displaystyle=(-1)^{r}d^{rm}(-1)^{d_{m}(d-1)}\Phi_{m}^{\ast}(0)^{d-1}=(-1)^{r+d_{m}(d-1)}d^{d_{m}}\Phi_{m}^{\ast}(0)^{d-1}$
(3.16)
$\displaystyle=(-1)^{r+d_{m}(d-1)}d^{d_{m}}\left(\prod_{k|m}\left.\left(f_{c}^{\circ
k}(z)-z\right)^{\mu(m/k)}\middle|_{z=0}\right.\right)^{d-1}$ (3.17)
$\displaystyle=(-1)^{r+d_{m}(d-1)}d^{d_{m}}\left(\prod_{k|m}\left(f_{c}^{\circ(k-1)}(c)\right)^{\mu(m/k)}\right)^{d-1}$
(3.18)
$\displaystyle=(-1)^{r+d_{m}(d-1)}d^{d_{m}}\left(\prod_{k|m}\left(c^{d^{k-1}}+(\text{terms
of lower degrees in }c)\right)^{\mu(m/k)}\right)^{d-1}$ (3.19)
$\displaystyle=(-1)^{r+d_{m}(d-1)}(d^{d}c^{d-1})^{\frac{d_{m}}{d}}+(\text{terms
of lower degrees in }c)$ (3.20)
$\displaystyle=(-1)^{\frac{d_{m}}{m}+d_{m}(d-1)}(d^{d}c^{d-1})^{\frac{d_{m}}{d}}+(\text{terms
of lower degrees in }c).$

∎

######  Remark 3.6.

We need to calculate the leading term of $\delta_{m}$, not of
$\delta_{m}^{m}$, to determine the leading term without the multiplication by
units. Thus, (i) of this proposition is not used in the latter discussion but
left in to make our argument parallel to the case of $z^{d+1}+cz$.

###  3.4. Integrality of parameters with periodic points of fixed multiplier

In this subsection, we prove the following theorem.

######  Theorem 3.7.

Let $d\geq 2$ be an integer and $f_{c}(z)=z^{d}+c$. Let $\mathcal{O}$ be an
integral domain.

  1. (i)

For a monic polynomial $\Phi(x)$ over an integral domain $\mathcal{O}$, the
quantity $\operatorname{Res}_{x}(\Phi(x),\delta_{f_{c},m}(x))$ is in
$\mathcal{O}[d^{d}c^{d-1}]$, and if $d\neq 0$ in $\mathcal{O}$, it is monic in
$d^{d}c^{d-1}$ up to multiplication by $\pm 1$.

  2. (ii)

For a parameter $\gamma\in\mathcal{O}$, assume that $f_{\gamma}(z)$ has an
$m$-periodic point $z_{0}\in\overline{\operatorname{Frac}\mathcal{O}}$ such
that its multiplier $(f_{\gamma}^{\circ m})^{\prime}(z_{0})$ is integral over
$\mathbb{Z}$. Then $d^{d}\gamma^{d-1}$ is integral over $\mathbb{Z}$ as well.

###### Proof.

(ii) follows from (i). We prove the assertion of (i). Let
$\Phi(x)=\sum_{i=0}^{N}a_{i}x^{i}\in\mathbb{Z}[x]$ be a monic polynomial.
Write $\delta_{m}(x)=\sum_{i=0}^{d_{m}}b_{i}x^{i}$, where
$d_{m}=\sum_{k|m}d^{k}\mu(m/k)$ and $b_{i}\in\mathbb{Z}[d^{d}c^{d-1}]$. Then
we have the equalities

(3.21) $\displaystyle 0$
$\displaystyle=\operatorname{Res}_{x}(\Phi(x),\delta_{m}(x))$ (3.22)
$\displaystyle=\begin{vmatrix}1&a_{N-1}&\cdots&\cdots&a_{0}&&&\\\
&1&a_{N-1}&\cdots&\cdots&a_{0}&&\\\ &&\ddots&&&&\ddots&\\\
&&&1&a_{N-1}&\cdots&\cdots&a_{0}\\\
b_{d_{m}}&b_{d_{m}-1}&\cdots&\cdots&b_{0}&&&\\\
&b_{d_{m}}&b_{d_{m}-1}&\cdots&\cdots&b_{0}&&\\\ &&\ddots&&&&\ddots&\\\
&&&b_{d_{m}}&b_{d_{m}-1}&\cdots&\cdots&b_{0}\end{vmatrix}$

By Proposition 3.5 (ii), the minimum valuation of coefficients of
$\delta_{m}(x)$ for $x$ is realized only in the constant term $b_{0}$, and
$\delta_{m}(x)\in\mathbb{Z}[d^{d}c^{d-1},x]$ is monic in $d^{d}c^{d-1}$ up to
multiplication by $\pm 1$. Hence
$\operatorname{Res}_{x}(\Phi(x),\delta_{m}(x))$ is also a monic polynomial in
$d^{d}c^{d-1}$. ∎

######  Corollary 3.8.

For an integer $d\geq 2$ and a parameter $\gamma\in\mathbb{C}$, assume that
$f_{\gamma}(z)=z^{d}+\gamma$ has a parabolic $m$-periodic point
$z_{0}\in\mathbb{C}$ for some $m\geq 1$. Then $d^{d}\gamma^{d-1}$ is an
algebraic integer. In particular, when we further assume that $\gamma$ is
rational, then $d^{d}\gamma^{d-1}$ is an integer.

###### Proof.

Since the multiplier of the parabolic periodic point is a root of unity by
definition, it is an algebraic integer by applying Theorem 3.7 in the case of
$R=\mathbb{C}$. The last statement follows from the normality of the ring
$\mathbb{Z}$. ∎

###  3.5. A remark on logistic maps

The map $g_{a}(z)=az(1-z)$ is called the logistic map. When the parameter $a$
varies in the interval $[0,4]$, there is a unique periodic cycle attracting
the critical point $1/2$. The periodic cycle attracting $1/2$ is plotted in
the following picture, where the horizontal axe is the parameter $a$. As a
corollary of Corollary 3.8, we get an algebraicity statement of the
bifurcation point in Figure 3.

Figure 3. Periodic cycle attracting $1/2$.

######  Corollary 3.9.

Let $g_{a}(z)$ be the logistic map $g_{a}(z)=az(1-z)$ with the parameter $a$.
When the parameter $a$ varies in the interval $[0,4]$, the periodic cycle of
$g_{a}(z)$ attracting $1/2$ bifurcates at only algebraic integers. Moreover,
if the periodic cycle of $g_{a}(z)$ bifurcates at a rational parameter
$a\in[0,4]$, then $a$ is equal to $1$ or $3$.

###### Proof.

It is well-known that the start points of the windows of the bifurcation
diagram are real parameters such that $g_{a}$ admits a parabolic periodic
point. Let $\phi(z)=-\frac{1}{a}(z-a/2)$. Then we have
$G_{a}(z)\coloneq\phi^{\circ(-1)}\circ
g_{a}\circ\phi(z)=z^{2}+\frac{1}{4}(-a^{2}+2a)$, where $\phi^{\circ(-1)}$ is
the compositional inverse of $\phi$. By Corollary 3.8, $-a^{2}+2a$ is an
algebraic integer, so $a$ is as well. Hence, the only candidates for such
parameters are $0,1,2,3$ or $4$. When $a$ is $1$ or $3$, the map $G_{a}(z)$
has a parabolic fixed point $1/2$ and $-1/2$, respectively. When $a$ is $0$,
$2$, or $4$, since the orbit of the unique critical point $0$ of $G_{a}(a)$ is
finite, the map $G_{a}(z)$ cannot have parabolic periodic points by Theorem
2.8. Thus, the map $g_{a}$ cannot have parabolic periodic points. ∎

##  4\. Multiplier polynomials for $z^{d+1}+cz$

In this section, we fix a positive integer $d$ and consider a polynomial map
$f(z)\coloneqq z^{d+1}+cz$ with an indeterminate $c$. We aim to prove Theorem
1.6.

###  4.1. Replacement of polynomials

Let $\tilde{f}(z)\coloneqq z^{d+1}-cz^{d}+c$ and
$\tilde{g}(z)\coloneqq(d+1)z-dc$. For a positive integer $m$, define

(4.1) $F_{m}(z)\coloneqq\prod_{i=0}^{m-1}\tilde{f}^{\circ
i}(z)-1,\quad\mathcal{F}_{m}(y,z)\coloneqq\prod_{i=0}^{m-1}\left(y+\tilde{f}^{\circ
i}(z)\right),$ (4.2)
$R_{k,m}(x)\coloneq\operatorname{Res}_{z}\left(F_{k}(z),x-(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,z\right)\right).$

Here we recall that we set $\tilde{f}^{\circ 0}(z)\coloneqq z$ in Section 1.
As the following lemma shows, these polynomials are helpful to study
multiplier polynomials.

