# On irreducibility of prefixed algebraic sets in moduli spaces of prime
degree polynomials

Niladri Patra  School of Mathematics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Navy Nagar, Colaba, Mumbai-400005<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Consider the moduli space, $\mathcal{M}_{d}$, of degree $d\geq 2$ polynomials
over ${\mathbbm{C}}$, with a marked critical point. Given $k\geq 0,\;p$ an odd
prime, we show that the set $\Sigma_{k,1,p}$ of conjugacy classes of degree
$p$ polynomials, for which the marked critical point is strictly
$(k,1)$-preperiodic, is an irreducible quasi-affine algebraic set.
Irreducibility of these sets was conjectured by Milnor, and has been proved
for $p=3$ by Buff, Epstein and Koch.

We prove that the subspaces of $\Sigma_{k,1,p}$, that arise by varying the
ramification index of the marked critical point all the way up to the
unicritical case, are all irreducible subvarieties. Finally, using the
irreducibility of $\Sigma_{k,1,p}$ we give a new and short proof of the fact
that the set of all unicritical points of $\Sigma_{k,1,p}$ form one Galois
orbit under the action of absolute Galois group of ${\mathbbm{Q}}$.

###### 2020 Mathematics Subject Classification:

Primary 11R09, Secondary 37F12, 37P45

##  1\. Introduction

Complex dynamics studies the behaviour of iterates of rational maps on the
Riemann sphere. Since the time of Fatou and Julia, it has been known that the
behaviour of critical points of a map, under iterations, dictates the
behaviour of the dynamical system generated by that map. Thurston ([DH93])
showed that the dynamical behaviour of the critical points of a polynomial map
determines the conjugacy class of the dynamical system, up to finitely many
choices.

Let $\mathcal{M}_{d}$ be the space of all degree $d$ polynomials over
${\mathbbm{C}}$ with a marked critical point, modulo affine conjugation. The
space $\mathcal{M}_{d}$ can be seen as $(d-1)$-dimensional affine variety.
Thurston’s work shows that by fixing the behaviour of _all_ the critical
points of a general degree $d$ polynomial map, one gets finitely many points
in $\mathcal{M}_{d}$. Instead of _all_ the critical points, fixing the
behaviour of _some_ of the critical points results in dynamically interesting
subfamilies of $\mathcal{M}_{d}$. These subfamilies have been studied since
the 1990s ([BH88], [M90], [BDK91], [BH92], [MP92], [R06], [LQ07], [Mil09],
[BKM10], [BEK18], [AK20]). A long-standing conjecture of Milnor ([Mil09])
states that:

######  Conjecture 1.1.

Fix $k\geq 0,n>0,d>2$. The family $\Sigma_{k,n,d}$ of conjugacy classes of
degree $d$ polynomials for which the marked critical point is strictly
$(k,n)$-preperiodic is an irreducible subset of $\mathcal{M}_{d}$, under the
Zariski topology.

Fixing the ramification index of the marked critical point to be equal to $d$,
we arrive at the unicritical case. In other words, polynomials with a unique
critical point in ${\mathbbm{C}}$ are called _unicritical polynomials_. Any
unicritical polynomial of degree $d\geq 2$ is conjugate to a polynomial of the
form $f_{c}(X)=X^{d}+c,c\in{\mathbbm{C}}$ ($0$ is the unique finite critical
point of $f_{c}$ with ramification index $d$). In the article [Mil14], Milnor
studied arithmetic properties of unicritical polynomials and made the
following conjecture:

######  Conjecture 1.2.

Fix $k\geq 0,n>0,d\geq 2$. The values of $c$ for which $0$ is strictly
$(k,n)$-preperiodic under $f_{c}$, form one Galois orbit under the action of
the absolute Galois group of ${\mathbbm{Q}}$.

Conjecture 1.2 has seen a lot of work in the last decade ([HT15], [FG15],
[BEK18], [Gok20], [G20], [BG23], [BGok22]). In particular, Vefa Goksel
([Gok20]) has proved Conjecture 1.2 for prime degree polynomials with $k\geq
0,n=1$, by studying arithmetic properties of the forward orbit of $0$ under
$f_{c}$. This result of Vefa Goksel has been used by Buff, Epstein and Koch
([BEK18]) to prove Conjecture 1.1 for cubic polynomials ($d=3$), with period
$n=1$. Arfeux and Kiwi ([AK20]) have proved Conjecture 1.1 in the strictly
periodic case, $k=0$, in the moduli space of cubic polynomials.

In this article, we go the other way around. We study Conjecture 1.1 for
general degree $d$. We show that (Theorem 5.9):

######  Theorem 1.3.

Fix $n=1$. Conjecture 1.1 is true for any odd prime degree $d$.

Our proof of Theorem 1.3 does not depend on the unicritical case (Conjecture
1.2). In contrast, we use Theorem 1.3 to give a new and short proof of
conjecture 1.2 for odd prime degree polynomials. In particular, we prove that
(Theorem 7.1):

######  Theorem 1.4.

Fix $n=1$. Conjecture 1.2 is true for any odd prime degree $d$.

###  1.1. Mixed critical cases:

In Conjecture 1.1, the ramification index of the marked critical point has
been allowed to vary freely between $2$ and $d$. Fixing a lower bound $j$ for
the ramification index and varying $j$ between $2$ and $d$, one obtains a
finite decreasing sequence of algebraic sets, with strictly decreasing
dimensions. We call these sets mixed critical cases. The advantage of studying
$\Sigma_{k,n,d}$ without any dependence on the unicritical case, is that it
allows us to prove irreducibility of mixed critical subfamilies of
$\mathcal{M}_{d}$. The mixed critical cases have not been studied much so far.
In [R06], Roesch studied the topology of the connectedness locus of a
particular mixed critical family of maps. Here, we consider algebraic
properties of mixed critical subfamilies of $\Sigma_{k,1,p}$ and show that
these are irreducible under Zariski topology on $\mathcal{M}_{p}$.

######  Theorem 1.5.

Fix $k\geq 0,p\text{ prime },p>3,\;2<j<p$. The family of conjugacy classes of
degree $p$ polynomials for which the marked critical point is strictly
$(k,1)$-preperiodic and has ramification index $\geq j$, is an irreducible
quasi-affine subvariety of $\mathcal{M}_{p}$.

Theorem 1.5 has been restated and proved under Theorem 6.1, in section 6. As a
corollary of Theorem 1.5, we get (Corollary 6.2),

######  Corollary 1.6.

Let $k\geq 0,p$ be an odd prime and $2\leq j\leq p-1$. The set of all points
of $\mathcal{M}_{p}$ for which the marked critical point is strictly
$(k,1)$-preperiodic and has ramification index $j$, is an irreducible quasi-
affine variety of dimension $p-j$.

Theorem 1.5 provides evidence for the following conjecture (Conjecture 6.3):

######  Conjecture 1.7.

Fix $k\geq 0,\;n>0,\;d>2,\;2<j<d$. The family of conjugacy classes of degree
$d$ polynomials for which the marked critical point is strictly
$(k,n)$-preperiodic, and has ramification index $\geq j$, is an irreducible
quasi-affine subvariety of $\mathcal{M}_{d}$.

Observe that, allowing $j$ to be $2$ in Conjecture 1.7, we obtain Conjecture
1.1. Similarly in Conjecture 1.7, allowing $j$ to be $d$ and restricting the
irreducibility to only over ${\mathbbm{Q}}$, we arrive at Conjecture 1.2.
Thus, Conjecture 1.7, along with Theorem 1.5, provides a bridge between
Conjectures 1.1 and 1.2.

###  1.2. Sketch of Proof.

####  1.2.1. Normal form.

To study the families mentioned in Theorem 1.3, we first require a concrete
description of $\mathcal{M}_{d}$ as an affine variety. By Thurston’s work, the
dynamical behaviour of any polynomial is dictated by the dynamical behaviour
of its critical points. This suggests representing polynomials by their finite
critical points and one finite critical value, modulo affine conjugation. We
introduce a normal form for monic, reduced degree $d\geq 2$ polynomials, which
is a modification of _Branner-Hubbard normal form_ appearing in ([BH88]). This
normal form also generalizes the unicritical normal form $f(X)=X^{d}+c$ used
in ([Mil14], [HT15], [Gok20], [G20], [BGok22], [BG23]) and each of the various
other normal forms considered in ([BH88], [Mil09], [LQ07], [FG15]).

This has two applications. First, we use this normal form to express
$\mathcal{M}_{d}$ explicitly as an affine variety, $(d-1)$-dimensional affine
space quotiented by the action of the group of $(d-1)$th roots of unity, and a
permutation group. Secondly, we use the normal form to construct polynomials
$h_{k,n,d},~{}k\geq 0,n>0,d>2,$ such that the families $\Sigma_{k,n,d}$
appearing in Conjecture 1.1 are Zariski dense in the algebraic sets of these
polynomials in $\mathcal{M}_{d}$.

####  1.2.2. Proof of Theorem 1.3: Irreducibility of $\Sigma_{k,1,p}$.

Let $k\geq 0,p$ be an odd prime. To show the irreducibility of
$\Sigma_{k,1,p}$, it suffices to show that the polynomials $h_{k,1,p}$ are
irreducible over ${\mathbbm{C}}$.

The proof of irreducibility uses arithmetical methods. We first observe that
the coefficients of these polynomials lie in the ring ${\mathbbm{Z}}_{(p)}$,
the localization of the rational integers ${\mathbbm{Z}}$ at the prime $p$. We
show that the polynomials $h_{k,1,p}$ are generalised Eisenstein-type
polynomials with respect to the prime $p$. For the resultant condition of the
generalised Eisenstein criterion, we need a weaker version of Thurston’s
rigidity, of which an elementary algebraic proof is given in Subsection 4.2.
This shows that $h_{k,1,p}$ polynomials are irreducible over ${\mathbbm{Q}}$.

To prove irreducibility over the complex numbers, we observe that the lowest
degree homogeneous part of $h_{k,1,p}$ factorizes into linear factors over
${\mathbbm{Q}}$, for any $k\geq 2,\;p$ an odd prime. As $h_{k,1,p}$
polynomials are irreducible over ${\mathbbm{Q}}$, this forces their
irreducibility over ${\mathbbm{C}}$, for any $k\geq 2$. The case $k=0$ is a
trivial case, as $h_{0,1,p}$ is linear. For $k=1$, a little modification of
the above proof shows that $h_{1,1,p}$ is also irreducible over
${\mathbbm{C}}$.

####  1.2.3. Proof of Theorem 1.5: Mixed critical case.

The variables of the polynomials $h_{k,1,p}$ stand for the critical points and
one critical value. The polynomials defining the mixed critical locus arise by
identifying some of the variables (critical points). The resulting polynomials
are once again generalised Eisenstein polynomials, and hence are irreducible
over ${\mathbbm{Q}}$. As long as we do not identify all the critical points,
the lowest degree homogeneous part of the resulting polynomials mentioned
above always has at least one simple linear factor defined over
${\mathbbm{Q}}$. This enforces their irreducibility over ${\mathbbm{C}}$,
completing the proof of irreducibility of all the mixed critical loci.

####  1.2.4. Proof of Theorem 1.4: Unicritical case.

Identifying all the finite critical points to $0$, we reach at the unicritical
case. The resulting polynomial we get from $h_{k,1,p}$ is a univariate
polynomial, which is an Eisenstein-type polynomial multiplied with some power
of the variable $\beta$ (critical value). Throwing away this power of $\beta$,
we see that the roots of these Eisentstein-type polynomials are the values of
$c$ mentioned in Conjecture 1.2. This allows us to complete the proof of
Theorem 1.4.

###  1.3. Outline of the paper.

Section 2 contains construction of normal form, moduli space $\mathcal{M}_{d}$
and the polynomials $h_{k,n,d}$. All the notations are collected in Section 3.
We gather some basic lemmas and tools, such as generalised Eisenstein
criterion, in Section 4. A short algebraic proof of a weaker version of
Thurston’s rigidity theorem is also included in Section 4. Proof of Theorem
1.3 is the content of Section 5. In Sections 6 and 7, we prove Theorem 1.5
(mixed critical case), and Theorem 1.4 (unicritical case), respectively.
Obstructions to direct generalization of our techniques to some other cases of
Conjecture 1.1 are studied in Sections 8 and 9.

###  1.4. Acknowledgement.

The author would like to thank C. S. Rajan, Sagar Shrivastava and Manodeep
Raha, for their valuable inputs and many discussions. The author expresses his
gratitude to Ashoka University for providing a productive workspace, where
this work was done. The author is also thankful to Kerala School of
Mathematics for his visit, where he had many helpful discussions with Plawan
Das, Subham Sarkar and M. M. Radhika.

