#  The size of arboreal images, I: exponential lower bounds for PCF and
unicritical polynomials

Carlo Pagano Vivatsgasse 7, 53111 Bonn, Germany<EMAIL_ADDRESS>Max
Planck Institute for Mathematics, Bonn  University of Glasgow, Glasgow

###### Abstract

Let $f$ be a polynomial over a global field $K$. For each $\alpha$ in $K$ and
$N$ in $\mathbb{Z}_{\geq 0}$ denote by $K_{N}(f,\alpha)$ the _arboreal field_
$K(f^{-N}(\alpha))$ and by $D_{N}(f,\alpha)$ its degree over $K$.

It is conjectured that $D_{N}(f,\alpha)$ should grow as a double exponential
function of $N$, unless $f$ is _post-critically finite_ (PCF), in which case
there are examples like $D_{N}(x^{2},\alpha)\leq 4^{N}$. There is evidence
conditionally on Vojta’s conjecture. However, before the present work, no
unconditional non-trivial lower bound was known for _post-critically infinite_
$f$. In the case $f$ is PCF, no non-trivial lower bound was known, not even
under Vojta’s conjecture.

In this paper we give two simple methods that turn the finiteness of the
critical orbit into an exploitable feature, also in the post-critically
infinite case. First, assuming GRH for number fields, we establish for all PCF
polynomials $f$ of degree at least $2$ and all $\alpha$ outside of the
critical orbits of $f$, the existence of a positive constant $c(f,\alpha)$
such that

$D_{N}(f,\alpha)\geq\text{exp}(c(f,\alpha)N),$

for all $N$, which is sharp up to, possibly, improve the constant
$c(f,\alpha)$.

Second, we show unconditionally that if $f$ is post-critically infinite over
any number field $K$ and unicritical, then, for each $\alpha$ in $K$, there
exists a positive constant $c(f,\alpha)$ such that

$D_{N}(f,\alpha)\geq\text{exp}(c(f,\alpha)N),$

for all $N$. The main input here is to work modulo a suitably chosen prime and
use a construction available for PCF unicritical polynomials with periodic
critical orbit.

##  1 Introduction

Arboreal Galois groups are among the central objects of study in modern
arithmetic dynamics. We refer to [6, Section 5] for a comprehensive list of
results and open problems on the subject. Arboreal Galois groups are typically
obtained by looking at the graph of pre-images

$T_{\infty}(f,\alpha):=\\{f^{-N}(\alpha)\\}_{N\in\mathbb{Z}_{\geq 0}}$

of an element $\alpha$ of a field $K$ under a separable polynomial $f\in
K[x]$: the edges are defined by connecting $\beta$ to $f(\beta)$. Under the
extra assumption that $\alpha$ is not a periodic point and is not in the orbit
of a critical point of $f$, one sees that $T_{\infty}(f,\alpha)$ is naturally
an infinite degree $d$ tree rooted at $\alpha$. 111Unless explicitly stated we
won’t make this extra assumption in what follows. Also we notice that it is
enough to assume that $\alpha$ is not of the form $f^{n}(\beta)$ for $\beta$ a
critical point and $n$ a _positive_ integer. The absolute Galois group $G_{K}$
acts by graph-preserving symmetries on $T_{\infty}(f,\alpha)$ and gives rise
to the _arboreal Galois group_

$G_{\infty}(f,\alpha):=\text{Gal}(K_{\infty}(f,\alpha)/K),$

where $K_{\infty}(f,\alpha)$ is the extension of $K$ generated by
$T_{\infty}(f,\alpha)$, and more generally
$K_{N}(f,\alpha):=K(f^{-N}(\alpha))$ for each $N$ in $\mathbb{Z}_{\geq 0}$, is
the splitting field over $K$ of the polynomial $f^{N}-\alpha$. In this paper
$f^{N}$ will always mean the $N$-th iterate of $f$ under composition.

Let us now consider the case that $K$ is a global field. It is generally
expected that arboreal Galois groups over $K$ should be _large_. The first
results in this direction are due to Odoni [25, 26]. Subsequently Cremona [8]
considered the polynomial $x^{2}+1$ in detail and conjectured that for
$K=\mathbb{Q}$ the arboreal Galois groups $G_{N}(x^{2}+1,0)$ are as large as
possible. This was proved by Stoll [32]. Stoll’s ingenious argument has been
recently extended in [11], which considered more general open subgroups of the
entire automorphism group of the tree $T_{\infty}(f,\alpha)$, which we denote
by $\Omega_{\infty}(f,\alpha)$. Examples of surjectivity for higher degree
polynomials can be found in [4, 23, 19, 31]. Some further explicit families of
unicritical polynomials yielding an open image have been exihibited in [7],
using the Chabauty–Coleman method and the Mordell–Weil sieve.

A precise conjecture, reflecting the general expectation on arboreal images
over global fields, is due to Jones, who conjectured [16, Conjecture 3.11]
that for quadratic polynomials $f$ over a number field, with $\alpha$ not
$f$-periodic and outside of the strict critical orbit of $f$, the group
$G_{\infty}(f,\alpha)$ is an open subgroup of $\Omega_{\infty}(f,\alpha)$
precisely when $f$ is _post-critically infinite_. We recall the definition of
post-critically (in)finite.