######  Lemma 4.1.

For a positive integer $m$, the following statements hold.

  1. (i)

Let $\mathbb{P}^{1}=\mathbb{P}^{1}_{\mathbb{Q}(c)}$ be the projective line
over $\mathbb{Q}(c)$ and
$\tau\colon\mathbb{P}^{1}\longrightarrow\mathbb{P}^{1}$ be an endomorphism
defined by $\tau(z)\coloneqq z^{d}+c$. Then, the diagram

(4.3)
${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$$\scriptstyle{\tau}$$\scriptstyle{f}$$\scriptstyle{\tilde{f}}$$\scriptstyle{\tau}$

commutes.

  2. (ii)

We have

(4.4) $\displaystyle\tilde{f}^{\circ m}(z)$
$\displaystyle=(z-c)\left(\prod_{i=0}^{m-1}\tilde{f}^{\circ
i}(z)\right)^{d}+c,$ (4.5) $\displaystyle f^{\circ m}(z)$
$\displaystyle=z\prod_{i=0}^{m-1}\tilde{f}^{\circ i}(\tau(z)),$ (4.6)
$\displaystyle\left(f^{\circ m}\right)^{\prime}(z)$
$\displaystyle=(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\tau(z)\right).$

  3. (iii)

For a positive integer $k$, we have

(4.7) $\operatorname{Res}_{z}\left(f^{\circ k}(z)-z,x-\left(f^{\circ
m}\right)^{\prime}(z)\right)=(x-c^{m})R_{k,m}(x)^{d}.$

In particular, we have

(4.8) $\delta_{m}(x)^{m}=(x-c^{m})^{\varepsilon_{m}}\left(\prod_{k\mid
m}R_{k,m}(x)^{\mu(m/k)}\right)^{d},$

where

(4.9) $\varepsilon_{m}\coloneqq\begin{cases}1&\text{ if }m=1,\\\ 0&\text{ if
}m\neq 1.\end{cases}$

  4. (iv)

The map $\tilde{f}$ induces a permutation of $Z(F_{m})$, which is the set of
roots of $F_{m}$.

  5. (v)

The degree of $F_{m}(z)$ with respect to $c$ is $\frac{(d+1)^{m-1}-1}{d}$. In
particular, the polynomial $d^{\frac{(d+1)^{m-1}-1}{d}}F_{m}(z)$ is an element
of $\mathbb{Z}[dc,x]$.

###### Proof.

(i) follows from a direct calculation.

(ii). The first equation holds for $m=1$ since $\tilde{f}(z)=(z-c)z^{d}+c$. If
it holds for $m$, then we have

(4.10) $\tilde{f}^{\circ(m+1)}(z)=\left(\tilde{f}^{\circ
m}(z)-c\right)\tilde{f}^{\circ
m}(z)^{d}+c=(z-c)\left(\prod_{i=0}^{m-1}\tilde{f}^{\circ
i}(z)\right)^{d}\tilde{f}^{\circ m}(z)^{d}+c.$

Thus, we obtain the first equation.

Next, we prove the second equation. By (i), we have $\tilde{f}^{\circ
m}\left(\tau(z)\right)=\tau\left(f^{\circ m}(z)\right)=f^{\circ m}(z)^{d}+c$.
Thus, by combining the first equation, we obtain

(4.11) $\displaystyle f^{\circ m}(z)^{d}$ $\displaystyle=\tilde{f}^{\circ
m}\left(\tau(z)\right)-c=\left(\tau(z)-c\right)\left(\prod_{i=0}^{m-1}\tilde{f}^{\circ
i}\left(\tau(z)\right)\right)^{d}=\left(z\prod_{i=0}^{m-1}\tilde{f}^{\circ
i}\left(\tau(z)\right)\right)^{d}.$

Thus, we obtain the second equation.

Finally, we prove the last equation. The chain rule implies $\left(f^{\circ
m}\right)^{\prime}(z)=\prod_{i=0}^{m-1}f^{\prime}\left(f^{\circ i}(z)\right)$.
Since $f^{\prime}(z)=(d+1)z^{d}+c$ and $f^{\circ m}(z)^{d}=\tilde{f}^{\circ
m}\left(\tau(z)\right)-c$, we have

(4.12) $f^{\circ m}(z)^{\prime}=\prod_{i=0}^{m-1}\left((d+1)\tilde{f}^{\circ
i}(z)^{d}+c\right)=\prod_{i=0}^{m-1}\left((d+1)\tilde{f}^{\circ
i}(\tau(z))-dc\right)=(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\tau(z)\right).$

(iii). By (ii), we have

(4.13) $\displaystyle\operatorname{Res}_{z}\left(f^{\circ k}(z)-z,x-f^{\circ
m}(z)^{\prime}\right)$ (4.14) $\displaystyle=\,$
$\displaystyle\operatorname{Res}_{z}\left(zF_{k}\left(\tau(z)\right),x-(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\tau(z)\right)\right)$
(4.15) $\displaystyle=\,$
$\displaystyle\left(x-(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\tau(0)\right)\right)\operatorname{Res}_{z}\left(F_{k}\left(z^{d}+c\right),x-(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,z^{d}+c\right)\right).$

Here we have
$(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\tau(0)\right)=c^{m}$ since
$\tilde{f}^{\circ i}\left(\tau(0)\right)=\tau\left(f^{\circ
i}(0)\right)=\tau(0)=c$ for each $0\leq i<m$ by (i). In addition, for any
polynomials $F(z)$ and $G(z)$, we have
$\operatorname{Res}\left(F\left(z^{d}+c\right),G\left(z^{d}+c\right)\right)=\operatorname{Res}(F,G)^{d}$
by Lemma 2.1 (ii). Thus, we obtain the first equation. The last equation
follows from the first equation and the Möbius inversion formula.

(iv). By (i), we have $\tilde{f}^{\circ
m}(z)-z=(z-c)\left(\left(\prod_{i=0}^{m-1}\tilde{f}^{\circ
i}(z)\right)^{d}-1\right)$. Since $F_{m}(z)$ divides this, for each $\alpha\in
Z(F_{m})$ we have $\tilde{f}^{\circ m}(\alpha)=\alpha$. Thus, we obtain

(4.16) $F_{m}\left(\tilde{f}(\alpha)\right)=\prod_{i=1}^{m}\tilde{f}^{\circ
i}(\alpha)-1=\prod_{i=0}^{m-1}\tilde{f}^{\circ i}(\alpha)-1=F_{m}(\alpha)=0.$

This implies $\tilde{f}(\alpha)\in Z(F_{m})$. Since $\tilde{f}^{\circ
m}(\alpha)=\alpha$, $\tilde{f}$ induces a permutation of $Z(F_{m})$.

(v) follows (ii) and induction on $m$. ∎

By Lemma 4.1 (iii), Theorem 1.6 follows from the following proposition.

######  Proposition 4.2.

For positive integers $k$ and $m$, the polynomial
$d^{m\frac{(d+1)^{k-1}-1}{d}}R_{k,m}(x)$ is an element of $\mathbb{Z}[dc,x]$
and monic in $dc$ up to multiplication by $\pm 1$.

In the following subsections, we prove this proposition.

###  4.2. Integrality

The first statement in Proposition 4.2 follows from the following proposition.

######  Proposition 4.3.

For positive integers $k$ and $m$, the degree of
$\operatorname{Res}_{z}\left(F_{k}(z),x-\mathcal{F}_{m}\left(y,z\right)\right)$
with respect to $c$ is $m\frac{(d+1)^{k-1}-1}{d}$. In particular, we have
$d^{m\frac{(d+1)^{k-1}-1}{d}}R_{k,m}(x)\in\mathbb{Z}[dc,x]$.

To prove this proposition, we study $c^{-1}$-adic valuations of roots of
$F_{k}(z)$.

We denote by $v$ the $c^{-1}$-adic valuation on
$\overline{\mathbb{Q}((c^{-1}))}$. Following the method in Section 2.2, we
extend $v$ to $\overline{\mathbb{Q}((c^{-1}))}(x,y)$, and the extension is
also denoted by $v$. Recall that $v$ has the following properties.

  * •

$v=-\deg_{c}$ on $\mathbb{Q}[c,x,y]$,

  * •

$v(x+f)=\min\\{0,v(f)\\}$ for all $f\in\overline{\mathbb{Q}((c^{-1}))}(y)$,
and

  * •

$v(y+\alpha)=\min\\{0,v_{c^{-1}}(\alpha)\\}$ for all
$\alpha\in\overline{\mathbb{Q}((c^{-1}))}$.