##  2\. Preliminaries

Let $f$ be a polynomial over ${\mathbbm{C}}$. Let $f^{m},m\in{\mathbbm{N}}$,
denote the iteration $f^{m}:=f\circ f^{m-1}$, and $f^{0}$ is defined to be the
identity map. For any point $x\in{\mathbbm{C}}$, the _forward orbit_ of $x$ is
defined to be the set,
$\\{x,f(x),f^{2}(x),\cdot\cdot\cdot,f^{m}(x),\cdot\cdot\cdot\\}=\\{f^{m}(x)|m\in{\mathbbm{N}}\cup\\{0\\}\\}$.
Recall that the roots of the derivative $f^{\prime}$ of $f$, are called the
_finite critical points_ or _ramified points_ of $f$. Since the time of Fatou
and Julia, it has been known that behaviour of a complex dynamical system
(which means forward orbits of $x$, where $x$ varies over ${\mathbbm{C}}$) is
largely governed by the forward orbits of the critical points of the map.
Hence, in order to understand the dynamical system generated by $f$, one needs
to study the forward orbits of critical points of $f$.

###  2.1. Periodic and preperiodic points

Let $g$ be a polynomial defined over ${\mathbbm{C}}$. Let $k\geq 0,n>0$ be
integers. A point $x\in{\mathbbm{C}}$ is called a _periodic point of period
$n$_ (or, _$n$ -periodic point_) if and only if $g^{n}(x)=x$, and called
_strictly $n$-periodic point_ (or, _periodic point of exact period_ $n$) if
and only if $n$ is the smallest natural number such that $g^{n}(x)=x$.

A point $x\in{\mathbbm{C}}$ is called a $(k,n)$-_preperiodic point_ if and
only if $g^{k+n}(x)=g^{k}(x)$, and _strictly $(k,n)$-preperiodic point_ if and
only if $g^{k+n}(x)=g^{k}(x)$ and for all $0\leq l\leq k,\;0\leq m\leq n$,
$(l,m)\neq(k,n)\in{\mathbbm{Z}}^{2}$, we have $g^{l+m}(x)\neq g^{l}(x)$.

###  2.2. Affine conjugation

Consider
$Aut({\mathbbm{C}})=\\{h\in{\mathbbm{C}}[z]~{}|~{}h(z)=az+b,a\in{\mathbbm{C}}\backslash\\{0\\},b\in{\mathbbm{C}}\\}$,
the group of holomorphic automorphisms of ${\mathbbm{C}}$. Two maps
$g,\tilde{g}\in{\mathbbm{C}}[z]$ are called _affine conjugate_ to each other
if and only if there exists a map $h\in Aut({\mathbbm{C}})$, such that
$\tilde{g}=h\circ g\circ h^{-1}$.

###  2.3. Normal form and moduli spaces of polynomials with a marked critical
point.

For an arbitrary but fixed integer $d>2$, let $S_{d}$ be the set of all
polynomials of degree $d$ with a marked critical point. Consider the _moduli
space_ $\mathcal{M}_{d}$, which is defined to be the quotient space of
$S_{d}$, obtained by identifying polynomials that are affine conjugate to each
other and the affine conjugation map sends the marked critical point of the
first to the marked critical point of the latter. We say a polynomial is
_reduced_ if the sum of its roots is zero, which is also equivalent to saying
that the coefficient of the second highest degree term of the polynomial is
zero. Any polynomial over ${\mathbbm{C}}$ is affine conjugate to a monic
reduced polynomial over ${\mathbbm{C}}$. Thus, the moduli space
$\mathcal{M}_{d}$ is the set of equivalence classes of monic reduced
polynomials of degree $d$ over ${\mathbbm{C}}$ with a marked critical point,
where the equivalence relation is given by affine conjugation.

Let $f$ be a monic, reduced, degree $d$ polynomial. We define _normal form_ of
$f$ to be a form of the polynomial $f$ in which each coefficient of $f$ is
expressed as polynomials in its finite critical points and one finite critical
value. Note that, any polynomial of degree $d$ has $(d-1)$ number of finite
critical points, counted with multiplicity.

######  Proposition 2.1.

Let $f$ be a monic and reduced element of ${\mathbbm{C}}[z]$ of degree $d$.
Let $\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1}$ be the finite
critical points of $f$, not necessarily distinct. Let
$f(\alpha_{1})=\beta,\;\bar{\alpha}:=(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1})$
and $s_{i}(\bar{\alpha})$ denote the $i$-th elementary symmetric polynomial in
$\alpha_{1},\cdot\cdot\cdot,\alpha_{d-1}$. Then, the polynomial $f(z)$ can be
written in a normal form as,

$z^{d}+\sum_{i=2}^{d-1}(-1)^{i}\frac{d}{d-i}\cdot s_{i}(\bar{\alpha})\cdot
z^{d-i}-\alpha_{1}^{d}-\sum_{i=2}^{d-1}(-1)^{i}\frac{d}{d-i}\cdot
s_{i}(\bar{\alpha})\cdot\alpha_{1}^{d-i}+\beta,$

with $\sum_{i=1}^{d-1}\alpha_{i}=0$.

###### Proof.

The derivative of $f$ can be written as,

$f^{\prime}(z)=c(z-\alpha_{1})(z-\alpha_{2})\cdot\cdot\cdot(z-\alpha_{d-1}),$

where $c\in{\mathbbm{C}},c$ constant. Since $f$ is reduced, $f^{\prime}$ is
also reduced. Hence, $\sum_{i=1}^{d-1}\alpha_{i}=0$. So,

$f^{\prime}(z)=c\left(z^{d-1}+\sum_{i=2}^{d-1}(-1)^{i}s_{i}(\bar{\alpha})z^{d-i-1}\right),$
(2.1)

where $\bar{\alpha}=(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1})$ and
$s_{i}(\bar{\alpha})$ is the $i$-th elementary symmetric polynomial in
$\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1}$. By integration,

$f(z)=c\left(\frac{z^{d}}{d}+\sum_{i=2}^{d-1}(-1)^{i}\frac{s_{i}(\bar{\alpha})}{d-i}z^{d-i}\right)+c^{\prime},$

where $c^{\prime}\in{\mathbbm{C}}$, $c^{\prime}$ constant. The polynomial $f$
being monic, we get $c=d$. Hence,

$f(z)=z^{d}+\sum_{i=2}^{d-1}(-1)^{i}\frac{d}{d-i}\cdot
s_{i}(\bar{\alpha})\cdot z^{d-i}+c^{\prime}.$

Equating $f(\alpha_{1})$ with $\beta$,

$\beta=f(\alpha_{1})=\alpha_{1}^{d}+\sum_{i=2}^{d-1}(-1)^{i}\frac{d}{d-i}\cdot
s_{i}(\bar{\alpha})\cdot\alpha_{1}^{d-i}+c^{\prime},$ $\implies
c^{\prime}=-\alpha_{1}^{d}-\sum_{i=2}^{d-1}(-1)^{i}\frac{d}{d-i}\cdot
s_{i}(\bar{\alpha})\cdot\alpha_{1}^{d-i}+\beta.$

Hence,

$f(z)=z^{d}+\sum_{i=2}^{d-1}(-1)^{i}\frac{d}{d-i}\cdot
s_{i}(\bar{\alpha})\cdot
z^{d-i}-\alpha_{1}^{d}-\sum_{i=2}^{d-1}(-1)^{i}\frac{d}{d-i}\cdot
s_{i}(\bar{\alpha})\cdot\alpha_{1}^{d-i}+\beta.$ (2.2)

∎

In this article, we will represent each point of $\mathcal{M}_{d}$ by a monic
reduced polynomial written in the normal form as in Equation (2.2), with
$\alpha_{1}$ as its marked critical point.

######  Remark 2.2.

The normal form mentioned above is a modification of the _Branner-Hubbard
normal form_ appearing in [BH88]. This also generalises various other normal
forms considered in [Mil09], [LQ07], [FG15], [Mil14].

###  2.4. Moduli space $\mathcal{M}_{d}$ is an affine variety

As $f$ varies over all monic reduced polynomials over ${\mathbbm{C}}$ of
degree $d$, $\alpha_{i},1\leq i\leq p-1,$ and $\beta$ vary over
${\mathbbm{C}}$, such that $\sum_{i=1}^{d-1}\alpha_{i}=0$. So, we can
parametrize the set of all monic, reduced polynomials over ${\mathbbm{C}}$ of
degree $d$ with a marked critical point, using
$\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2}$ as $(d-2)$ finite
critical points with $\alpha_{1}$ as its marked critical point and $\beta$
such that $f(\alpha_{1})=\beta$. From here on in this article, we will
consider $\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\beta$ as
variables over ${\mathbbm{C}}$, unless otherwise mentioned. For the rest of
this subsection, we will denote the polynomial $f$ in Equation (2.2) as
$f_{\bar{\alpha},\beta}$, where
$\bar{\alpha}=(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1})$.

Following the previous subsection and the above paragraph, $\mathcal{M}_{d}$
can be considered as a quotient space of ${\mathbbm{C}}^{d-1}$, quotiented by
affine conjugation relation, where the coordinates of ${\mathbbm{C}}^{d-1}$
are $\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\beta$. Next, we will
show that $\mathcal{M}_{d}$ is an affine variety.

######  Proposition 2.3.

For any $d\in{\mathbbm{N}},\;d>2$, the space $\mathcal{M}_{d}$ is an affine
variety.

###### Proof.

Consider the affine space ${\mathbbm{C}}^{d}$ with
$\alpha_{1},\cdot\cdot\cdot,\alpha_{d-1},\beta$ as coordinates. Consider the
hyperplane $V$ which is the zero set of $\sum_{i=1}^{d-1}\alpha_{i}$ in
${\mathbbm{C}}^{d}$. We have an affine isomorphism,

${\mathbbm{C}}^{d-1}\rightarrow
V,\;\;\;(\alpha_{1},\cdot\cdot\cdot,\alpha_{d-2},\beta)\mapsto(\alpha_{1},\cdot\cdot\cdot,\alpha_{d-2},-\sum_{i=1}^{d-2}\alpha_{i},\beta).$
(2.3)

As we have seen above, we have a surjective map,

$V\twoheadrightarrow\mathcal{M}_{d},\;\;\;\;(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\alpha_{d-1},\beta)\mapsto
f_{\bar{\alpha},\beta}.$ (2.4)

Now, two points
$(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\alpha_{d-1},\beta)$ and
$(\alpha^{\prime}_{1},\alpha^{\prime}_{2},\cdot\cdot\cdot,\alpha^{\prime}_{d-2},\alpha^{\prime}_{d-1},\beta^{\prime})$
in $V$ maps to the same point in $\mathcal{M}_{d}$ iff there is an affine
conjugation map $h$ such that $h\circ f_{\bar{\alpha},\beta}\circ
h^{-1}=f_{\bar{\alpha^{\prime}},\beta^{\prime}}$ and
$h(\alpha_{1})=\alpha^{\prime}_{1}$.

If $h$ is the identity map, then
$f_{\bar{\alpha},\beta}=f_{\bar{\alpha^{\prime}},\beta^{\prime}},\alpha_{1}=\alpha^{\prime}_{1}$
and $\beta=\beta^{\prime}$. Hence, the ordered set
$\\{\alpha^{\prime}_{2},\cdot\cdot\cdot,\alpha^{\prime}_{d-2},\alpha^{\prime}_{d-1}\\}$
is a permutation of
$\\{\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\alpha_{d-1}\\}$, and every such
permutation will map to the same point in $\mathcal{M}_{d}$.

Let $h$ be a non-identity map. As
$f_{\bar{\alpha},\beta},f_{\bar{\alpha^{\prime}},\beta^{\prime}}$ are both
reduced, monic and of degree $d$, the affine conjugation map between them has
to be $h(z)=\zeta\cdot z$, for some $(d-1)$th root of unity $\zeta$. A brief
computation shows that $h^{-1}\circ f_{\bar{\alpha},\beta}\circ
h=f_{\zeta^{-1}\cdot\bar{\alpha},\zeta^{-1}\cdot\beta}$.

Consider the following two finite subgroups of $GL_{d}({\mathbbm{C}})$,

$\mu_{d-1}:=\\{\zeta I\;|\;I\text{ is the identity matrix},\zeta\text{ is a
}(d-1)\text{th root of unity}\\}.$ $P:=\text{Group of all }d\times d\text{
permutation matrices which fix the first and last coordinate.}$

The hyperplane $V$ is closed under the action of both $P$ and $\mu_{d-1}$.
From Equation (2.4) and last four paragraphs, we get that there is a bijective
map,

$V/\langle P,\mu_{d-1}\rangle\;\;\;\longleftrightarrow\;\;\;\mathcal{M}_{d}.$
(2.5)

As $\mu_{d-1}$ and $P$ commute with each other, the group $\langle
P,\mu_{d-1}\rangle$ is a finite group. So, the quotient space $V/\langle
P,\mu_{d-1}\rangle$ is an affine variety. Putting Equations (2.3), (2.5)
together,

${\mathbbm{C}}^{d-1}\;\;\xrightarrow{\simeq}\;\;V\;\;\twoheadrightarrow\;\;V/\langle
P,\mu_{d-1}\rangle\;\;\longleftrightarrow\;\;\mathcal{M}_{d},$ (2.6)

we see that $\mathcal{M}_{d}$ is an affine variety. ∎

###  2.5. The reduced polynomial condition: $\sum_{i=1}^{d-1}\alpha_{i}=0$

The polynomial $f$ in Equation (2.2) lies in
${\mathbbm{Q}}[z,\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1},\beta]$.
In the later sections, we would want to apply the relation
$\sum_{i=1}^{d-1}\alpha_{i}=0$ on the polynomial $f$ and the multivariate
polynomials we form using $f$. Here we will study a ring homomorphism between
multivariate polynomial rings whose kernel is the ideal generated by
$\sum_{i=1}^{d-1}\alpha_{i}$.