######  Lemma 4.4.

For any positive integers $k$ and $m$ and any root $\alpha\in Z(F_{k})$, we
have $v\left(\mathcal{F}_{m}(y,\alpha)\right)\leq 0$ and
$v\left(x-\mathcal{F}_{m}(y,\alpha)\right)=v\left(\mathcal{F}_{m}(y,\alpha)\right)$.

###### Proof.

Since we have

(4.17)
$v\left(\mathcal{F}_{m}(y,\alpha)\right)=v(y+\alpha)+v\left(y+\tilde{f}(\alpha)\right)+\cdots+v\left(y+\tilde{f}^{\circ(m-1)}(\alpha)\right)$

and $\tilde{f}(\alpha),\dots,\tilde{f}^{\circ(m-1)}(\alpha)$ are in $Z(F_{k})$
by Lemma 4.1 (iv), it suffices to show $v(y+\alpha)\leq 0$ to prove the first
inequality. If $v(\alpha)>0$, we have $v(y+\alpha)\leq 0$ since
$v(y)=0<v(\alpha)$. If $v(\alpha)\leq 0$, we have $v(y+\alpha)\leq 0$ .The
last inequality follows from the first inequality. ∎

Finally, we prove Proposition 4.3.

###### Proof of Proposition 4.3.

By Lemma 4.4, we have

(4.18)
$\displaystyle\phantom{{}={}}-\deg_{a}\operatorname{Res}_{z}\left(F_{k}(z),x-\mathcal{F}_{m}\left(y,z\right)\right)$
(4.19)
$\displaystyle=v\left(\operatorname{Res}_{z}\left(F_{k}(z),x-\mathcal{F}_{m}\left(y,z\right)\right)\right)$
(4.20) $\displaystyle=\sum_{\alpha\in
Z(F_{k})}v\left(x-\mathcal{F}_{m}(y,\alpha)\right)$ (4.21)
$\displaystyle=\sum_{\alpha\in
Z(F_{k})}v\left(\mathcal{F}_{m}(y,\alpha)\right)$ (4.22)
$\displaystyle=v\left(\prod_{\alpha\in
Z(F_{k})}\prod_{i=0}^{m-1}\left(y+\tilde{f}^{\circ i}(\alpha)\right)\right).$

By Lemma 4.1 (iv) and (v), this equals to

(4.23) $v\left(F_{k}(y)^{m}\right)=-m\deg_{c}F_{k}=-m\frac{(d+1)^{k-1}-1}{d}.$

∎

###  4.3. Valuations of roots

To prove the last statement in Proposition 4.2, we study valuations of roots
of $F_{k}$ in this subsection. Our primary tool is Newton polygons.

######  Lemma 4.5.

For a positive integer $k$ and any roots $\alpha\in Z\left(\tilde{f}^{\circ
k}\right)\cup Z\left(F_{k}\right)$, we have $v(\alpha)\geq-1$.

###### Proof.

It suffices to show that Newton polygons of $\tilde{f}^{\circ k}(z)$ and
$F_{k}(z)$ have only slope at most $1$.

To begin with, we prove the statement for $\tilde{f}^{\circ k}(z)$ by
induction. For $m=1$, the Newton polygon of $\tilde{f}(z)=z^{d+1}-cz^{d}+c$ is
shown in Figure 5 and it has only slope at most $1$. Suppose the statements
for $\tilde{f}^{\circ i}(z)$ hold for $1\leq i\leq m$. Then,
$(z-c)\left(\prod_{i=0}^{k-1}\tilde{f}^{\circ i}(z)\right)^{d}$ has only roots
with valuations at most $-1$ and thus its Newton polygon takes the form as
shown in Figure 5. Thus, Newton polygon of $\tilde{f}^{\circ
k}(z)=(z-c)\left(\prod_{i=0}^{mk-1}\tilde{f}^{\circ i}(z)\right)^{d}+c$ also
takes the form as shown in Figure 5.

Finally, we prove the statement for $F_{k}(z)$. By the statement for
$\tilde{f}^{\circ k}(z)$, the polynomial $\prod_{i=0}^{k-1}\tilde{f}^{\circ
i}(z)$ has only roots with valuations at most $-1$ and thus its Newton polygon
takes the form as shown in Figure 5. Thus, the Newton polygon of
$F_{k}(z)=\prod_{i=0}^{k-1}\tilde{f}^{\circ i}(z)-1$ also takes the form as
shown in Figure 5. ∎

Figure 4. The Newton polygon of $\tilde{f}(z)$.

Figure 5.

######  Remark 4.6.

We can describe the Newton polygon of $\tilde{f}^{\circ k}(z)$ explicitly. See
Figure 8.

######  Proposition 4.7.

For positive integers $k$ and $m$ and any roots $\alpha\in
Z\left(F_{k}\right)$, we have $v((d+1)\alpha-dc)=-1$ and
$v\left((d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\alpha\right)\right)=-m$.

###### Proof.

The second equality follows from the first equality because

(4.24)
$(d+1)^{k}\mathcal{F}_{k}\left(-\frac{d}{d+1}c,\alpha\right)=\prod_{i=0}^{k-1}\left((d+1)\tilde{f}^{\circ
i}(\alpha)-dc\right)$

and $\tilde{f}(\alpha),\dots,\tilde{f}^{\circ(k-1)}(\alpha)$ are roots of
$F_{k}(z)$ by Lemma 4.1 (v).

We prove the first equality. Since $v(\alpha)\geq-1$ for any $\alpha\in
Z\left(F_{k}\right)$ by Lemma 4.5, we have

(4.25) $v((d+1)\alpha-dc)\geq\min\\{v(\alpha),-1\\}=-1.$

Thus, the Newton polygon of

(4.26)
$G_{k}(x)\coloneqq\operatorname{Res}_{z}\left(F_{k}(z),x-(d+1)z+dc\right)=\prod_{\alpha\in
Z(F_{k})}\left(x-(d+1)\alpha+dc\right)$

is of the form shown in Figure 5. On the other hand, its constant term equals
to

(4.27)
$G_{k}(0)=\operatorname{Res}_{z}\left(F_{k}(z),-(d+1)z+dc\right)=(-d-1)^{\deg_{z}F_{k}}F_{k}\left(\frac{d}{d+1}c\right)=(-dc)^{\deg_{z}F_{k}}+(\text{
lower terms })$

since $F_{k}(z)$ is monic in $z$. Thus, its valuation is
$v(G_{k}(0))=-\deg_{z}F_{k}=-\deg_{x}G_{k}(x)$. Therefore, the Newton polygon
of $G_{k}(x)$ takes the form shown in Figure 6. It implies all roots of
$G_{k}(x)$ has valuation $v=-1$. ∎

Figure 6.

###  4.4. Monicness

In conclusion of this section, we prove the following proposition, which
implies the last statement in Proposition 4.2, and finish a proof of it.

######  Proposition 4.8.

For positive integers $k$ and $m$, the leading term of $R_{k,m}(x)$ in $c$ is

(4.28)
$(-1)^{\frac{(m+1)((d+1)^{k}-1)+m((d+1)^{k-1}-1)}{d}}d^{m(d+1)^{k-1}}c^{m\frac{(d+1)^{k}-1}{d}}$

Moreover, $d^{m\frac{(d+1)^{k-1}-1}{d}}R_{k,m}(x)$ is monic in $dc$ up to
multiplication by $\pm 1$.

Here, we remark that the last statement in this proposition follows from the
first statement and Proposition 4.2. We need the following lemma to prove
Proposition 4.8. In the following, we denote the leading term of a polynomial
$P(c)$ by $\operatorname{LT}_{c}(P)$.