Fix an integer $d>2$. Let $R$ be a subring of ${\mathbbm{C}}$ containing $1$.
Consider the ring homomorphism,

$\Psi_{d,R}:R[z,\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1},\beta]\longrightarrow
R[z,\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\beta],$ (2.7)

which is identity map on
$R[z,\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\beta]$ and maps
$\alpha_{d-1}$ to $-(\sum_{i=1}^{d-2}\alpha_{i})$. Kernel of this map is the
ideal generated by $\sum_{i=1}^{d-1}\alpha_{i}$ in
$R[z,\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1},\beta]$. For any $f\in
R[z,\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1},\beta]$, we denote the
element $\Psi_{d,R}(f)$ as $\hat{f}$.

For two polynomials $f,g\in
R[z,\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1},\beta]$, we define
$\circ$ to be the composition with respect to $z$, i.e. $g\circ
h:=g(h(z,\alpha_{1},\cdot\cdot\cdot,\alpha_{d-1},\beta),\alpha_{1},\cdot\cdot\cdot,\alpha_{d-1},\beta)$
and
$\hat{g}\circ\hat{h}:=\hat{g}(\hat{h}(z,\alpha_{1},\cdot\cdot\cdot,\alpha_{d-2},\beta),\alpha_{1},\cdot\cdot\cdot,\alpha_{d-2},\beta)$.
Note that, $\Psi_{d,R}$ being a ring homomorphism,

$\widehat{g\circ h}=\hat{g}\circ\hat{h}.$ (2.8)

###  2.6. The polynomials $h_{k,n,d}$

In this subsection, we will form a polynomial $h_{k,n,d}$ in
${\mathbbm{Q}}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\beta]$.
Then we will show that to prove irreducibility of $\Sigma_{k,n,d}$ in
$\mathcal{M}_{d}$, it is enough to show that $h_{k,n,d}$ is irreducible over
${\mathbbm{C}}$.

From Equation (2.2), let

$f(z)=z^{d}+\sum_{i=2}^{d-1}(-1)^{i}\frac{d}{d-i}\cdot
s_{i}(\bar{\alpha})\cdot
z^{d-i}-\alpha_{1}^{d}-\sum_{i=2}^{d-1}(-1)^{i}\frac{d}{d-i}\cdot
s_{i}(\bar{\alpha})\cdot\alpha_{1}^{d-i}+\beta,$

with $\sum_{i=1}^{d-1}\alpha_{i}=0$. Let us assume that $\alpha_{1}$ is
strictly $(k,n)$-preperiodic under $f$. Then, we get
$f^{k+n}(\alpha_{1})-f^{k}(\alpha_{1})=0$. From the definition of $f$, the
polynomial

$f_{k,n,d}:=f^{k+n}(\alpha_{1})-f^{k}(\alpha_{1}),$

lies in
${\mathbbm{Q}}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1},\beta]$.
Consider $\widehat{f_{k,n,d}}$ and
$\hat{f}_{k,n,d}:=(\hat{f})^{k+n}(\alpha_{1})-(\hat{f})^{k}(\alpha_{1})$,
where $\;\hat{}\;$ is as defined in the previous subsection. From Equation
(2.8), we get

$\widehat{f_{k,n,d}}=\hat{f}_{k,n,d}.$ (2.9)

The set of solutions of the polynomial $\hat{f}_{k,n,d}$ in
${\mathbbm{C}}^{d-1}$ contains all the points in ${\mathbbm{C}}^{d-1}$ for
which $\alpha_{1}$ is (not necessarily strictly) $(k,n)$-preperiodic under
$\hat{f}$. Consider $\hat{f}_{l,m,d}$, defined similarly as $\hat{f}_{k,n,d}$,
for any non-negative integers $l,m$ such that $l\leq
k,m|n,(l,m)\neq(k,n)\in{\mathbbm{Z}}^{2}$. We will show in Lemma 4.1 that for
any such $(l,m)$, the polynomial $\hat{f}_{l,m,d}$ divides $\hat{f}_{k,n,d}$
in ${\mathbbm{Q}}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\beta]$.

Let,

$h_{k,n,d}:=\frac{\hat{f}_{k,n,d}}{\prod_{i}g_{i}^{a_{i}}}=\frac{\widehat{f_{k,n,d}}}{\prod_{i}g_{i}^{a_{i}}},$
(2.10)

where $g_{i}$ varies over all irreducible factors of
$\hat{f}_{l,m,d}=\widehat{f_{l,m,d}}$ in
${\mathbbm{Q}}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\beta]$, for
all $0\leq l\leq k,1\leq m\leq n,m|n,(l,m)\neq(k,n)\in{\mathbbm{Z}}^{2}$, and
$a_{i}$ is the highest power of $g_{i}$ that divides $\widehat{f_{k,n,d}}$.

######  Lemma 2.4.

Let $k\in{\mathbbm{N}}\cup\\{0\\},n,d\in{\mathbbm{N}},d>2$. Let us assume that
$h_{k,n,d}$ is irreducible over ${\mathbbm{C}}$. Then, the set of points of
the form $(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\beta)$ in
${\mathbbm{C}}^{d-1}$ for which $\alpha_{1}$ is strictly $(k,n)$-preperiodic
under $\hat{f}$, is open and Zariski dense in the algebraic set of
$h_{k,n,d}$.

###### Proof.

Let us denote the algebraic set of $h_{k,n,d}$ in ${\mathbbm{C}}^{d-1}$ as
$V(h_{k,n,d})$. Any point
$(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\beta)\in{\mathbbm{C}}^{d-1}$
for which $\alpha_{1}$ is strictly $(k,n)$-preperiodic under $\hat{f}$, is a
solution to the equation $\hat{f}_{k,n,d}$ but not a solution of any of the
$g_{i}$’s in Equation (2.10). Hence, all such points lie in $V(h_{k,n,d})$.
Set of all these points are precisely the set
$U:=V(h_{k,n,d})\setminus\cup_{g_{i}}V(h_{k,n,d},g_{i})$, where $g_{i}$ varies
over all irreducible factors of $\hat{f}_{l,m,d},$ for all $0\leq l\leq
k,1\leq m\leq n,m|n,(l,m)\neq(k,n)\in{\mathbbm{Z}}^{2}$. By definition of
$h_{k,n,d}$, the polynomial $h_{k,n,d}$ is coprime to $g_{i}$ over
${\mathbbm{Q}}$, for every $g_{i}$. Hence, they are coprime over
${\mathbbm{C}}$ too. So, $U$ is nonempty and open. As $h_{k,n,d}$ is
irreducible, $U$ is Zariski dense in $V(h_{k,n,d})$. ∎

Let us define $\Sigma_{k,n,d}$ to be the set of all points in
$\mathcal{M}_{d}$ for which the marked critical point is strictly
$(k,n)$-preperiodic. From Lemma 2.4, we get the following corollary,

######  Corollary 2.5.

If $h_{k,n,d}$ is irreducible over ${\mathbbm{C}}$, then $\Sigma_{k,n,d}$ is
an irreducible quasi-affine subvariety of $\mathcal{M}_{d}$. ∎

######  Remark 2.6.

Our first goal in this article is to prove the irreducibility of
$\Sigma_{k,1,p},k\geq 0,p$ odd prime. Let us summarize our method. Consider
the diagram below. Let
${\mathbbm{C}}[d]:={\mathbbm{C}}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d},\beta],$
for any $d\in{\mathbbm{N}}$. The polynomial $f_{k,n,d}\in{\mathbbm{C}}[d-1]$
kills all $(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1},\beta)$ for
which $\alpha_{1}$ is (not necessarily strictly) $(k,n)$-preperiodic under $f$
as in Equation (2.2). We take the image of $f_{k,n,d}$ under $\;\hat{}\;$ map.
From $\hat{f}_{k,n,d}\in{\mathbbm{C}}[d-2]$, we factor out all common
irreducible factors of $\hat{f}_{k,n,d}$ and $\hat{f}_{l,m,d}$, $0\leq l\leq
k,1\leq m\leq n,m|n,(l,m)\neq(k,n)\in{\mathbbm{Z}}^{2}$, with each factor
raised to the highest power that divides $\hat{f}_{k,n,d}$. We call the
resulting polynomial $h_{k,n,d}\in{\mathbbm{C}}[d-2]$. Assuming $h_{k,n,d}$ is
irreducible, we get that $V(h_{k,n,d})\subseteq{\mathbbm{C}}^{d-1}$ is
irreducible. The algebraic set $V(h_{k,n,d})$ is irreducible implies the set
$S$ of all points of ${\mathbbm{C}}^{d-1}$ for which $\alpha_{1}$ is strictly
$(k,n)$-preperiodic, is non-empty and open in $V(h_{k,n,d})$. Hence, $S$ is
irreducible. So the image of $S$ under the horizontal sequence of arrows in
the diagram is irreducible, which is $\Sigma_{k,n,d}$.

${{\mathbbm{C}}[d-1]}$${{\mathbbm{C}}[d-2]}$${\nicefrac{{{\mathbbm{C}}[d-1]}}{{(\sum_{i=1}^{d-1}\alpha_{i})}}}$${{\mathbbm{C}}^{d-1}}$${V=V(\sum_{i=1}^{d-1}\alpha_{i})}$${V/\langle
P,\mu_{d-1}\rangle}$${\mathcal{M}_{d}}$$\scriptstyle{\widehat{}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

In Section 5, we will show that $h_{k,1,p}$ polynomials are irreducible over
${\mathbbm{C}}$, for any $k\geq 0$ and odd prime $p$. By Corollary 2.5, this
shows that $\Sigma_{k,1,p},k\geq 0,p$ odd prime, is irreducible, completing
the proof of Theorem 1.3 stated in the introduction.

##  3\. Notations

We will use the following notations for the rest of the article. Let $R$ be a
ring and $d\in{\mathbbm{N}}$.

  * •

$R[d]:=R[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d},\beta],$ the
multivariate polynomial ring over $R$ in variables
$\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d},\beta$.

Let $d>2$. Let $g,h$ be elements of $R[d-1].$

  * •

By $\hat{g}$, we denote the image of $g$ under the map
$\Psi_{d}:R[d-1]\rightarrow R[d-2]$, that is the identity map on
$R[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2},\beta]$ and maps
$\alpha_{d-1}$ to $-(\sum_{i=1}^{d-2}\alpha_{i})$.

  * •

By saying $g$ is monic in
$R[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1}][\beta]$, we mean $g$ is
monic as a polynomial in $\beta$ over the ring
$R[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1}]$.

  * •

By $\mbox{Res}(g,h)$, we denote the resultant of $g$ and $h$, both considered
as polynomials in $\beta$ with coefficients coming from the integral domain
$R[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1}]$. So,
$\mbox{Res}(g,h)\in R[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1}]$.

Consider the polynomial $f$ as defined in Equation (2.2). For any non-negative
integers $k,n$, with $n>0$,

  * •

$f^{0}:=$ identity map, $f^{1}:=f,f^{n}:=f^{n-1}\circ f,$ for all
$n\in{\mathbbm{N}}$.

  * •

$f^{\prime}$ denote the derivative of $f$ w.r.t $z$.

  * •

$f_{k,n,d}=f_{k,n,d}(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1},\beta):=f^{k+n}(\alpha_{1})-f^{k}(\alpha_{1})$.

  * •

$\hat{f}_{k,n,d}=\widehat{f_{k,n,d}}:=\Psi_{d,{\mathbbm{C}}}(f_{k,n,d}),$
where $\Psi_{d,{\mathbbm{C}}}$ is as defined in Equation (2.7).

  * •

$h_{k,n,d}=h_{k,n,d}(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-1},\beta):=\hat{f}_{k,n,d}/\prod_{i}g_{i}^{a_{i}}$,
where $g_{i}$’s vary over all distinct irreducible factors of
$\hat{f}_{l,m,d}$, where $l\leq k,m|n,(l,m)\neq(k,n)\in{\mathbbm{Z}}^{2}$, and
for each $i$, $a_{i}$ is the highest power of $g_{i}$ that divides
$\hat{f}_{k,n,d}$ over ${\mathbbm{Q}}$.