######  Lemma 4.9.

For a positive integer $k$, we have

(4.29)
$\displaystyle\operatorname{LT}_{c}\left((d+1)^{\deg_{z}\tilde{f}^{\circ
k}}\tilde{f}^{\circ k}\left(\frac{d}{d+1}c\right)\right)$
$\displaystyle=(-d^{d}c^{d+1})^{(d+1)^{k-1}},$ (4.30)
$\displaystyle\operatorname{LT}_{c}\left((d+1)^{\deg_{z}F_{k}}F_{k}\left(\frac{d}{d+1}c\right)\right)$
$\displaystyle=(-1)^{\frac{(d+1)^{k-1}-1}{d}}d^{(d+1)^{k-1}}c^{\frac{(d+1)^{k}-1}{d}}.$

###### Proof.

Let $\tilde{f}_{k}\coloneqq(d+1)^{\deg_{z}\tilde{f}^{\circ k}}\tilde{f}^{\circ
k}\left(\frac{d}{d+1}c\right)$ and
$l_{k}\coloneqq\operatorname{LT}_{c}\left(\tilde{f}_{k}\right)$. Then, we have

(4.31)
$l_{1}=\operatorname{LT}_{c}\left((d+1)^{d+1}\left(\left(\frac{d}{d+1}c\right)^{d}\left(\frac{d}{d+1}c-c\right)+c\right)\right)=-d^{d}c^{d+1}$

and

(4.32) $\displaystyle\tilde{f}_{k+1}$
$\displaystyle=(d+1)^{(d+1)\deg_{z}\tilde{f}^{\circ k}}\left(\tilde{f}^{\circ
k}\left(\frac{d}{d+1}c\right)^{d+1}-c\tilde{f}^{\circ
k}\left(\frac{d}{d+1}c\right)^{d}+c\right)$ (4.33)
$\displaystyle=\tilde{f}_{k}^{d+1}-(d+1)^{\deg_{z}\tilde{f}^{\circ
k}}c\tilde{f}_{k}^{d}+(d+1)^{(d+1)\deg_{z}\tilde{f}^{\circ k}}c.$

These equalties imply $\deg_{c}\tilde{f}_{k}\geq 2$, and thus,

(4.34)
$l_{k+1}=\operatorname{LT}_{c}\left(\tilde{f}_{k+1}\right)=\operatorname{LT}_{c}\left(\tilde{f}_{k}^{d+1}\right)=l_{k}^{d+1}.$

Thus, we obtain $l_{k}=(-d^{d}c^{d+1})^{(d+1)^{k-1}}$. Since
$F_{k}(z)\coloneqq z\tilde{f}(z)\cdots\tilde{f}^{\circ(k-1)}(z)-1$ , we have

(4.35)
$\operatorname{LT}_{c}\left((d+1)^{\deg_{z}F_{k}}F_{k}\left(\frac{d}{d+1}c\right)\right)=\operatorname{LT}_{c}\left(dc\right)l_{1}\cdots
l_{k-1}=(-1)^{\frac{(d+1)^{k-1}-1}{d}}d^{(d+1)^{k-1}}c^{\frac{(d+1)^{k}-1}{d}}.$

∎

###### Proof of Proposition 4.8.

The degree of $R_{k,m}(x)$ with respect to $c$ is

(4.36) $-\sum_{\alpha\in
Z(F_{k})}v\left(x-(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\alpha\right)\right).$

By Proposition 4.7, this equals to

(4.37) $-\sum_{\alpha\in Z(F_{k})}(-m)=m\frac{(d+1)^{k}-1}{d}.$

Since $v(x)=0$ and
$v\left((d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\alpha\right)\right)=-m$
for any $\alpha\in Z(F_{k})$ by Proposition 4.7, the minimum valuation of the
terms of $\displaystyle\prod_{\alpha\in
Z(F_{k})}\left(x-(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\alpha\right)\right)$
is realized only in the constant term for $x$. Hence we have

(4.38) $\operatorname{LT}_{c}\left(\prod_{\alpha\in
Z(F_{k})}\left(x-(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\alpha\right)\right)\right)=\operatorname{LT}_{c}\left(\prod_{\alpha\in
Z(F_{k})}\left(-(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\alpha\right)\right)\right)$

Thus, we can calculate the leading term as

(4.39) $\displaystyle\operatorname{LT}_{c}\left(R_{k,m}(x)\right)$
$\displaystyle=\operatorname{LT}_{c}\left(\prod_{\alpha\in
Z(F_{k})}\left(x-(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\alpha\right)\right)\right)$
(4.40) $\displaystyle=\operatorname{LT}_{c}\left(\prod_{\alpha\in
Z(F_{k})}\left(-(d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\alpha\right)\right)\right)\text{\qquad\qquad
by \eqref{eq:non-unicritical_valuation_with_indeterminates}}$ (4.41)
$\displaystyle=\operatorname{LT}_{c}\left(\prod_{\alpha\in
Z(F_{k})}(-1)\prod_{i=0}^{m-1}\left((d+1)\tilde{f}^{\circ
i}(\alpha)-dc\right)\right)$ (4.42)
$\displaystyle=(-1)^{\deg_{z}F_{k}}\operatorname{LT}_{c}\left(\prod_{\alpha\in
Z(F_{k})}\left((d+1)\alpha-dc\right)\right)^{m}$ (4.43)
$\displaystyle=(-1)^{\deg_{z}F_{k}}\operatorname{LT}_{c}\left((-d-1)^{\deg_{z}F_{k}}F_{k}\left(\frac{d}{d+1}c\right)\right)^{m}\text{\qquad
by \lx@cref{creftype~refnum}{lem:non_unicritical_tilde_f}
\lx@cref{creftypeplural~refnum}{item:lem:non_unicritical_tilde_f:4}
and~\lx@cref{refnum}{item:lem:non_unicritical_tilde_f:5}}$ (4.44)
$\displaystyle=(-1)^{\deg_{z}F_{k}}\left((-1)^{\deg_{z}F_{k}}\cdot(-1)^{\frac{(d+1)^{k-1}-1}{d}}d^{(d+1)^{k-1}}c^{\frac{(d+1)^{k}-1}{d}}\right)^{m}\text{\qquad
by \lx@cref{creftype~refnum}{lem:non_unicritical_leading_term_f^k}}$ (4.45)
$\displaystyle=(-1)^{\frac{(m+1)((d+1)^{k}-1)+m((d+1)^{k-1}-1)}{d}}d^{m(d+1)^{k-1}}c^{m\frac{(d+1)^{k}-1}{d}},$

which was what we wanted. ∎

##  5\. Height bound of the parabolic parameters

This section establishes a uniform upper bound for the naive height of
parabolic parameters in polynomial families $z^{d}+c$ and $z^{d+1}+cz$. In the
evaluation at finite places, we use the results on integrality discussed
earlier, while in the evaluation at infinite places, we rely on the
boundedness of Mandelbrot-type sets.

Throughout this section, the set of critical points of a polynomial
$f\in\mathbb{C}[z]$ is denoted by $\mathrm{Crit}(f)$:

(5.1) $\mathrm{Crit}(f)=\\{c\in\mathbb{C}\colon f^{\prime}(c)=0\\}.$

###  5.1. Height bound for $z^{d}+c$

In this subsection, we fix $f_{c}(z)=z^{d}+c$ with $d\geq 2$. Recall that the
Multibrot set is defined by
$\mathbb{M}_{d}\coloneq\\{\gamma\in\mathbb{C}\colon(f_{\gamma}^{\circ
n}(0))_{n=0}^{\infty}\text{ is bounded}\\}$. The following fact might be well-
known to experts, but we describe it for the reader’s convenience.

######  Lemma 5.1.

Let $\mathbb{M}_{d}$ be the Multibrot set defined in Section 2.3. The set
$\mathbb{M}_{d}$ is contained in the disk $\\{c\in\mathbb{C}\colon|c|\leq
2^{1/(d-1)}\\}$. In particular, if $f_{c}=z^{d}+c$ has a parabolic periodic
point, then the inequality $|c|\leq 2^{1/(d-1)}$ holds.

###### Proof.

The first statement is equivalent to that if $|c|>2^{1/(d-1)}$,
$\lim_{n\to\infty}|f_{c}^{\circ n}(0)|=\infty$ holds. If $|c|>2^{1/(d-1)}$,
then there exists a positive real number $\varepsilon>0$ which satisfies
$|c|\geq(2(1+\varepsilon))^{1/(d-1)}$. For a complex number $z\in\mathbb{C}$
which satisfies $|z|\geq|c|$, we have

(5.2)
$|f_{c}(z)|\geq|z|\left(|z|^{d-1}-1\right)\geq|z|\left(|c|^{d-1}-1\right)\geq(1+\varepsilon)|z|.$

By induction, $|f_{c}^{\circ(n+1)}(0)|=|f_{c}^{\circ
n}(c)|\geq(1+\varepsilon)^{n}|c|\to\infty$. The second statement is a
consequence of $\mathrm{Crit}(f_{c})=\\{0\\}$, the first statement, and
Theorem 2.8. ∎

###### Proof of Theorem 1.3.

Let $\gamma\in\mathbb{C}$ as in the assertion. By theorem Theorem 1.2,
$d^{d}\gamma^{d-1}$ is an algebraic integer, in particular, the parameter
$\gamma$ is in $\overline{\mathbb{Q}}$. Let $K=\mathbb{Q}(\gamma)$ be the
field generated by $\gamma$ and $r=[K:\mathbb{Q}]$ be the extension degree.
Since $\|d^{d}\gamma^{d-1}\|_{v}\leq 1$ for all finite places $v\in
M_{K}^{0}$, we have

(5.3) $\prod_{v\in M_{K}^{0}}\max\\{1,\|\gamma\|_{v}\\}\leq\prod_{v\in
M_{K}^{0}}\|d^{d}\|_{v}^{-1/(d-1)}=\prod_{v\in
M_{K}^{\infty}}\|d^{d}\|_{v}^{1/(d-1)}=(d^{d/(d-1)})^{r}.$

The second equation follows from the product formula (Theorem 2.3). On the
other hand, since the inequality $|\gamma|\leq 2^{1/(d-1)}$ holds by Lemma
5.1, we obtain

(5.4) $\prod_{v\in
M_{K}^{\infty}}\max\\{1,\|\gamma\|_{v}\\}\leq\left(2^{1/(d-1)}\right)^{r}.$

By combining these two estimations,

(5.5) $H(\gamma)^{r}=H_{K}(\gamma)\leq(2d^{d})^{r/(d-1)}.$

∎

###  5.2. Height bound for $z^{d+1}+cz$

In this subsection, we fix the following notation.