  * •

${\mathbbm{C}}^{n}:=$ complex affine space of dimension $n$.

  * •

$\mathcal{M}_{d}:=$ the moduli space of degree $d$ polynomials with a marked
critical point.

  * •

$\Sigma_{k,n,d}:=$ the set of all points of $\mathcal{M}_{d}$ for which the
marked critical point is strictly $(k,n)$-preperiodic.

  * •

$\Phi_{n}(z,w):=\frac{z^{n}-w^{n}}{z-w}$, for all $n\in{\mathbbm{N}}$.

  * •

$G_{{\mathbbm{Q}}}$ denote the absolute Galois group of ${\mathbbm{Q}}$.

##  4\. Basic lemmas and Tools

###  4.1. Divisibility properties of $\hat{f}_{k,n,d}$

Let $d$ be a natural number, $d>2$. In this subsection, we will study some
divisibility properties of $\hat{f}_{k,n,d}$, for non-negative integers $k,n$,
with $n>0$.

######  Lemma 4.1.

Let $k,l\in{\mathbbm{N}}\cup\\{0\\},n,m\in{\mathbbm{N}}$ such that $l\leq k$
and $m|n$, the polynomial $f_{l,m,d}$ divides $f_{k,n,d}$ in
${\mathbbm{Q}}[d-1]$. Hence, $\hat{f}_{l,m,d}$ divides $\hat{f}_{k,n,d}$ in
${\mathbbm{Q}}[d-2]$.

###### Proof.

To prove the lemma, it is enough to show that $f_{k-1,n,d}$ divides
$f_{k,n,d}$ and $f_{k,m,d}$ divides $f_{k,n,d}$ in ${\mathbbm{Q}}[d-1]$, for
any $k,n,m\in{\mathbbm{N}},$ such that $m|n$.

Consider the polynomial $f$ in Equation (2.2) as a polynomial in a single
variable, $z$, over the integral domain ${\mathbbm{Q}}[d-1]$. Observe that,
$(z-w)$ divides $f(z)-f(w)$. Putting $z=f^{k+n-1}(\alpha_{1})$,
$w=f^{k-1}(\alpha_{1})$, one gets that $f_{k-1,n,d}$ divides $f_{k,n,d}$ in
${\mathbbm{Q}}[d-1]$.

Hence, for any non-negative integers $l,k,n,$ such that $l\leq k,n>0$, the
polynomial $f_{l,n,d}$ divides $f_{k,n,d}$ in ${\mathbbm{Q}}[d-1]$. Let
$n=mt,t\in{\mathbbm{N}}$. Then,

$\displaystyle f_{k,n,d}$ $\displaystyle=f_{k,mt,d}$
$\displaystyle=f_{k+m(t-1),m,d}+f_{k+m(t-2),m,d}+\cdot\cdot\cdot+f_{k,m,d}$
$\displaystyle=\sum_{i=0}^{t-1}f_{k+mi,m,d}.$

As $f_{k,m,d}$ divides $f_{k+mi,m,d},$ for $i\geq 0$, the polynomial
$f_{k,m,d}$ divides $f_{k,n,d}$, whenever $m$ divides $n$. The lemma is
proved. ∎

######  Lemma 4.2.

Let $g$ be an irreducible element of ${\mathbbm{Q}}[d-2]$, monic as a
polynomial in
${\mathbbm{Q}}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{d-2}][\beta]$.
Let $k,l,m,n$ be non-negative integers with $m,n$ non-zero, $l\leq k,$ g.c.d
$(m,n)=r$. If $g$ divides both $\hat{f}_{k,n,d}$ and $\hat{f}_{l,m,d}$ in
${\mathbbm{Q}}[d-2]$, then $g$ divides $\widehat{f_{l,r,d}}$ in
${\mathbbm{Q}}[d-2]$.

###### Proof.

Let us consider the algebraic set $V(g)$ in ${\mathbbm{C}}^{d-1}$. Let
$(\alpha_{1}^{0},\alpha_{2}^{0},\cdot\cdot\cdot,\alpha_{d-2}^{0},\beta^{0})\in
V(g)$. As $g$ divides both $\hat{f}_{k,n,d}$ and $\hat{f}_{l,m,d}$,
$\alpha_{1}^{0}$ is both $(k,n)$-preperiodic and $(l,m)$-preperiodic, for any
polynomial corresponding to the point
$(\alpha_{1}^{0},\alpha_{2}^{0},\cdot\cdot\cdot,\alpha_{d-1}^{0},\beta^{0})$.
So, $\alpha_{1}^{0}$ is $(l,r)$-preperiodic. Hence, $V(g)\subseteq
V(\hat{f}_{l,r,d})$. So in ${\mathbbm{C}}[d-2]$,

$rad(g)=I(V(g))\supseteq
I(V(\hat{f}_{l,r,d}))=rad(\hat{f}_{l,r,d})\supseteq(\hat{f}_{l,r,d})$.

Now, $g$ being irreducible in ${\mathbbm{Q}}[d-2]$, $g$ is seperable over
${\mathbbm{C}}$. So, we have $(g)=rad(g)\supseteq(\hat{f}_{l,r,d})$. So, $g$
divides $\hat{f}_{l,r,d}$ in ${\mathbbm{C}}[d-2]$. Hence $g$ divides
$\hat{f}_{l,r,d}$ in ${\mathbbm{Q}}[d-2]$, as both $g$ and $\hat{f}_{l,r,d}$
are in ${\mathbbm{Q}}[d-2]$. ∎

From Lemmas 4.1 and 4.2, one directly obtains the following corollary,

######  Corollary 4.3.

Let $k,l,m,n$ be non-negative integers with $m,n$ non-zero, $l\leq k,$ g.c.d
$(m,n)=r$. Then, $\hat{f}_{l,r,d}$ divides g.c.d
$(\hat{f}_{k,n,d},\hat{f}_{l,m,d})$ in ${\mathbbm{Q}}[d-2]$. Moreover, the
radical ideals of the ideal generated by $\hat{f}_{l,r,d}$ and the ideal
generated by g.c.d $(\hat{f}_{k,n,d},\hat{f}_{l,m,d})$ in ${\mathbbm{C}}[d-2]$
are same. ∎

From here on, we will consider $d$ to be an odd prime power. Let $p$ be an odd
prime and $e\in{\mathbbm{N}}$, both chosen arbitrarily but fixed. Consider the
polynomial expression in Equation $(\ref{b})$ with $d=p^{e}$,

$f(z)=z^{p^{e}}+\sum_{i=2}^{p^{e}-1}(-1)^{i}\frac{p^{e}}{p^{e}-i}\cdot
s_{i}(\bar{\alpha})\cdot
z^{p^{e}-i}-\alpha_{1}^{p^{e}}-\sum_{i=2}^{p^{e}-1}(-1)^{i}\frac{p^{e}}{p^{e}-i}\cdot
s_{i}(\bar{\alpha})\cdot\alpha_{1}^{p^{e}-i}+\beta.$ (4.1)

Considering $f$ as a polynomial in variables
$z,\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-1},\beta$, the
coeffiecients of $f$ are not necessarily integers. However the coefficients of
$f$ belongs to ${\mathbbm{Z}}_{(p)}$, the localisation of ${\mathbbm{Z}}$ at
the prime $p$. The polynomial $f$ is monic as a polynomial in
${\mathbbm{Z}}_{(p)}[z,\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-1}][\beta]$.
Similarly, the polynomials $f_{k,n,p^{e}},h_{k,n,p^{e}}$ lie in the polynomial
ring
${\mathbbm{Z}}_{(p)}[p^{e}-1]:={\mathbbm{Z}}_{(p)}[\alpha_{1},\cdot\cdot\cdot,\alpha_{p^{e}-1},\beta]$,
${\mathbbm{Z}}_{(p)}[p^{e}-2]$, respectively and both are monic as polynomials
in $\beta$.

Note that,

$\displaystyle f(z)-f(w)$
$\displaystyle=(z^{p^{e}}-w^{p^{e}})+\sum_{i=2}^{p^{e}-1}(-1)^{i}\frac{p^{e}}{p^{e}-i}\cdot
s_{i}(\bar{\alpha})\cdot(z^{p^{e}-i}-w^{p^{e}-i})$
$\displaystyle=(z-w)\left(\Phi_{p^{e}}(z,w)+\sum_{i=2}^{p^{e}-1}(-1)^{i}\frac{p^{e}}{p^{e}-i}\cdot
s_{i}(\bar{\alpha})\cdot\Phi_{p^{e}-i}(z,w)\right),$ (4.2)

where $\Phi_{d}(z,w)=(z^{d}-w^{d})/(z-w),d\in{\mathbbm{N}}$.

###  4.2. A weak version of Thurston’s rigidity theorem for
$\mathcal{M}_{p^{e}}$

Thurston’s rigidity theorem studies transversality of the intersection of
equations for two different critical points to be preperiodic, in moduli
spaces of rational maps (see [DH93]). In this article, we need a much weaker
version of Thurston’s rigidity for moduli spaces of polynomials with a marked
critical point. While the proof of the general Thurston’s rigidity theorem
requires deep tools, the version we need here (Theorem 4.4) can be proved
using elementary algebraic methods. Let $f$ be the polynomial as defined in
Equation (4.1).

######  Theorem 4.4.

Fix $i\in\\{2,\cdot\cdot\cdot,p^{e}-1\\}$. Fix
$k_{1},k_{i}\in{\mathbbm{N}}\cup\\{0\\}$, and $n_{1},n_{i}\in{\mathbbm{N}}$.
Then, the polynomials

$f^{k_{1}+n_{1}}(\alpha_{1})-f^{k_{1}}(\alpha_{1})\text{ and
}f^{k_{i}+n_{i}}(\alpha_{i})-f^{k_{i}}(\alpha_{i})$

are coprime in
${\mathbbm{C}}[p^{e}-1]:={\mathbbm{C}}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-1},\beta]$.
Moreover, the polynomials

$\hat{f}^{k_{1}+n_{1}}(\hat{\alpha_{1}})-\hat{f}^{k_{1}}(\hat{\alpha_{1}})\text{
and
}\hat{f}^{k_{i}+n_{i}}(\hat{\alpha_{i}})-\hat{f}^{k_{i}}(\hat{\alpha_{i}})$

are coprime in ${\mathbbm{C}}[p^{e}-2]$.

###### Proof.

As all the polynomials in the statement of the theorem are defined over
${\mathbbm{Q}}$, it is enough to show that they are coprime over
${\mathbbm{Q}}$.

More precisely, all the polynomials mentioned above are defined over
${\mathbbm{Z}}_{(p)}$. Let $l\in{\mathbbm{N}}$. Consider $f^{l}(\alpha_{j})$,
where $j\in\\{1,2,\cdot\cdot\cdot,p^{e}-1\\}$. The term in $f^{l}(\alpha_{j})$
containing the highest power of $\beta$ is $\beta^{p^{(l-1)e}}$. As
$n_{1},n_{i}\in{\mathbbm{N}}$, we have $k_{1}+n_{1}>k_{1}\geq 0$ and
$k_{i}+n_{i}>k_{i}\geq 0$. Hence,
$f^{k_{1}+n_{1}}(\alpha_{1})-f^{k_{1}}(\alpha_{1})$ and
$f^{k_{i}+n_{i}}(\alpha_{i})-f^{k_{i}}(\alpha_{i})$ are both monic polynomials
in $\beta$ over the integral domain
${\mathbbm{Z}}_{(p)}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-1}]$.
Hence, to show that $f^{k_{1}+n_{1}}(\alpha_{1})-f^{k_{1}}(\alpha_{1})$ and
$f^{k_{i}+n_{i}}(\alpha_{i})-f^{k_{i}}(\alpha_{i})$ are coprime over
${\mathbbm{Q}}$, it is enough to show that they are coprime over
${\mathbbm{Z}}_{(p)}$.

Observe that $\;\hat{}\;$ being a ring homomorphism,

$\displaystyle\hat{f}^{k_{1}+n_{1}}(\hat{\alpha_{1}})-\hat{f}^{k_{1}}(\hat{\alpha_{1}})$
$\displaystyle=\widehat{f^{k_{1}+n_{1}}(\alpha_{1})-f^{k_{1}}(\alpha_{1})}$
$\displaystyle\mbox{and}\quad\hat{f}^{k_{i}+n_{i}}(\hat{\alpha_{i}})-\hat{f}^{k_{i}}(\hat{\alpha_{i}})$
$\displaystyle=\widehat{f^{k_{i}+n_{i}}(\alpha_{i})-f^{k_{i}}(\alpha_{i})}.$

As $\hat{\beta}=\beta$, we get that
$\hat{f}^{k_{1}+n_{1}}(\hat{\alpha_{1}})-\hat{f}^{k_{1}}(\hat{\alpha_{1}})$
and
$\hat{f}^{k_{i}+n_{i}}(\hat{\alpha_{i}})-\hat{f}^{k_{i}}(\hat{\alpha_{i}})$
are both monic polynomials in $\beta$ over
${\mathbbm{Z}}_{(p)}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-2}]$.
Similarly as above, to show that
$\hat{f}^{k_{1}+n_{1}}(\hat{\alpha_{1}})-\hat{f}^{k_{1}}(\hat{\alpha_{1}})$
and
$\hat{f}^{k_{i}+n_{i}}(\hat{\alpha_{i}})-\hat{f}^{k_{i}}(\hat{\alpha_{i}})$
are coprime over $Q$, it is enough to show that they are coprime over
${\mathbbm{Z}}_{(p)}$.