  * •

$f_{c}(z)=z^{d+1}+cz$,

  * •

$r(d)=(2d+2)^{1/d}\left(1+\frac{1}{d}\right)(>1)$.

######  Lemma 5.2.

For a positive integer $d$, the following statements hold.

  1. (i)

If two complex numbers $c,z\in\mathbb{C}$ satisfy $|z|>(2|c|)^{1/d}$ and
$|c|>1$, then $\displaystyle\lim_{n\to\infty}|f_{c}^{\circ n}(z)|=\infty$.

  2. (ii)

For $c\in\mathbb{C}$, assume that $f_{c}(z)=z^{d+1}+cz$ has a parabolic
periodic point, then $|c|\leq r(d)$.

###### Proof.

(i) If two complex numbers $c,z\in\mathbb{C}$ satisfy the relation
$|z|>(2|c|)^{1/d}$ and $|c|>1$, then there exists a positive real number
$\varepsilon>0$ which satisfies $|z|\geq(2(1+\varepsilon)|c|)^{1/d}$.
Therefore,

(5.6)
$|f_{c}(z)|\geq|z|\left(|z|^{d}-|c|\right)\geq|z|\left(2(1+\varepsilon)|c|-|c|\right)\geq(1+\varepsilon)|z|.$

By induction, $|f_{c}^{\circ
n}(z)|\geq(1+\varepsilon)^{n}|z|\rightarrow\infty$.

(ii) If $|c|>r(d)$, then
$d\left|\frac{c}{d+1}\right|^{1+\frac{1}{d}}>(2|c|)^{1/d}$ and $|c|>1$. On the
other hand, the set of critical points of $f_{c}$ is described as
$\mathrm{Crit}(f_{c})=\left\\{\zeta_{d}^{i}\left(\frac{-c}{d+1}\right)^{1/d}\colon
0\leq i\leq d-1\right\\}$ and
$\left|f_{c}\left(\zeta_{d}^{i}\left(\frac{-c}{d+1}\right)^{1/d}\right)\right|=d\left|\frac{c}{d+1}\right|^{1+\frac{1}{d}}$.
Therefore, every critical orbit of $f_{c}$ is attracted into $\infty$ by (i).
By Theorem 2.8, $f_{c}$ have no parabolic periodic point. ∎

###### Proof of Theorem 1.7.

The same argument as in the proof of Theorem 1.3 works in this situation.
Indeed, we can prove

(5.7) $\prod_{v\in M_{K}^{0}}\max\\{1,\|\gamma\|_{v}\\}\leq d^{r}$

by Theorem 1.6 and

(5.8) $\prod_{v\in M_{K}^{\infty}}\max\\{1,\|\gamma\|_{v}\\}\leq r(d)^{r}$

by Lemma 5.2 for each $\gamma$ such that $f_{\gamma}$ has a parabolic periodic
point. ∎

##  6\. Determination of rational parabolic parameters

In this section, we give a set of candidates of parabolic rational parameters
for $f_{c}(z)=z^{2}+c$.

###### Proof of Theorem 1.5.

For a rational number $\gamma\in\mathbb{Q}$, assume that the polynomial
$f_{\gamma}(z)=z^{2}+\gamma$ has a parabolic periodic point in $\mathbb{C}$.
Then, the parameter $\gamma$ is in the interval $[-2,\frac{1}{4}]$ since the
intersection of Mandelbrot set and the real line is given by
$\mathbb{M}_{2}\cap\mathbb{R}=[-2,\frac{1}{4}]$. On the other hand, by
Corollary 3.8, $4\gamma$ is an integer. Therefore, the parameter $\gamma$ is
in the set
$\\{-2,-\frac{7}{4},-\frac{3}{2},-\frac{5}{4},-1,-\frac{3}{4},-\frac{1}{2},-\frac{1}{4},0,\frac{1}{4}\\}$.
However, if $\gamma$ is either $-1,-\frac{1}{2},-\frac{1}{4},0$, then
$f_{\gamma}$ has an attracting periodic point as shown in Table 1. In Table 1,
the numbers $\alpha_{1},\alpha_{2},\alpha_{3}$ are roots of the polynomial
$8z^{3}+4z^{2}-18z-1$. In addition, every periodic point of $z^{2}-2$ is
repelling. This fact follows from the discussion on the Chebyshev polynomial
(See, for example, [Sil07, Proposition 6.8.]). ∎

Table 1. attracting/parabolic periodic point and multiplier $\gamma$ | $-2$ | $-\frac{7}{4}$ | $-\frac{3}{2}$ | $-\frac{5}{4}$ | $-1$ | $-\frac{3}{4}$ | $-\frac{1}{2}$ | $-\frac{1}{4}$ | $0$ | $\frac{1}{4}$
---|---|---|---|---|---|---|---|---|---|---
periodic point(s) | - | $\alpha_{1},\alpha_{2},\alpha_{3}$ | ? | $\frac{-1\pm\sqrt{2}}{2}$ | $0,-1$ | $-\frac{1}{2}$ | $\frac{1-\sqrt{3}}{2}$ | $\frac{1-\sqrt{2}}{2}$ | $0$ | $\frac{1}{2}$
period | - | $3$ | ? | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$
multiplier | - | $1$ | ? | $-1$ | $0$ | $-1$ | $1-\sqrt{3}$ | $1-\sqrt{2}$ | $0$ | $1$

In principle, we can give candidates for parabolic rational parameters for the
family $(f_{d}(z)=z^{d}+c)_{c}$ for every integer $d\geq 2$ in this way.

######  Remark 6.1.

In [MV95], it is conjectured that the polynomial $\Delta_{n,m}(c)$ for the
family $f_{c}(z)=z^{2}+c$ is irreducible over $\mathbb{Q}$ for all $m|n$. In
addition, [MV95, Corollary 3.3.] asserts the following:

(6.1) $\deg_{c}\Delta_{n,m}=\varphi(n/m)d_{m}/2\text{ for }m|n,n\neq m,$ (6.2)
$\deg_{c}\Delta_{n,n}=d_{n}/2-\sum_{m|n,m\neq n}\varphi(n/m)d_{m}/2.$

By this formula, the $\Delta_{n,m}$ with degree $1$ are limited to
$\Delta_{1,1}=4c-1,\Delta_{2,1}=4c+3,\Delta_{3,3}=4c+7$, and
$\Delta_{4,2}=-4c-5$. Combining these facts, $-3/2$ is NOT a parabolic
parameter of the family $f_{c}(z)=z^{2}+c$ if the irreducibility conjecture
for $\Delta_{n,m}$ holds.

Generally, proving whether a specific parameter $c$ is in the Mandelbrot set
is not easy. As is mentioned in Remark $8.8$ and page 268 of [Mil06], the
value $c=-3/2$ is one such example.

##  7\. Remarks on other polynomial families

In this section, we give some remarks on other polynomial families.

###  7.1. The family $(z-c)z^{d}+c$

The method in Section 4 works for the polynomial family
$(\tilde{f}_{c}(z)=(z-c)z^{d}+c)_{c}$. We use the same notation in Section 4
to see this. In addition, we define the dynatomic polynomial for
$\tilde{f}_{c}$ and the multiplier polynomial for $\tilde{f}_{c}$ by

(7.1) $\displaystyle\tilde{\Phi}_{m}^{*}(z)$
$\displaystyle:=\prod_{k|m}\left(\tilde{f}_{c}^{\circ
k}(z)-z\right)^{\mu(m/k)},$ (7.2) $\displaystyle\tilde{\delta}_{m}(x)^{m}$
$\displaystyle:=\mathrm{Res}_{z}\left(\tilde{\Phi}_{m}(z),x-(\tilde{f}_{c}^{\circ
m})^{\prime}(z)\right).$

Then, the following theorem holds.

######  Theorem 7.1.

For a positive integer $m$, the polynomial
$d^{(d-1)\varepsilon_{m}+\deg_{z}\tilde{\Phi}_{m}/(d+1)}\tilde{\delta}_{m}(x)$
is an element of $\mathbb{Z}[(dc)^{d},x]$ and is monic in $(dc)^{d}$ up to
multiplication by $\pm 1$.

The quantity $\varepsilon_{m}$ in Theorem 7.1 is defined as in
LABEL:{sec:non_unicritical}. Namely,

(7.3)
$\displaystyle\varepsilon_{m}\coloneqq\sum_{k|n}\mu(n/k)=\begin{cases}1&\text{
if }m=1,\\\ 0&\text{ if }m\neq 1.\end{cases}$

For the family $\left(\tilde{f}_{c}(z)=(z-c)z^{d}+c\right)_{c}$, the following
diagram commutes

(7.4)
${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$$\scriptstyle{\nu}$$\scriptstyle{\tilde{f}_{c}}$$\scriptstyle{\tilde{f}_{\zeta
c}}$$\scriptstyle{\nu}$

for $\nu:\mathbb{P}\longrightarrow\mathbb{P};z\mapsto\zeta_{d}z$, where
$\zeta_{d}$ is a primitive $d$-th root of unity. Thus the discussion in the
proof of Lemma 3.2 (ii) shows $\mathrm{Res}_{z}(\tilde{f}_{c}^{\circ
k}(z)-z,x-(\tilde{f}_{c}^{\circ m})^{\prime}(z))\in\mathbb{Z}[c^{d},x]$. So,
it is enough to show that
$d^{\deg_{z}\tilde{\Phi}_{m}/(d+1)}\tilde{\delta}_{m}(x)$ is an element of
$\mathbb{Z}[dc,x]$ and monic in $dc$ up to multiplication by $\pm 1$.

By Lemma 4.1 (ii), we have $\tilde{f}_{c}^{\circ
k}(z)-z=(z-c)\left((F_{k}(z)+1)^{d}-1\right)$ and

(7.5) $(\tilde{f}_{c}^{\circ
m})^{\prime}(z)=(F_{m}(z)+1)^{d-1}(d+1)^{m}\mathcal{F}_{m}\left(-\frac{dc}{d+1},z\right).$

We define a poyninomial $H_{k}(z)$ by $H_{k}(z)=(F_{k}(z)+1)^{d}-1$. Then, we
have

(7.6) $\displaystyle\tilde{\delta}_{m}(x)^{m}$
$\displaystyle=\mathrm{Res}_{z}(\tilde{\Phi}_{m}^{*}(z),x-(\tilde{f}_{c}^{\circ
m})^{\prime}(z))$ (7.7)
$\displaystyle=\prod_{k|m}\mathrm{Res}_{z}\left((z-c)H_{k}(z),x-(F_{k}(z)+1)^{d-1}(d+1)^{m}\mathcal{F}_{m}\left(-\frac{dc}{d+1},z\right)\right)^{\mu(m/k)}$
(7.8)
$\displaystyle=\prod_{k|m}\left((x-c^{md})\mathrm{Res}_{z}\left(H_{k}(z),x-(F_{k}(z)+1)^{d-1}(d+1)^{m}\mathcal{F}_{m}\left(-\frac{dc}{d+1},z\right)\right)\right)^{\mu(m/k)}$
(7.9)
$\displaystyle=(x-c^{md})^{\varepsilon_{m}}\prod_{k|m}\operatorname{Res}_{z}\left(H_{k}(z),x-(F_{k}(z)+1)^{d-1}(d+1)^{m}\mathcal{F}_{m}\left(-\frac{dc}{d+1},z\right)\right)^{\mu(m/k)}.$

Putting
$\tilde{R}_{k,m}(x)=\operatorname{Res}_{z}\left(H_{k}(z),x-(F_{k}(z)+1)^{d-1}(d+1)^{m}\mathcal{F}_{m}\left(-\frac{dc}{d+1},z\right)\right)$,
the statement Theorem 7.1 is reduced to the following lemma.

######  Lemma 7.2.

The following statements hold for positive integers $k$ and $m$, which satisfy
$k|m$.

  1. (i)

$d^{m((d+1)^{k-1}-1)}\tilde{R}_{k,m}(x)\in\mathbb{Z}[dc,x]$.

  2. (ii)

The leading coefficient of $\tilde{R}_{k,m}(x)$ in $c$ is $\pm
d^{md(d+1)^{k-1}}$.

###### Proof.

Since $F_{m}(z)+1=\prod_{i=0}^{m-1}\tilde{f}_{c}^{\circ i}(z)$,
$H_{k}(z)=\left(\prod_{i=0}^{k-1}\tilde{f}_{c}^{\circ i}(z)\right)^{d}-1$, and
$Z(H_{k})\subset Z(\tilde{f}_{c}^{\circ k}(z)-z)$, the element
$\xi_{\alpha}:=(F_{k}(\alpha)+1)^{d-1}$ is a $d$-th root of unity for each
$\alpha\in Z(H_{k})$. Therefore, we have the expression

(7.10)
$\displaystyle\operatorname{Res}_{z}\left(H_{k}(z),x-(F_{k}(z)+1)^{d-1}(d+1)^{m}\mathcal{F}_{m}\left(y,z\right)\right)$
(7.11) $\displaystyle=$ $\displaystyle\prod_{\alpha\in
Z(H_{k})}\left(x-\xi_{\alpha}(d+1)^{m}\mathcal{F}_{m}\left(y,\alpha\right)\right).$

Therefore, the same argument as in Proposition 4.3 shows that

(7.12)
$\deg_{c}\operatorname{Res}_{z}\left(H_{k}(z),x-(F_{k}(z)+1)^{d-1}(d+1)^{m}\mathcal{F}_{m}\left(y,z\right)\right)=m((d+1)^{k-1}-1).$

This assertion proves (i). By using the same discussion as in Proposition 4.7,
we get the equality

(7.13)
$v\left((d+1)^{m}\mathcal{F}_{m}\left(-\frac{d}{d+1}c,\alpha\right)\right)=-m$

for each $\alpha\in Z(H_{k})$. Therefore, we can prove

(7.14) $\deg_{c}\tilde{R}_{k,m}(x)=m((d+1)^{k}-1)$

and

(7.15)
$\operatorname{LT}_{c}\left(\tilde{R}_{k,m}(x)\right)=\pm\operatorname{LT}_{c}\left((d+1)^{\deg_{z}H_{k}}H_{k}\left(\frac{dc}{d+1}\right)\right)^{m}=\pm
d^{d(d+1)^{k-1}}c^{m((d+1)^{k}-1)}.$

∎

###  7.2. The family $z^{d+2}+cz^{2}$

Throughout this subsection, we put $f_{c}(z)\coloneq z^{d+2}+cz^{2}$ for an
integer $d\geq 3$. In this subsection, we prove Theorem 1.9. More precisely,
we prove the following proposition and then observe that the same claim as in
Theorem 1.1 does not hold in this situation.

######  Proposition 7.3.

The first multiplier polynomial for $f_{c}(z)=z^{d+2}+cz^{2}$ is given by the
formula

(7.16) $\delta_{1}(x)=x\left((x-(d+2))^{d+1}+c(-cd)^{d}(x-2)\right).$

In particular, for each integer $n>1$, the following statement holds.

(7.17) $\operatorname{LT}_{c}(\Delta_{n,1}(c))=\pm
d^{d\varphi(n)}\Phi^{\mathrm{cyc}}_{n}(2)c^{(d+1)\varphi(n)}.$

Although we give the computation only for $n=1$, a better observation might be
obtained by computing the $n$-th multiplier polynomial in general.

###### Proof of Proposition 7.3.

A direct computation shows that if $\alpha$ is a fixed point of
$f_{c}(z)=z^{d+2}+cz^{2}$, then $\alpha=0$ or $\alpha\in Z(z^{d+1}+cz-1)$. By
the definition of the multiplier, we obtain $\omega_{1}(0)=0$ and
$\omega_{1}(\alpha)=(d+2)\alpha^{d+1}+2c\alpha=(d+2)-cd\alpha$. Hence, we can
explicitly calculate $\delta_{1}(x)$ as follows.