Now, we will show that $f^{k_{1}+n_{1}}(\alpha_{1})-f^{k_{1}}(\alpha_{1})$ and
$f^{k_{i}+n_{i}}(\alpha_{i})-f^{k_{i}}(\alpha_{i})$ are coprime over
${\mathbbm{Z}}_{(p)}$. As they are both monic polynomials in $\beta$ over
${\mathbbm{Z}}_{(p)}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-1}]$,
their g.c.d. must be monic polynomial in $\beta$ over
${\mathbbm{Z}}_{(p)}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-1}]$
too, upto associates. Hence,
$f^{k_{1}+n_{1}}(\alpha_{1})-f^{k_{1}}(\alpha_{1})$ and
$f^{k_{i}+n_{i}}(\alpha_{i})-f^{k_{i}}(\alpha_{i})$ are coprime over
${\mathbbm{Z}}_{(p)}$ if they are coprime modulo $p{\mathbbm{Z}}_{(p)}$.
Observe that,

$f(z)\equiv z^{p^{e}}-\alpha_{1}^{p^{e}}+\beta\;(\text{mod }p).$

So, $f^{l}(\alpha_{1})=\alpha_{1}\;(\text{or, }\beta)$ if $l=0\;(\text{or,
}1)$, and for $l>1$,

$\displaystyle f^{l}(\alpha_{1})$
$\displaystyle\equiv(\beta^{p^{(l-1)e}}-\alpha_{1}^{p^{(l-1)e}})+\cdot\cdot\cdot+(\beta^{p^{e}}-\alpha_{1}^{p^{e}})+\beta$
$\displaystyle\equiv\beta+\sum_{j=1}^{l-1}(\beta-\alpha_{1})^{p^{je}}\;(\text{mod
}p).$

Hence,

$f^{k_{1}+n_{1}}(\alpha_{1})-f^{k_{1}}(\alpha_{1})\equiv\sum_{j=k_{1}}^{k_{1}+n_{1}-1}(\beta-\alpha_{1})^{p^{je}}\;(\text{mod
}p).$

Similarly, $f^{0}(\alpha_{i})=\alpha_{i}$, and for $l\in{\mathbbm{N}}$

$f^{l}(\alpha_{i})\equiv(\alpha_{i}-\alpha_{1})^{p^{le}}+(\beta-\alpha_{1})^{p^{(l-1)e}}+\cdot\cdot\cdot+(\beta-\alpha_{1})^{p^{e}}+\beta\;(\text{mod
}p).$

So modulo $p$,

$f^{k_{i}+n_{i}}(\alpha_{i})-f^{k_{i}}(\alpha_{i})=\left((\alpha_{i}-\alpha_{1})^{p^{(k_{i}+n_{i})e}}-(\alpha_{i}-\alpha_{1})^{p^{k_{i}e}}\right)+\left(\sum_{j=k_{i}}^{k_{i}+n_{i}-1}(\beta-\alpha_{1})^{p^{je}}\right).$

Modulo $p$, the highest degree homogeneous part of
$f^{k_{1}+n_{1}}(\alpha_{1})-f^{k_{1}}(\alpha_{1})$ is

$(\beta-\alpha_{1})^{p^{(k_{1}+n_{1}-1)e}}$

and the highest degree homogeneous part of
$f^{k_{i}+n_{i}}(\alpha_{i})-f^{k_{i}}(\alpha_{i})$ is

$(\alpha_{i}-\alpha_{1})^{p^{(k_{i}+n_{i})e}}.$

The highest degree homogeneous part of g.c.d. of two polynomials must divide
the highest degree homogeneous part of each of them. As $(\beta-\alpha_{1})$
and $(\alpha_{i}-\alpha_{1})$ are two distinct irreducible polynomials over
${\mathbbm{F}}_{p}$, the polynomials
$f^{k_{1}+n_{1}}(\alpha_{1})-f^{k_{1}}(\alpha_{1})$ and
$f^{k_{i}+n_{i}}(\alpha_{i})-f^{k_{i}}(\alpha_{i})$ are coprime modulo
$p{\mathbbm{Z}}_{(p)}$. Hence, They are coprime over ${\mathbbm{Z}}_{(p)}$
too.

In the last paragraph, replacing
$f^{k_{1}+n_{1}}(\alpha_{1})-f^{k_{1}}(\alpha_{1})$ and
$f^{k_{i}+n_{i}}(\alpha_{i})-f^{k_{i}}(\alpha_{i})$ with
$\hat{f}^{k_{1}+n_{1}}(\hat{\alpha_{1}})-\hat{f}^{k_{1}}(\hat{\alpha_{1}})$
and
$\hat{f}^{k_{i}+n_{i}}(\hat{\alpha_{i}})-\hat{f}^{k_{i}}(\hat{\alpha_{i}})$,
respectively, every argument follows verbatim and one obtains the second part
of the theorem. ∎

######  Remark 4.5.

We will require the above theorem in later sections. Whenever we mention
“Thurston’s rigidity” from here on, we will mean the above theorem (Theorem
4.4).

###  4.3. Generalised Eisenstein Irreducibility criterion

######  Theorem 4.6.

Let $p,e\in{\mathbbm{N}},p$ odd prime. Let ${\mathbbm{Z}}_{(p)}$ be the
localisation of ${\mathbbm{Z}}$ at the prime $p$. Let $g,h$ be non-constant
elements of ${\mathbbm{Z}}_{(p)}[p^{e}-2]$, both monic as elements of
${\mathbbm{Z}}_{(p)}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-2}][\beta]$.
Let $\mbox{Res}(g,h)$ denote the resultant of $g,h$ both considered as
polynomials in $\beta$ over the integral domain
${\mathbbm{Z}}_{(p)}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-2}]$.
Suppose the following conditions hold,
1) $g\equiv h^{n}\;(\text{mod }p)$, for some $n\in{\mathbbm{N}}$.
2) $h\;(\text{mod }p)$ is irreducible in ${\mathbbm{F}}_{p}[p^{e}-2]$.
3) $\mbox{Res}(g,h)\not\equiv 0\;(\text{mod }p^{2\cdot
deg(h)}{\mathbbm{Z}}_{(p)})$, where $deg(h)$ is the degree of $h$ as a
polynomial in $\beta$ over
${\mathbbm{Z}}_{(p)}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-2}]$.

Then, $g$ is irreducible in ${\mathbbm{Q}}[p^{e}-2]$.

###### Proof.

We will prove by contradiction. Let us assume that $g$ is reducible in
${\mathbbm{Q}}[p^{e}-2]$. As $g$ is monic in $\beta$, by Gauss lemma, let
$g=g_{1}\cdot g_{2},$ such that $g_{1},g_{2}\in{\mathbbm{Z}}_{(p)}[p^{e}-2]$
both monic as elements of
${\mathbbm{Z}}_{(p)}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-2}][\beta]$
and none of them is a unit. By conditions $1$ and $2$ of the theorem,
$g_{1}\equiv h^{n_{1}}\;(\textnormal{mod }p)$, and $g_{2}\equiv
h^{n_{2}}\;(\textnormal{mod }p)$, such that
$n_{1},n_{2}\in{\mathbbm{N}},n_{1}+n_{2}=n$. So, there exist polynomials
$x_{1},x_{2}\in{\mathbbm{Z}}_{(p)}[p^{e}-2]$, such that
$g_{1}=h^{n_{1}}+p\cdot x_{1}$ and $g_{2}=h^{n_{2}}+p\cdot x_{2}$. Then,

$\displaystyle\mbox{Res}(g,h)$
$\displaystyle=\mbox{Res}(g_{1},h)\cdot\mbox{Res}(g_{2},h)$
$\displaystyle=\mbox{Res}(p\cdot x_{1},h)\cdot\mbox{Res}(p\cdot x_{2},h)$
$\displaystyle=p^{2\cdot
deg(h)}\cdot\mbox{Res}(x_{1},h)\cdot\mbox{Res}(x_{2},h).$

But by the third assumption of the theorem, $p^{2\cdot deg(h)}$ does not
divide the $\mbox{Res}(g,h)$. Hence, we arrive at a contradiction. ∎

##  5\. Irreducibility of $\Sigma_{k,1,p}$

Let the degree of the polynomial $f$ in Equation (2.2) be an odd prime $p$.
Then,

$f(z)=z^{p}+\sum_{i=2}^{p-1}(-1)^{i}\frac{p\cdot
s_{i}(\bar{\alpha})}{p-i}z^{p-i}-\alpha_{1}^{p}-\sum_{i=2}^{p-1}(-1)^{i}\frac{p\cdot
s_{i}(\bar{\alpha})}{p-i}\alpha_{1}^{p-i}+\beta.$ (5.1)

We recall a few notations and a fact from previous sections.

  * •

For any ring $R$ and $d\in{\mathbbm{N}}$, we define
$R[d]:=R[\alpha_{1},\alpha_{2},..,\alpha_{d},\beta]$.

  * •

$f_{k,1,p}$ lies in ${\mathbbm{Z}}_{(p)}[p-1],\hskip
5.0pt\hat{f}_{k,1,p},h_{k,1,p}$ lie in ${\mathbbm{Z}}_{(p)}[p-2]$.

  * •

for $g,h\in{\mathbbm{Z}}_{(p)}[p-1]$, $g\equiv h\;(\textnormal{mod }p)$
implies that $\hat{g}\equiv\hat{h}\;(\textnormal{mod }p)$.

Let $k\in{\mathbbm{N}}$. We will show that $h_{k,1,p}$ is a generalised
Eisenstein polynomial. First, we study the polynomial $h_{0,1,p}$.

######  Lemma 5.1.

The polynomial $h_{0,1,p}$ is $\beta-\alpha_{1}$.

###### Proof.

By definition of $h_{0,1,p}$, we have
$h_{0,1,p}=\hat{f}_{0,1,p}=\hat{f}(\alpha_{1})-\alpha_{1}=\beta-\alpha_{1}$. ∎

Now we will show $h_{k,1,p}$ satisfies condition $1$ of generalised Eisenstein
irreducibility criterion (Theorem 4.6), with respect to the polynomial
$h_{0,1,p}$.

######  Lemma 5.2.

For any $k\in{\mathbbm{N}}$, $h_{k,1,p}\equiv
h_{0,1,p}^{N_{k,p}}\>(\textnormal{mod }p)$, for some
$N_{k,p}\in{\mathbbm{N}}$.

###### Proof.

Let $k\in{\mathbbm{N}}$. From Equation (5.1), we have $f(z)\equiv
z^{p}-\alpha_{1}^{p}+\beta$ $(\textnormal{mod }p)$. So,

$f^{k}(z)\equiv
z^{p^{k}}-\alpha_{1}^{p^{k}}+\beta^{p^{k-1}}-\alpha_{1}^{p^{k-1}}+\cdot\cdot\cdot+\beta^{p}-\alpha_{1}^{p}+\beta\;\;(\textnormal{mod
}p),$ $\displaystyle f_{k,1,p}$
$\displaystyle=f^{k+1}(\alpha_{1})-f^{k}(\alpha_{1})$
$\displaystyle=f^{k}(\beta)-f^{k-1}(\beta)$
$\displaystyle\equiv\beta^{p^{k}}-\alpha_{1}^{p^{k}}(\textnormal{mod }p)$
$\displaystyle\equiv(\beta-\alpha_{1})^{p^{k}}(\textnormal{mod }p).$

Hence,
$\hat{f}_{k,1,p}=\widehat{f_{k,1,p}}\equiv(\beta-\alpha_{1})^{p^{k}}\;(\textnormal{mod
}p)$. As $h_{k,1,p}$ divides $\hat{f}_{k,1,p}$ and
$h_{0,1,p}=\beta-\alpha_{1}$ is irreducible over ${\mathbbm{F}}_{p}$, we get
$h_{k,1,p}\equiv h_{0,1,p}^{N_{k,p}}\;(\textnormal{mod }p)$, for some
$N_{k,p}\in{\mathbbm{N}}$. This proves the lemma. ∎

We see from Lemma 5.1 that $h_{0,1,p}$ satisfies condition $2$ of generalised
Eisenstein irreducibility criterion (Theorem 4.6). For the resultant criterion
(condition $3$) of Theorem 4.6, we need to study the lowest degree homogeneous
part of $h_{k,1,p}$.

######  Lemma 5.3.

For any natural number $k\geq 2$, the polynomial $h_{k,1,p}$ is of the form

$h_{k,1,p}=\frac{\hat{f}_{k,1,p}}{\hat{f}_{k-1,1,p}\cdot(\beta-\alpha_{1})},$

and the lowest degree homogeneous part of $h_{k,1,p}$ is
$p(\beta+\sum_{i=1}^{p-2}\alpha_{i})\prod_{i=2}^{p-2}(\beta-\alpha_{i})$.

###### Proof.

To obtain $h_{k,1,p}$, we need to factor out all the irreducible factors of
$\hat{f}_{k-1,1,p}$ from $\hat{f}_{k,1,p}$, each raised to the highest power
that divides $\hat{f}_{k,1,p}$. First, let us consider the polynomial

$\frac{\hat{f}_{k,1,p}}{\hat{f}_{k-1,1,p}}=\frac{\widehat{f_{k,1,p}}}{\widehat{f_{k-1,1,p}}}.$

From Equation (4.1),

$\frac{f(z)-f(w)}{z-w}=\Phi_{p}(z,w)+\sum_{i=2}^{p-1}(-1)^{i}\frac{p}{p-i}\cdot
s_{i}(\bar{\alpha})\cdot\Phi_{p-i}(z,w).$ (5.2)

Putting $z=f^{k}(\alpha_{1}),w=f^{k-1}(\alpha_{1})$ in above, we get,

$\frac{\hat{f}_{k,1,p}}{\hat{f}_{k-1,1,p}}=\Phi_{p}\left(\hat{f}^{k}(\alpha_{1}),\hat{f}^{k-1}(\alpha_{1})\right)+\sum_{i=2}^{p-1}(-1)^{i}\frac{p}{p-i}\cdot\widehat{s_{i}(\bar{\alpha})}\cdot\Phi_{p-i}\left(\hat{f}^{k}(\alpha_{1}),\hat{f}^{k-1}(\alpha_{1})\right).$

Next, we need to figure out what are the irreducible polynomials in
${\mathbbm{Z}}_{(p)}[p-2]$ that divide both $\hat{f}_{k-1,1,p}$ and
$\hat{f}_{k,1,p}/\hat{f}_{k-1,1,p}$. Note that,

$\frac{f(z)-f(w)}{z-w}\equiv f^{\prime}(w)\;(\text{mod }z-w),$ (5.3)

where $f^{\prime}(w)$ is the derivative of $f(w)$ with respect to $w$.
Replacing $z$ with $f^{k}(\alpha_{1})$ and $w$ with $f^{k-1}(\alpha_{1})$ in
Equation (5.3), we get

$\frac{f_{k,1,p}}{f_{k-1,1,p}}\equiv
f^{\prime}\left(f^{k-1}(\alpha_{1})\right)\equiv
p\left(\prod_{i=1}^{p-1}\left(f^{k-1}(\alpha_{1})-\alpha_{i}\right)\right)\text{
(mod }f_{k-1,1,p}).$ $\text{Hence,
}\frac{\hat{f}_{k,1,p}}{\hat{f}_{k-1,1,p}}\equiv
p\left(\prod_{i=1}^{p-1}\left(\hat{f}^{k-1}(\alpha_{1})-\hat{\alpha_{i}}\right)\right)\text{
(mod }\hat{f}_{k-1,1,p}).$ (5.4)

Let $g$ be an irreducible polynomial in ${\mathbbm{Z}}_{(p)}[p-2]$ that
divides both $\hat{f}_{k-1,1,p}$ and $\hat{f}_{k,1,p}/\hat{f}_{k-1,1,p}$.
Then, $g$ divides
$p\prod_{i=1}^{p-1}(\hat{f}^{k-1}(\alpha_{1})-\hat{\alpha_{i}})$. By
thurston’s rigidity theorem (Theorem 4.4), $\hat{f}_{k-1,1,p}$ and
$\hat{f}^{k-1}(\alpha_{1})-\hat{\alpha_{i}}$ are coprime over
${\mathbbm{Z}}_{(p)}$, for any $i,2\leq i\leq p-1$. So, $g$ divides
$\hat{f}^{k-1}(\alpha_{1})-\hat{\alpha_{1}}$ (we can remove $p$, as all the
polynomials considered here are monic in $\beta$ over
${\mathbbm{Z}}_{(p)}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p-2}]$).
Note that, $\hat{f}^{k-1}(\alpha_{1})-\hat{\alpha_{1}}=\hat{f}_{0,k-1,p}$. As
$g$ divides both $\hat{f}_{k-1,1,p}$ and $\hat{f}_{0,k-1,p}$, from Lemma 4.2,
$g$ divides $\hat{f}_{0,1,p}=\beta-\alpha_{1}$. Assuming
$t_{k}\in{\mathbbm{Z}}_{\geq 0}$ is the highest power of $\beta-\alpha_{1}$
that divides $\hat{f}_{k,1,p}/\hat{f}_{k-1,1,p}$, we get

$h_{k,1,p}=\frac{\hat{f}_{k,1,p}}{\hat{f}_{k-1,1,p}(\beta-\alpha_{1})^{t_{k}}},\>\>\>\text{
for any }k\in{\mathbbm{N}}.$

We will now show that $\beta-\alpha_{1}$ divides $f_{k,1,p}/f_{k-1,1,p}$. Note
that,

$f^{k}(\alpha_{1})\equiv\alpha_{1}\;\;(\text{mod }\beta-\alpha_{1}),\text{ for
any }k\in{\mathbbm{N}}.$

Therefore, modulo $\beta-\alpha_{1}$,

$\displaystyle\frac{f_{k,1,p}}{f_{k-1,1,p}}$
$\displaystyle=\Phi_{p}\left(f^{k}(\alpha_{1}),f^{k-1}(\alpha_{1})\right)+\sum_{i=2}^{p-1}(-1)^{i}\frac{p}{p-i}s_{i}(\bar{\alpha})\Phi_{p-i}\left(f^{k}(\alpha_{1}),f^{k-1}(\alpha_{1})\right)$
$\displaystyle=\Phi_{p}(\alpha_{1},\alpha_{1})+\sum_{i=2}^{p-1}(-1)^{i}\frac{p}{p-i}s_{i}(\bar{\alpha})\Phi_{p-i}(\alpha_{1},\alpha_{1})$
$\displaystyle=p\left(\alpha_{1}^{p-1}+\sum_{i=2}^{p-1}(-1)^{i}s_{i}(\bar{\alpha})\alpha_{1}^{p-i-1}\right)$
$\displaystyle=f^{\prime}(\alpha_{1})$ $\displaystyle=0$

Hence, $\beta-\alpha_{1}$ divides $f_{k,1,p}/f_{k-1,1,p}$ in
${\mathbbm{Z}}_{(p)}[p-1]$.

Let us compute the lowest degree homogeneous part of
$f_{k,1,p}/\left(f_{k-1,1,p}(\beta-\alpha_{1})\right)$. Observe that, the
lowest degree homogeneous part of $f^{l}(\alpha_{1})$ is $\beta$, for any
$l\in{\mathbbm{N}}$. So, for any $k\in{\mathbbm{N}},k\geq 2$ (note that in
this proof this is the first time we excluded the case $k=1$), the lowest
degree homogeneous part of $f_{k,1,p}/f_{k-1,1,p}$ is

$\Phi_{p}(\beta,\beta)+\sum_{i=2}^{p-1}(-1)^{i}\frac{p}{p-i}\cdot
s_{i}(\bar{\alpha})\cdot\Phi_{p-i}(\beta,\beta)=f^{\prime}(\beta)=p\prod_{i=1}^{p-1}(\beta-\alpha_{i}).$

Hence, the lowest degree homogeneous part of
$f_{k,1,p}/\left(f_{k-1,1,p}(\beta-\alpha_{1})\right)$ is
$p\prod_{i=2}^{p-1}(\beta-\alpha_{i})$ and the lowest degree homogeneous part
of $\hat{f}_{k,1,p}/\left(\hat{f}_{k-1,1,p}(\beta-\alpha_{1})\right)$ is

$p\prod_{i=2}^{p-1}(\beta-\hat{\alpha_{i}})=p(\beta+\sum_{i=1}^{p-2}\alpha_{i})\prod_{i=2}^{p-2}(\beta-\alpha_{i}).$

As $\beta-\alpha_{1}$ does not divide the lowest degree homogeneous part of
$\hat{f}_{k,1,p}/\left(\hat{f}_{k-1,1,p}(\beta-\alpha_{1})\right)$, we get
that $\beta-\alpha_{1}$ does not divide
$\hat{f}_{k,1,p}/\left(\hat{f}_{k-1,1,p}(\beta-\alpha_{1})\right)$. Hence,
both parts of the lemma are proved. ∎

As a corollary of Lemma 5.3, we get that the resultant condition of
generalised Eisenstein irreducibility criterion (Theorem 4.6) is satisfied by
$h_{k,1,p}$ and $h_{0,1,p}$, for any $k\geq 2$.

######  Corollary 5.4.

For any $k\in{\mathbbm{N}},k\geq 2$, the resultant
$\mbox{Res}\;(h_{k,1,p},h_{0,1,p})\not\equiv 0\;(\text{mod }p^{2})$.

###### Proof.

Let $k\in{\mathbbm{N}},k\geq 2$. The resultant of $h_{k,1,p}$ and $h_{0,1,p}$
is

$\mbox{Res}(h_{k,1,p},h_{0,1,p})=\mbox{Res}(h_{k,1,p},\beta-\alpha_{1})=h_{k,1,p}(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p-2},\alpha_{1}).$

From Lemma 5.3, the lowest degree homogeneous part of
$\mbox{Res}(h_{k,1,p},h_{0,1,p})$ is

$p(\alpha_{1}+\sum_{i=1}^{p-2}\alpha_{i})\prod_{i=2}^{p-2}(\alpha_{1}-\alpha_{i})\not\equiv
0\;\;\;(\text{mod }p^{2}).$

Hence, the corollary is proved. ∎

Next we will study the resultant $\mbox{Res}(h_{1,1,p},h_{0,1,p})$. For that,
we first need to figure out what $h_{1,1,p}$ looks like.

######  Lemma 5.5.

The following equality holds,

$h_{1,1,p}=\frac{\hat{f}_{1,1,p}}{(\beta-\alpha_{1})^{2}}.$

###### Proof.

From the proof of Lemma 5.3, we know that $h_{1,1,p}$ divides

$\frac{\hat{f}_{1,1,p}}{\hat{f}_{0,1,p}(\beta-\alpha_{1})}=\frac{\hat{f}_{1,1,p}}{(\beta-\alpha_{1})^{2}}.$

We need to check whether higher powers of $(\beta-\alpha_{1})$ divides
$\hat{f}_{1,1,p}$. As $f^{\prime}(\alpha_{1})=0$, we get from Taylor series
expansion $f$ at $\alpha_{1}$,

$\frac{f(z)-f(\alpha_{1})}{(z-\alpha_{1})^{2}}\equiv\frac{f^{\prime\prime}(\alpha_{1})}{2}\;(\text{mod
}z-\alpha_{1}).$

Putting $z=\beta$ in above,

$\frac{f_{1,1,p}}{(\beta-\alpha_{1})^{2}}=\frac{f(\beta)-f(\alpha_{1})}{(\beta-\alpha_{1})^{2}}\equiv\frac{f^{\prime\prime}(\alpha_{1})}{2}\;(\text{mod
}\beta-\alpha_{1}).$ (5.5)

As $f^{\prime\prime}(\alpha_{1})$ is not identically zero, we get from
Equation (5.5),

$h_{1,1,p}=\frac{\hat{f}_{1,1,p}}{(\beta-\alpha_{1})^{2}}.$ (5.6)

∎

######  Remark 5.6.

Brief calculation shows that explicit form of $h_{1,1,p}$ is as follows,

$h_{1,1,p}:=\sum_{j=0}^{p-1}\alpha_{1}^{j}\Phi_{p-1-j}(\beta,\alpha_{1})+\sum_{i=2}^{p-1}(-1)^{i}\frac{p\cdot
s_{i}(\bar{\alpha})}{p-i}\sum_{j=0}^{p-i-1}\alpha_{1}^{j}\Phi_{p-i-1-j}(\beta,\alpha_{1}).$

We will not require this explicit form in what follows.

Now, we can study the resultant of $h_{1,1,p}$ and $h_{0,1,p}$.

######  Corollary 5.7.

The resultant $\mbox{Res}(h_{1,1,p},h_{0,1,p})\not\equiv 0\;(\text{mod
}p^{2})$.

###### Proof.

From Equations (5.5) and (5.6),
$\mbox{Res}(h_{1,1,p},h_{0,1,p})=\frac{\widehat{f^{\prime\prime}(\alpha_{1})}}{2}\not\equiv
0\;(\text{mod }p^{2})$. ∎

We are ready to prove irreducibility of $h_{k,1,p}$ over ${\mathbbm{Q}}$.