(7.18) $\displaystyle\delta_{1}(x)$
$\displaystyle=\prod_{f_{c}(\alpha)=\alpha}(x-\omega_{n}(\alpha))$ (7.19)
$\displaystyle=x\prod_{\alpha\in Z(z^{d+1}+cz-1)}(x-(d+2)+cd\alpha)$ (7.20)
$\displaystyle=x(-cd)^{d+1}\prod_{\alpha\in
Z(z^{d+1}+cz-1)}\left(\frac{x-(d+2)}{-cd}-\alpha\right)$ (7.21)
$\displaystyle=x(-cd)^{d+1}\left(\left(\frac{x-(d+2)}{-cd}\right)^{d+1}+c\frac{x-(d+2)}{-cd}-1\right)$
(7.22) $\displaystyle=x\left((x-(d+2))^{d+1}+c(-cd)^{d}(x-2)\right).$

By the explicit calculation of the multiplier polynomial $\delta_{1}(x)$, we
can easily calculate the leading term of $\Delta_{n,1}(c)$ as follows.

(7.23) $\displaystyle\operatorname{LT}_{c}(\Delta_{n,1}(c))$
$\displaystyle=\operatorname{LT}_{c}\left(\prod_{\Phi_{n}^{\mathrm{cyc}}(\zeta)=0}\delta_{n}(\zeta)\right)$
(7.24) $\displaystyle=\prod_{\Phi_{n}^{\mathrm{cyc}}(\zeta)=0}\zeta
c(-cd)^{d}(\zeta-2)$ (7.25) $\displaystyle=\pm
d^{d\varphi(n)}\Phi_{n}^{\mathrm{cyc}}(2)c^{(d+1)\varphi(n)}.$

∎

Bang’s theorem [Ban86] on cyclotomic numbers shows that
$\Phi_{n}^{\mathrm{cyc}}(2)$ has a prime divisor $q$ such that $q\equiv
1\text{ mod }n$ except for $\Phi_{6}^{\mathrm{cyc}}(2)=3$. Therefore, the
uniform integrality cannot be guaranteed as in other examples.

##  Appendix A Equalities on dynatomic modular curves

The first equality in Proposition 1.12 follows from the following lemma.

######  Lemma A.1.

For coprime integers $l$ and $n$, we have

(A.1) $\prod_{d\mid l}\Phi_{f,dn}^{\ast}(z)=\Phi_{f^{\circ l},n}^{\ast}(z).$

###### Proof.

For a divisor $d$ of $l$, we have

(A.2) $\Phi_{f,dn}^{\ast}(z)=\prod_{e\mid d}\prod_{e^{\prime}\mid
n}\left(f^{\circ
ee^{\prime}}(z)-z\right)^{\mu(d/e)\mu(n/e^{\prime})}=\prod_{e\mid
d}\Phi_{f^{\circ e},n}^{\ast}(z)^{\mu(d/e)}$

since $d$ and $n$ are coprime. Thus, we have

(A.3) $\prod_{d\mid l}\Phi_{f,dn}^{\ast}(z)=\prod_{d\mid l}\prod_{e\mid
d}\Phi_{f^{\circ e},n}^{\ast}(z)^{\mu(d/e)}.$

By letting $e^{\prime}\coloneqq d/e$, this equals to

(A.4) $\prod_{d\mid l}\prod_{e\mid l/e}\Phi_{f^{\circ
e},n}^{*}(z)^{\mu(e^{\prime})}=\prod_{e\mid l}\Phi_{f^{\circ
e},n}^{\ast}(z)^{\sum_{e^{\prime}\mid l/e}\mu(e^{\prime})}=\Phi_{f^{\circ
l},n}^{*}(z).$

∎

###### Proof of Proposition 1.12.

It suffices to show the second equality. We have

(A.5) $\Phi_{f^{\circ\widetilde{m}},m^{\prime}}^{\ast}(z)=\prod_{d\mid
m^{\prime}}\left(f^{\circ\widetilde{m}d}(z)-z\right)^{\mu(m^{\prime}/d)}=\prod_{d\mid
m^{\prime}}\left(\frac{f^{\circ\widetilde{m}d}(z)-z}{z-z_{0}}\right)^{\mu(m^{\prime}/d)}.$

By taking a limit as $z\to z_{0}$, we obtain

(A.6) $\Phi_{f^{\circ\widetilde{m}},m^{\prime}}^{\ast}(z_{0})=\prod_{d\mid
m^{\prime}}\left(\left(f^{\circ\widetilde{m}d}\right)^{\prime}(z_{0})-1\right)^{\mu(m^{\prime}/d)}=\prod_{d\mid
m^{\prime}}\left(\lambda^{\widetilde{m}d/k}-1\right)^{\mu(m^{\prime}/d)}=\Phi^{\mathrm{cyc}}_{m^{\prime}}\left(\lambda^{\widetilde{m}/k}\right).$

∎

##  Appendix B Lists of $\delta_{m}(x)$ and $\Delta_{n,m}$ for our polynomials

This section shows computational results on $\delta_{m}$ and $\Delta_{n,m}$.

For $f_{c}(z)=z^{d}+c$. Then, for integers $d\geq 2$ and $m\geq 1$, the
polynomial $\Psi(x,C)$ such that $\delta_{m}(x)=\Psi(x,d^{d}c^{d-1})$ are
explicitly written as follows.

$d$ | $m$ | $\deg_{z}\Phi_{m}^{*}$ |  $\Psi(x,C)$ such that $\delta_{m}(x)=\Psi(x,d^{d}c^{d-1})$ for $f_{c}(z)=z^{d}+c$.
---|---|---|---
$2$ | $1$ | $2$ |  $C+x^{2}-2x$
$2$ | $2$ | $2$ |  $-C+x-4$
$2$ | $3$ | $6$ |  $C^{3}+8C^{2}-2Cx+16C+x^{2}-16x+64$
$3$ | $1$ | $3$ |  $-C+x^{3}-6x^{2}+9x$
$3$ | $2$ | $6$ |  $-C^{2}+6Cx-54C+x^{3}-27x^{2}+243x-729$
$3$ | $3$ | $24$ |  $C^{8}+C^{7}x+162C^{7}+C^{6}x^{2}+108C^{6}x+10935C^{6}-2C^{5}x^{3}+54C^{5}x^{2}+2187C^{5}x+393660C^{5}-2C^{4}x^{4}+162C^{4}x^{3}-1458C^{4}x^{2}-196830C^{4}x+9034497C^{4}-2C^{3}x^{5}-324C^{3}x^{4}+23328C^{3}x^{3}-157464C^{3}x^{2}-11691702C^{3}x+172186884C^{3}+C^{2}x^{6}+162C^{2}x^{5}-19683C^{2}x^{4}+551124C^{2}x^{3}+1594323C^{2}x^{2}-258280326C^{2}x+2711943423C^{2}+Cx^{7}-108Cx^{6}+2187Cx^{5}+196830Cx^{4}-13286025Cx^{3}+344373768Cx^{2}-4261625379Cx+20920706406C+x^{8}-216x^{7}+20412x^{6}-1102248x^{5}+37200870x^{4}-803538792x^{3}+10847773692x^{2}-83682825624x+282429536481$
$4$ | $1$ | $4$ |  $C+x^{4}-12x^{3}+48x^{2}-64x$
$4$ | $2$ | $12$ |  $C^{3}-C^{2}x^{2}-48C^{2}x+768C^{2}-Cx^{4}+1536Cx^{2}-32768Cx+196608C+x^{6}-96x^{5}+3840x^{4}-81920x^{3}+983040x^{2}-6291456x+16777216$
$5$ | $1$ | $5$ |  $-C+x^{5}-20x^{4}+150x^{3}-500x^{2}+625x$
$5$ | $2$ | $20$ |  $C^{4}-20C^{3}x^{2}-500C^{3}x+12500C^{3}-2C^{2}x^{5}-150C^{2}x^{4}+9375C^{2}x^{3}+31250C^{2}x^{2}-5859375C^{2}x+58593750C^{2}+20Cx^{7}-2500Cx^{6}+112500Cx^{5}-1562500Cx^{4}-39062500Cx^{3}+1757812500Cx^{2}-24414062500Cx+122070312500C+x^{10}-250x^{9}+28125x^{8}-1875000x^{7}+82031250x^{6}-2460937500x^{5}+51269531250x^{4}-732421875000x^{3}+6866455078125x^{2}-38146972656250x+95367431640625$

Let $f_{c}(z)=z^{d+1}+cz$. Then, for integers $d\geq 1$ and $m\geq 1$, the
polynomials $\Psi(x,C)$ such that
$d^{(deg_{z}\Phi_{m}^{*})/(d+1)}\delta_{m}(x)=\Psi(x,dc)$ are explicitly
written as follows.