######  Theorem 5.8.

For any $k\in{\mathbbm{N}}$, the polynomial $h_{k,1,p}$ is irreducible over
${\mathbbm{Q}}$.

###### Proof.

Fix $e=1,\;g=h_{k,1,p},\;h=h_{0,1,p}=\beta-\alpha_{1}$, in the generalised
Eisenstein irreducibility criterion (Theorem 4.6). The Lemmas 5.1, 5.2 and
Corollaries 5.4, 5.7 show that all the three conditions of Theorem 4.6 holds.
Hence, $h_{k,1,p}$ is $p$-Eisenstein with respect to the polynomial
$h_{0,1,p}$, for every $k\in{\mathbbm{N}}$. So, $h_{k,1,p}$ is irreducible
over ${\mathbbm{Q}}$, for every choice of $k\in{\mathbbm{N}}$. ∎

Next, we extend the irreducibility of $h_{k,1,p}$ from over ${\mathbbm{Q}}$ to
over ${\mathbbm{C}}$, which, by Corollary 2.5, is enough to prove the
following theorem.

######  Theorem 5.9.

For any $k\in{\mathbbm{N}}\cup\\{0\\}$, the quasi-affine algebraic set
$\Sigma_{k,1,p}$ is irreducible.

###### Proof.

By Corollary 2.5, it is enough to show that $h_{k,1,p}$ is irreducible over
${\mathbbm{C}}$.

For $k=0$, this is trivial.

Let $k=1$. Recall that the derivative of $f$ is of the form,

$f^{\prime}(z)=p\prod_{i=1}^{p-1}(z-\alpha_{i}).$

From Equation (5.6), we get

$h_{1,1,p}\equiv\frac{\widehat{f^{\prime\prime}(\alpha_{1})}}{2}\equiv\frac{p}{2}\prod_{i=2}^{p-1}(\alpha_{1}-\hat{\alpha_{i}})\;(\text{mod
}\beta-\alpha_{1}).$

So,

$h_{1,1,p}=\frac{p}{2}\prod_{i=2}^{p-1}(\beta-\hat{\alpha_{i}})+(\beta-\alpha_{1})t_{p},$

for some polynomial $t_{p}\in{\mathbbm{Z}}_{(p)}[p-2]$. Let us assume that
$h_{1,1,p}$ is reducible over ${\mathbbm{C}}$. Up to a rearrangement of
$\alpha_{i}$’s, $2\leq i\leq p-1$, there exists an irreducible factor $g$ of
$h_{1,1,p}$ of the form

$g=\left(\prod_{i=2}^{l}(\beta-\hat{\alpha_{i}})\right)+(\beta-\alpha_{1})t,$

for some $l\in{\mathbbm{N}},2\leq l\leq p-1,$ and $t\in{\mathbbm{C}}[p-2]$.

Consider $G_{{\mathbbm{Q}}}$, the absolute Galois group of ${\mathbbm{Q}}$.
For any $\sigma\in G_{{\mathbbm{Q}}}$, we have $\sigma(g)$ divides
$h_{1,1,p}$. Observe that the first summand in the expression of $g$ above is
defined over ${\mathbbm{Q}}$. Hence, if $\sigma(g)$ is a constant multiple of
$g$ then $\sigma(g)=g$. Otherwise, if $\sigma(g)\neq g$, then $g$ being
irreducible, $g\cdot\sigma(g)$ divides $h_{1,1,p}$. But that is not possible
as the first summand in the expression of $g$ has all simple factors and they
are all defined over ${\mathbbm{Q}}$. So, $\sigma(g)=g$. As $\sigma\in
G_{{\mathbbm{Q}}}$ was arbitrarily chosen, we get that
$g\in{\mathbbm{Q}}[p-2]$. But, $h_{1,1,p}$ is irreducible over
${\mathbbm{Q}}$, by Theorem 5.8. We arrive at a contradiction. Hence,
$h_{1,1,p}$ is irreducible over ${\mathbbm{C}}$.

For $k\geq 2$, from Lemma 5.3, we have that the lowest degree homogeneous part
of $h_{k,1,p}$ is

$p\prod_{i=2}^{p-1}(\beta-\hat{\alpha_{j}}).$

Observe that the lowest degree homogeneous part of $h_{k,1,p}$ has all simple
factors and they are all defined over ${\mathbbm{Q}}$. Similarly as the $k=1$
case, one sees that any irreducible factor of $h_{k,1,p}$ must lie in
${\mathbbm{Q}}[p-2]$. As $h_{k,1,p}$ is irreducible over ${\mathbbm{Q}}$ by
Theorem 5.8, it is also irreducible over ${\mathbbm{C}}$. ∎

##  6\. mixed critical cases: The cases in between the unicritical and
separably critical case

Let $k\geq 0,p>3,p$ prime. For a degree $p$ polynomial $f$ with marked
critical point $\alpha_{1}$, we define _ramification index_ of the pair
$(f,\alpha_{1})$ to be the ramification index of $\alpha_{1}$ under the map
$f$, which is the multiplicity of $\alpha_{1}$ as a root of the equation
$f(z)-f(\alpha_{1})$. As $\alpha_{1}$ is a critical point of a degree $p$
polynomial $f$, ramification index of the pair $(f,\alpha_{1})$ varies between
$2$ and $p$. By definition, $\mathcal{M}_{p}$ is the space of affine conjugacy
classes of pairs $(f,\alpha_{1})$, where $f$ is a degree $p$ polynomial and
$\alpha_{1}$ is one of its finite critical points. Ramification index stays
invariant under affine conjugation of polynomials with a marked critical
point. Hence, for any point in $\mathcal{M}_{p}$, its _ramification index_ is
well-defined and it lies between $2$ and $p$.

For any point
$(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p-2},\beta)\in\mathcal{M}_{p}$,
we call it _separably critical_ if its finite critical points
$\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p-2},\alpha_{p-1}:=-(\sum_{i=1}^{p-2}\alpha_{i})$
are all distinct and _unicritical_ if
$\alpha_{1}=\alpha_{2}=\cdot\cdot\cdot.=\alpha_{p-1}=0$. A separably critical
point in $\mathcal{M}_{p}$ has ramification index $2$, but the converse is not
true. A point in $\mathcal{M}_{p}$ is unicritical iff its ramification index
is $p$.

Observe that the set of all separably critical points of $\Sigma_{k,1,p}$ is
non-empty and open in $\Sigma_{k,1,p}$. As $\Sigma_{k,1,p}$ is irreducible and
quasi-affine, the set of all separably critical points in $\Sigma_{k,1,p}$ is
also an irreducible quasi-affine variety. In this section, we will study
similar irreducibility questions for mixed critical cases. A point in
$\mathcal{M}_{p}$ is called _mixed critical_ iff its ramification index $e$
satisfies $2<e<p$.

Fix $j,\;2\leq j\leq p-2$. Let $\Sigma_{j}:=\Sigma_{k,1,p}\cap V_{j},\text{
where
}V_{j}:=\\{(\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p-2},\beta)\in\mathcal{M}_{p}\;|\;\alpha_{1}=\alpha_{2}=\cdot\cdot\cdot=\alpha_{j}\\}$.
The set $\Sigma_{j}$ is the set of all points of $\Sigma_{k,1,p}$ for which
the ramification index is $\geq j+1$. This is a quasi-affine subset of the
affine variety $V_{j}$. Here, we will show that $\Sigma_{j}$ is irreducible,
for every $j,\;2\leq j\leq p-2$.

######  Theorem 6.1.

Let $k,p,j\in{\mathbbm{N}},p\text{ prime },p>3,2\leq j\leq p-2$. The quasi-
affine algebraic set $\Sigma_{j}:=\Sigma_{k,1,p}\cap V_{j}$ is an irreducible
subset of $\mathcal{M}_{p}$.

###### Proof.

Similar to the separably critical case, to prove irreducibility of
$\Sigma_{j}$ it is enough to show that the ideal

$(h_{k,1,p},\alpha_{1}-\alpha_{2},\alpha_{1}-\alpha_{3},\cdot\cdot\cdot,\alpha_{1}-\alpha_{j}),$

is prime in
${\mathbbm{C}}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p-2},\beta]$. For
that, it is enough to show that the polynomial

$h_{k,1,p}^{j}:=h_{k,1,p}(\alpha_{1},\alpha_{1},\cdot\cdot\cdot,\alpha_{1},\alpha_{j+1},\cdot\cdot\cdot,\alpha_{p-2},\beta),$

is irreducible in
${\mathbbm{C}}[\alpha_{1},\alpha_{j+1},\alpha_{j+2},\cdot\cdot\cdot,\alpha_{p-2},\beta]$.

Now, $h_{k,1,p}$ being $p$-Eisenstein with respect to the polynomial
$\beta-\alpha_{1}$, the polynomial $h_{k,1,p}^{j}$ is so too. Hence,
$h_{k,1,p}^{j}$ is irreducible over ${\mathbbm{Q}}$. As $j<p-1$, Lemma 5.3
shows that the lowest degree homogeneous part of $h_{k,1,p}^{j}$ has at least
one simple linear factor defined over ${\mathbbm{Q}}$, for example:

$l:=(\beta+j\alpha_{1}+\alpha_{j+1}+\alpha_{j+2}+\cdot\cdot\cdot+\alpha_{p-2}).$

We argue similarly as in the proof of Theorem 5.9. By irreducibility of
$h_{k,1,p}^{j}$ over ${\mathbbm{Q}}$, the lowest degree homogeneous part of
any factor of $h_{k,1,p}^{j}$ over ${\mathbbm{C}}$ must have $l$ as one of its
factor. As $l$ is a simple factor of the lowest degree homogeneous part of
$h_{k,1,p}^{j}$, the polynomial $h_{k,1,p}^{j}$ is irreducible over
${\mathbbm{C}}$. ∎

Theorem 6.1 is equivalent to Theorem 1.5, mentioned in the introduction. Fix
$k\geq 0,p>3,p$ prime. Theorem 6.1 provides us with a chain of quasi-affine
subvarities of $\mathcal{M}_{p}$ of strictly decreasing dimensions.

$\mathcal{M}_{p}\supset\Sigma_{k,1,p}\supset\Sigma_{2}\supset\Sigma_{3}\supset\cdot\cdot\cdot\supset\Sigma_{p-2}\supset\text{
\\{unicritical points\\} }.$ (6.1)

As $\mathcal{M}_{p}$ is a $(p-1)$-dimensional affine variety, dimension
decreases by exactly $1$ in each stage of the above chain. Let $2\leq j\leq
p-2$. Similar to the separably critical case, as $\Sigma_{j}$ is irreducible,
the set of all points of $\Sigma_{k,1,p}$ for which ramification index is
exactly $j+1$ is open and dense in $\Sigma_{j}$. This shows that,

######  Corollary 6.2.

Let $k\geq 0,p$ be an odd prime and $2\leq j\leq p-1$. The set of all points
of $\mathcal{M}_{p}$ for which the marked critical point is strictly
$(k,1)$-preperiodic and has ramification index $j$, is an irreducible quasi-
affine variety of dimension $p-j$.

Theorem 6.1 provides evidence for extending Milnor’s conjecture to the mixed
critical case. More precisely,

######  Conjecture 6.3.

Fix $k\geq 0,\;n>0,\;d>2,\;2<j<d$. The family of conjugacy classes of degree
$d$ polynomials for which the marked critical point is strictly
$(k,n)$-preperiodic, and has ramification index $\geq j$, is an irreducible
quasi-affine subvariety of $\mathcal{M}_{d}$.

######  Remark 6.4.

Observe that, the method above can be generalised for any $\mathcal{M}_{d}$,
$d\in{\mathbbm{N}},d>2$. Let $k\in{\mathbbm{N}}\cup\\{0\\},n\in{\mathbbm{N}}$.
Let us assume that $h_{k,n,d}$ is irreducible over ${\mathbbm{C}}$. Then, one
can prove irreducibility in the mixed critical cases, by showing the
irreducibility of
$h_{k,n,d}(\alpha_{1},\cdot\cdot\cdot,\alpha_{1},\alpha_{j+1},\cdot\cdot\cdot,\alpha_{d-2},\beta)$
over ${\mathbbm{C}}$, for $1<j<d-1$.

##  7\. The unicritical case

Putting $\alpha_{1}=\alpha_{2}=\cdot\cdot\cdot=\alpha_{d-1}=0$ in Equation
(2.2), we obtain a normal form for degree $d\geq 2$ unicritical polynomials,

$f(z)=z^{d}+\beta.$ (7.1)

Let $k,n\in{\mathbbm{Z}},k\geq 0,n>0$. Let $R_{k,n,d}$ be the polynomial in
${\mathbbm{Q}}[\beta]$, whose roots are exactly the values of $\beta$ for
which $0$ is strictly $(k,n)$-preperiodic under $f(z)=z^{d}+\beta$. _Milnor’s
conjecture_ in the unicritical case (Conjecture 1.2), is equivalent to
$R_{k,n,d}$ polynomials being either constant or irreducible over
${\mathbbm{Q}}$, for every $k\geq 0,n>0.$

Putting $\alpha_{1}=\alpha_{2}=\cdot\cdot\cdot=\alpha_{d-1}=0$ in Equation
(2.10), we get that $R_{k,n,d}$ divides
$h_{k,n,d}(0,0,\cdot\cdot\cdot,0,\beta)$ in ${\mathbbm{Q}}[\beta]$. In this
section, we will prove the irreducibility of $R_{k,1,p}$ over ${\mathbbm{Q}}$,
where $k\geq 0,p$ an odd prime.

######  Theorem 7.1.

For any $k,p\in{\mathbbm{Z}},k\geq 0,p$ odd prime, The polynomial $R_{k,1,p}$
is either constant or irreducible over ${\mathbbm{Q}}$.

###### Proof.

The polynomial $R_{0,1,p}$ is $\beta$, hence irreducible. For any
$n\in{\mathbbm{N}}$, if $0$ is $(1,n)$-preperiodic under $f$ as in Equation
(7.1), then it is also $n$-periodic. Hence, the polynomial $R_{1,1,p}$ is
constant.

Let $k\geq 2$. From section 5, the polynomial $h_{k,1,p}$ is $p$-Eisenstein
with respect to the polynomial $\beta-\alpha_{1}$, and its lowest degree
homogeneous part is

$p(\beta+\sum_{i=1}^{p-2}\alpha_{i})\prod_{i=2}^{p-2}(\beta-\alpha_{i}).$

Hence, the lowest degree term of $h_{k,1,p}(0,0,\cdot\cdot\cdot,0,\beta)$ is
$p\beta^{p-2}$, and the polynomial

$\dfrac{h_{k,1,p}(0,0,\cdot\cdot\cdot,0,\beta)}{\beta^{p-2}},$

is $p$-Eisenstein with respect to $\beta$. By definition of $R_{k,1,p}$,
$\;\beta$ divides $R_{k,1,p}$ iff $k=0$. Hence, for any $k\geq 2$, if
$R_{k,1,p}$ is non-constant polynomial, then it is $p$-Eisenstein with respect
to $\beta$, for every $k\in{\mathbbm{N}}$. Therefore $R_{k,1,p}$ is either
constant or irreducible over ${\mathbbm{Q}}$, for any $k\in{\mathbbm{Z}}_{\geq
0}$. ∎

######  Remark 7.2.

Theorem 7.1 is equivalent to Theorem 1.4 stated in the introduction. A
different proof of this theorem can be found in [Gok20, Corollary 1.1.i]. A
parallel but weaker version of Theorem 7.1, obtained by choosing the normal
form of degree $d$ unicritical polynomial to be $g(z)=az^{d}+1$, is proved by
Buff, Epstein and Koch in [BEK18, Theorem 22].

##  8\. A study of $\Sigma_{k,2,p}$

For this section, $f$ is as defined in Equation (5.1).

$f(z)=z^{p}+\sum_{i=2}^{p-1}(-1)^{i}\frac{p\cdot
s_{i}(\bar{\alpha})}{p-i}z^{p-i}-\alpha_{1}^{p}-\sum_{i=2}^{p-1}(-1)^{i}\frac{p\cdot
s_{i}(\bar{\alpha})}{p-i}\alpha_{1}^{p-i}+\beta.$

######  Lemma 8.1.

The polynomial $h_{0,2,p}(\text{mod }p)$ is reducible in
${\mathbbm{F}}_{p}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p-2},\beta]$,
for all prime $p>3$.

###### Proof.

We have,

$\displaystyle f_{0,2,p}$ $\displaystyle=f^{2}(\alpha_{1})-\alpha_{1}$
$\displaystyle=(\beta^{p}-\alpha_{1}^{p})+\sum_{i=2}^{p-1}(-1)^{i}\frac{p\cdot
s_{i}(\bar{\alpha})}{p-i}(\beta^{p-i}-\alpha_{1}^{p-i})+(\beta-\alpha_{1})$
$\displaystyle\equiv(\beta-\alpha_{1})^{p}+(\beta-\alpha_{1})\;(\textnormal{mod
}p).$

As $f_{0,1,p}=\beta-\alpha_{1}$, we have

$h_{0,2,p}\equiv(\beta-\alpha_{1})^{p-1}+1\;(\textnormal{mod }p).$

For $p\equiv 1(\text{mod }4)$, $-1$ being a square modulo $p$, $h_{0,2,p}$ is
reducible over ${\mathbbm{F}}_{p}$. For $p\equiv 3(\text{mod }4),p\neq 3$, the
integer $p-1$ has at least one odd prime factor, say $q$. Then
$(\beta-\alpha_{1})^{q}+1$ divides $h_{0,2,p}$ modulo $p$. So,
$h_{0,2,p}\;(\text{mod }p)$ is reducible for any prime $p>3$. ∎

######  Remark 8.2.

Lemma 8.1 shows that modulo $p$, the polynomial $h_{0,2,p}$ is reducible, for
any prime $p>3$. So, one can not extend the proof for $(k,1,p)$ case directly
to this case.

##  9\. A study of $\Sigma_{k,1,p^{e}}$

Let $p,e\in{\mathbbm{N}},p$ odd prime. Putting $d=p^{e}$ in Equation (2.2),
the expression of $f$ becomes,

$f(z)=z^{p^{e}}+\sum_{i=2}^{p^{e}-1}(-1)^{i}\frac{p^{e}}{p^{e}-i}\cdot
s_{i}(\bar{\alpha})\cdot
z^{p^{e}-i}-\alpha_{1}^{p^{e}}-\sum_{i=2}^{p^{e}-1}(-1)^{i}\frac{p^{e}}{p^{e}-i}\cdot
s_{i}(\bar{\alpha})\cdot\alpha_{1}^{p^{e}-i}+\beta.$

Here,
$f\in{\mathbbm{Z}}_{(p)}[z,\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-1},\beta],\hat{f}_{k,1,p^{e}},h_{k,1,p^{e}}\in{\mathbbm{Z}}_{(p)}[\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{p^{e}-2},\beta]$.
One can prove the following lemmas similarly as they were proved in the
$(k,1,p)$ case.

######  Lemma 9.1.

The polynomial $h_{0,1,p^{e}}$ is $\beta-\alpha_{1}$.

######  Lemma 9.2.

For any $k\in{\mathbbm{N}}$, we have $h_{k,1,p^{e}}\equiv
h_{0,1,p^{e}}^{N_{k,p^{e}}}(\text{mod }p)$, for some
$N_{k,p^{e}}\in{\mathbbm{N}}$.

######  Lemma 9.3.

For any $k\in{\mathbbm{N}},k\geq 2,$ the following equality holds,

$h_{k,1,p^{e}}=\frac{\hat{f}_{k,1,p^{e}}}{\hat{f}_{k-1,1,p^{e}}(\beta-\alpha_{1})}$

and the lowest degree homogeneous part of $h_{k,1,p^{e}}$ is

$p^{e}(\beta+\sum_{i=1}^{p^{e}-2}\alpha_{i})\prod_{i=2}^{p^{e}-2}(\beta-\alpha_{i})$

.

######  Remark 9.4.

Due to Lemma 9.3, we get that the resultant

$\displaystyle\mbox{Res}(h_{k,1,p^{e}},h_{0,1,p^{e}})$
$\displaystyle=p^{e}(2\alpha_{1}+\alpha_{2}+\cdot\cdot\cdot+\alpha_{p^{e}-2})\prod_{i=2}^{p^{e}-2}(\alpha_{1}-\alpha_{i})$
$\displaystyle\equiv 0\;(\text{mod }p^{2}),$

for any $e\in{\mathbbm{N}},e\geq 2$. This shows the obstruction to directly
generalise the proof of irreducibility of $\Sigma_{k,1,p}$ directly to the
case of $\Sigma_{k,1,p^{e}},e\geq 2$. The lowest degree homogeneous part of
$h_{k,1,p^{e}}$ suggests that one needs to replace $h_{0,1,p^{e}}$ with
another polynomial whose degree with respect to $\beta$ is $p^{e}$. A prime
candidate would be $h_{1,1,p^{e}}$. But $h_{1,1,p^{e}}$ is reducible modulo
$p$, as one sees directly from Lemma 9.2. In general, for any replacement of
$h_{0,1,p^{e}}$, a major obstacle is figuring out the resultant
$\mbox{Res}(h_{k,1,p^{e}},h_{1,1,p^{e}})$.

## References

  * [AK20] Arfeux, Matthieu and Jan Kiwi. Irreducibility of periodic curves in cubic polynomial moduli space. (2020), Proceedings of London Mathematical Society, to appear. https://api.semanticscholar.org/CorpusID:229923967
  * [BDK91] Blanchard, Paul, Robert L. Devaney and Linda Keen. The dynamics of complex polynomials and automorphisms of the shift. Inventiones mathematicae 104 (1991): 545-580. https://doi.org/10.1007/BF01245090
  * [BEK18] Buff, Xavier, Adam L. Epstein, and Sarah Koch. Prefixed curves in moduli space. American Journal of Mathematics 144, no. 6 (2022): 1485-1509. doi:10.1353/ajm.2022.0036.
  * [BG23] Benedetto, Robert L. and Vefa Goksel. Misiurewicz polynomials and dynamical units, part I. International Journal of Number Theory: Vol. 19, No. 06, pp. 1249-1267 (2023). https://doi.org/10.1142/S1793042123500616
  * [BGok22] Benedetto, Robert L. and Vefa Goksel. Misiurewicz polynomials and dynamical units, Part II. (2022) ArXiv. https://doi.org/10.48550/arXiv.2203.14431
  * [BH88] Branner, Bodil & Hubbard, John. (1988). The iteration of cubic polynomials Part I: The global topology of parameter space. Acta Mathematica. 160. 143-206. 10.1007/BF02392275.
  * [BH92] Branner, Bodil & Hubbard, John. (1992). The iteration of cubic polynomials Part II: patterns and parapatterns. Acta Mathematica. 169. 229-325. 10.1007/BF02392761.
  * [BKM10] Bonifant, Araceli & Kiwi, Jan & Milnor, John. (2010). Cubic polynomial maps with periodic critical orbit, Part II. Conformal Geometry and Dynamics of The American Mathematical Society. 14. 68-112. 10.1090/S1088-4173-10-00204-3.
  * [DH93] Douady, Adrien & Hubbard, John. (1993). A proof of Thurston’s topological characterization of rational functions. Acta Mathematica. 171. 263-297. 10.1007/BF02392534.
  * [FG15] Favre, C., Gauthier, T. Distribution of postcritically finite polynomials. Isr. J. Math. 209, 235–292 (2015). https://doi.org/10.1007/s11856-015-1218-0
  * [Gok20] Goksel, Vefa. On the orbit of a post-critically finite polynomial of the form $x^{d}+c$, Functiones et Approximatio Commentarii Mathematici, Funct. Approx. Comment. Math. 62(1), 95-104, (March 2020) https://api.semanticscholar.org/CorpusID:119131690
  * [G20] Goksel, Vefa. (2020). A note on Misiurewicz polynomials. Journal de Théorie des Nombres de Bordeaux. 32. 373-385. 10.5802/jtnb.1126.
  * [HT15] Hutz, Benjamin & Towsley, Adam. (2015). Misiurewicz points for polynomial maps and transversality. New York Journal of Mathematics. 21. 297-319. https://nyjm.albany.edu/j/2015/21-13v.pdf
  * [LQ07] Lü, Jing & Qiu, Wei-Yuan. (2007). Local connectivity of Julia sets for a family of biquadratic polynomials. Science in China Series A: Mathematics. 50. 1477-1492. 10.1007/s11425-007-0114-4.
  * [M90] Milnor, John W. Remarks on Iterated Cubic Maps. Exp. Math. 1 (1990): 5-24.
  * [MP92] Milnor, John W., and Alfredo Poirier. Hyperbolic components in spaces of polynomial maps. arXiv preprint math/9202210 (1992).
  * [Mil09] Milnor, John. (2009). Cubic Polynomial Maps with Periodic Critical Orbit, Part I. Complex Dynamics. 10.1201/b10617-13.
  * [Mil14] Milnor, John. Arithmetic of Unicritical Polynomial Maps. Frontiers in Complex Dynamics: In Celebration of John Milnor’s 80th Birthday, Princeton: Princeton University Press, 2014, pp. 15-24. https://doi.org/10.1515/9781400851317-003
  * [R06] Roesch, Pascale. Hyperbolic components of polynomials with a fixed critical point of maximal order. Annales Scientifiques De L Ecole Normale Superieure 40 (2006): 901-949. doi : 10.1016/j.ansens.2007.10.001.