$d$ | $m$ | $\deg_{z}\Phi_{m}^{\ast}$ |  $\Psi(x,C)$ such that $d^{(deg_{z}\Phi_{m}^{*})/(d+1)}\delta_{m}(x)=\Psi(x,dc)$ for $f_{c}(z)=z^{d+1}+cz$
---|---|---|---
$2$ | $1$ | $3$ |  $\left(-C+2x\right)\left(C+x-3\right)^{2}$
$2$ | $2$ | $6$ |  $\left(C^{2}+2x-18\right)^{2}\left(-C^{2}-6C+x-9\right)$
$2$ | $3$ | $12$ |  $C^{12}+3C^{11}-72C^{10}+C^{9}x-216C^{9}+1539C^{8}-99C^{7}x+4617C^{7}-12C^{6}x^{2}-108C^{6}x-5832C^{6}-36C^{5}x^{2}+2916C^{5}x-17496C^{5}+396C^{4}x^{2}+5832C^{4}x-131220C^{4}+4C^{3}x^{3}+972C^{3}x^{2}-14580C^{3}x-393660C^{3}+48C^{2}x^{3}-1296C^{2}x^{2}-34992C^{2}x+944784C^{2}-144Cx^{3}+11664Cx^{2}-314928Cx+2834352C+16x^{4}-1728x^{3}+69984x^{2}-1259712x+8503056$
$3$ | $1$ | $4$ |  $\left(-C+3x\right)\left(C+x-4\right)^{3}$
$3$ | $2$ | $12$ |  $\left(-C^{4}-4C^{3}-2C^{2}x-16C^{2}-12Cx+192C+3x^{2}-96x+768\right)^{3}$
$4$ | $1$ | $5$ |  $\left(-C+4x\right)\left(C+x-5\right)^{4}$
$4$ | $2$ | $20$ |  $\left(-C^{2}-10C+x-25\right)^{2}\left(-C^{4}-3C^{2}x-25C^{2}+4x^{2}-200x+2500\right)^{4}$
$5$ | $1$ | $6$ |  $(-C+5x)(C+x-6)^{5}$
$5$ | $2$ | $30$ |  $(C^{6}+6C^{5}+3C^{4}x+36C^{4}+24C^{3}x+216C^{3}-9C^{2}x^{2}+288C^{2}x+1296C^{2}-30Cx^{2}+2160Cx-38880C+5x^{3}-540x^{2}+19440x-233280)^{5}$

Let $f_{c}(z)=(z-c)z^{d}+c$. Then, for integers $d\geq 1$ and $m\geq 1$, there
is a polynomial $\Psi(x,C)$ such that
$d^{(d-1)\varepsilon_{m}+(deg_{z}\Phi_{m}^{*})/(d+1)}\delta_{m}(x)=\Psi(x,(dc)^{d})$,
where $\varepsilon_{m}$ is $1$ when $m=1$, and is $0$ otherwise. The
polynomials $\Psi(x,C)$ for some pairs $(d,m)$ are explicitly written as
follows.

$d$ | $m$ | $\deg_{z}\Phi_{m}^{\ast}$ |  $\Psi(x,C)$ such that $d^{(d-1)\varepsilon_{m}+(deg_{z}\Phi_{m}^{*})/(d+1)}\delta_{m}(x)=\Psi(x,(dc)^{d})$ for $f_{c}(z)=(z-c)z^{d}+c$
---|---|---|---
$2$ | $1$ | $3$ |  $\left(-C+4x\right)\left(-C+x^{2}-6x+9\right)$
$2$ | $2$ | $6$ |  $\left(C+2x-18\right)\left(-C^{2}+Cx+27C+2x^{2}-36x+162\right)$
$3$ | $1$ | $4$ |  $\left(-C+27x\right)\left(C+x^{3}-12x^{2}+48x-64\right)$
$3$ | $2$ | $12$ |  $-C^{4}-12C^{3}x+704C^{3}-26C^{2}x^{3}+240C^{2}x^{2}+5376C^{2}x-151552C^{2}-324Cx^{4}+13824Cx^{3}-165888Cx^{2}+7077888C+27x^{6}-2592x^{5}+103680x^{4}-2211840x^{3}+26542080x^{2}-169869312x+452984832$
$4$ | $1$ | $5$ |  $\left(-C+256x\right)\left(-C+x^{4}-20x^{3}+150x^{2}-500x+625\right)$
$4$ | $2$ | $20$ |  $(C^{2}-17Cx^{2}+250Cx-5625C+16x^{4}-1600x^{3}+60000x^{2}-1000000x+6250000)(C^{3}+18C^{2}x^{2}-100C^{2}x-6250C^{2}+33Cx^{4}+500Cx^{3}-71250Cx^{2}+562500Cx+9765625C+16x^{6}-2400x^{5}+150000x^{4}-5000000x^{3}+93750000x^{2}-937500000x+3906250000)$

Let $f_{c}(z)=z^{d+2}+cz^{2}$. Related to $\Delta_{n,m}$, the values
$\operatorname{Res}_{x}(\Phi^{\mathrm{cyc}}_{n},\delta_{m})$ for integers
$d,n,m\geq 1$ are explicitly computed as follows. One can see that they are
polynomials in $c^{d+1}$ as the other polynomials we studied in this paper.
Can the readers find out some patterns in the leading coefficients?

$d$ | $n$ | $m$ | Leading coefficient |  $\operatorname{Res}_{x}(\Phi^{\mathrm{cyc}}_{n},\delta_{m})$ for $f_{c}(z)=z^{d+2}+cz^{2}$
---|---|---|---|---
$2$ | $1$ | $1$ | $-2^{2}$ |  $-4c^{3}-27$
$2$ | $1$ | $2$ | $2^{10}\cdot 3$ |  $(12c^{3}+125)(256c^{9}+3328c^{6}-19872c^{3}+91125)$
$2$ | $2$ | $1$ | $2^{2}\cdot 3$ |  $12c^{3}+125$
$2$ | $2$ | $2$ | $2^{10}\cdot 5$ |  $5120c^{12}+125184c^{9}+340608c^{6}-2967452c^{3}+24137569$
$2$ | $3$ | $1$ | $2^{4}\cdot 7$ |  $112c^{6}+1980c^{3}+9261$
$2$ | $3$ | $2$ | $2^{20}\cdot 3\cdot 7$ |  $22020096c^{24}+1068761088c^{21}+15903686656c^{18}+47302946816c^{15}-293551023104c^{12}+3011458104576c^{9}+19218341274768c^{6}-104174224739676c^{3}+413976684737889$
$3$ | $1$ | $1$ | $-3^{3}$ |  $-27c^{4}-256$
$3$ | $1$ | $2$ | $2^{12}\cdot 3^{23}$ |  $14281868906496(c^{4}+16)^{4}(27c^{4}+256)$
$3$ | $2$ | $1$ | $3^{4}$ |  $81(c^{4}+16)$
$3$ | $2$ | $2$ | $2^{12}\cdot 3^{8}\cdot 5$ |  $4096(9c^{4}+169)^{3}(3645c^{8}+77328c^{4}+456976)$
$3$ | $3$ | $1$ | $3^{6}\cdot 7$ |  $5103c^{8}+131517c^{4}+923521$
$3$ | $3$ | $2$ | $2^{4}\cdot 3^{31}\cdot 7$ |  $3486784401(27c^{8}+495c^{4}+2401)(144c^{8}+5196c^{4}+47089)^{3}(5103c^{8}+131517c^{4}+923521)$
$4$ | $1$ | $1$ | $-2^{8}$ |  $-256c^{5}-3125$
$4$ | $1$ | $2$ | $2^{48}\cdot 3$ |  $(768c^{5}+16807)(1099511627776c^{25}+157582350090240c^{20}+7170620588556288c^{15}+81096927846400000c^{10}-943965155000000000c^{5}+8620460479736328125)$
$4$ | $2$ | $1$ | $2^{8}\cdot 3$ |  $768c^{5}+16807$
$4$ | $2$ | $2$ | $2^{48}\cdot 5$ |  $1407374883553280c^{30}+237348276553121792c^{25}+14256316535097262080c^{20}+331738425127665139712c^{15}+1211789484758147465216c^{10}-21435361197289473187072c^{5}+333446267951815307088493$
$4$ | $3$ | $1$ | $2^{16}\cdot 7$ |  $458752c^{10}+15544576c^{5}+147008443$
$4$ | $3$ | $2$ | $2^{96}\cdot 3\cdot 7$ |  Too long

##  Appendix C Exact description of Newton polygons for $z^{d+1}+cz$ and
$z^{d+1}-cz+c$

For a positive integer $n$ and $f(z)=z^{d+1}+cz$ or $f(z)=z^{d+1}-cz+c$, we
show Newton polygons of $f^{\circ n}(z)$ in Figures 8 and 8. The proof is by
induction on $n$.

Figure 7. The Newton polygon of $f^{\circ n}(z)$ for $f(z)=z^{d+1}+cz$.

Figure 8. The Newton polygon of $f^{\circ n}(z)$ for $f(z)=z^{d+1}-cz+c$.

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