######  Definition 1.1.

Let $K$ be a field and $d$ be in $\mathbb{Z}_{\geq 2}$. Let $f\in K[x]$ be a
monic polynomial of degree $d$. We say that $f$ is _post-critically finite_
(abbreviated _PCF_) in case for each $\gamma\in\bar{K}$ with
$f^{\prime}(\gamma)=0$, we have that

$\\#\\{f^{n}(\gamma)\\}_{n\in\mathbb{Z}_{\geq 0}}<\infty.$

If $f$ is not post-critically finite, we say that it is _post-critically
infinite_.

We call each of the orbits $\\{f^{n}(\gamma)\\}_{n\in\mathbb{Z}_{\geq 0}}$
(respectively $\\{f^{n}(\gamma)\\}_{n\in\mathbb{Z}_{\geq 1}}$), with $\gamma$
a zero of $f^{\prime}(x)$, a critical orbit of $f$ (respectively a strict
critical orbit). The notion of PCF can be extended to all rational maps and in
a precise sense PCF rational maps over a number field are not abundant: [5]
shows that, aside from the set of Lattès maps, they form a set of bounded
height in the moduli space of rational maps.

Jones’ conjecture would in particular imply that the finite arboreal Galois
groups attached to post-critically infinite quadratic polynomials are very
large in size. Namely that, as long as $(f,\alpha)$ obeys the above
assumptions, for

$G_{N}(f,\alpha):=\text{Gal}(K_{N}(f,\alpha)/K)$

one has that there exists a positive constant $c(f,\alpha)$ such that

$\\#G_{N}(f,\alpha)\geq c(f,\alpha)\cdot 2^{2^{N}}$

as $N$ varies among the positive integers. 222Dropping the assumptions on
$\alpha$, but keeping that $f$ is post-critically infinite, would still give a
double exponential lower bound, with an adaptation very similar to the one
that we will give at the end of the proof of Theorem 1.2.Under Vojta’s
conjectures for number fields there is substantial evidence for Jones’
conjecture [14], and over function fields one can, in some cases, use these
techniques to prove unconditionally that the representation is surjective [9].
Under the abc conjecture [18, Theorem 3] shows Jones’ conjecture in the case
of eventually stable pairs $(f,\alpha)$. Here eventually stable means that
$f^{N}-\alpha$ factorizes in a number of irreducibles that is eventually
constant in $N$. Similar results were obtained in [12]. Some unconditional
evidence for Jones’ conjecture over number fields has been provided in the
recent work [10].

For the PCF case, proving lower bounds for $\\#G_{N}(f,\alpha)$ seems at first
a more elusive problem, already in the quadratic case and even if one assumes
strong diophantine conjectures. Indeed, when $f$ is PCF, all the standard sub-
extensions that one typically constructs out of the critical orbit of $f$,
amount to a finite multi-quadratic subextension of $K_{\infty}(f,\alpha)/K$.

In the present paper we give two simple methods that turn the finiteness of
the critical orbit into a _feature_ that allows to prove lower bounds for
$\\#G_{N}(f,\alpha)$, both in the PCF and, perhaps surprisingly, in the post-
critically infinite case. In the PCF case, our lower bound for
$\\#G_{N}(f,\alpha)$ are basically sharp. For the first method, however,
instead of relying on diophantine conjectures we have to rely on conjectures
from analytic number theory. In contrast our second method leads to an
unconditional result. We recall that the pair $(f,\alpha)$ is said to be
_exceptional_ in case $T_{\infty}(f,\alpha)$ is a finite set. A sufficient
condition to be non exceptional is that $\alpha$ is not in any critical orbit
of $f$, which excludes only finitely many values in case $f$ is PCF. Our first
main theorem is as follows.

######  Theorem 1.2.

Let $K$ be a number field and let $f$ be a post-critically finite polynomial
over $K$ of degree at least $2$. Let $\alpha$ be in $K$ and suppose
$(f,\alpha)$ is not exceptional. Assume that GRH holds. Then there exists a
positive constant $c(f,\alpha)$ such that

$\\#G_{N}(f,\alpha)\geq\textup{exp}(c(f,\alpha)\cdot N),$

for all $N$ in $\mathbb{Z}_{\geq 1}$.

We recall that one has examples such as

$\\#G_{N}(x^{2},\alpha)\leq 4^{N}\ \text{and}\ \\#G_{N}(x^{2}-2,0)\leq 2^{N},$

hence showing that our exponential lower bound for PCF is sharp, apart from
possibly improving on the constant $c(f,\alpha)$. That said, exponential
behavior might likely be very rare even among PCF and it might be that for
arboreal Galois groups that are sufficiently non-abelian a much better lower
bound holds. However, currently we do not know how to achieve this. Abelian
arboreal Galois groups are the subject of a very recent conjecture [3], where
the conjecture has been also proved for quadratic stable polynomials over
$\mathbb{Q}$ [3, Theorem 1]. Later on, the conjecture has been settled for
general quadratic polynomials over $\mathbb{Q}$ in [10, Theorem 3.12]. In this
work it was also shown that for a quadratic polynomial $f$ over a number field
$K$ the associated arboreal representation is abelian for some $\alpha$ in $K$
only if $f$ is PCF [10, Theorem 3.8].

In the proof of Theorem 1.2, we actually give as a lower bound
$c(\epsilon,f,\alpha)\cdot(d^{\frac{1}{2}-\epsilon})^{N}$, for any
$\epsilon>0$, where the constant $c(\epsilon,f,\alpha)$ depends, aside from
$\epsilon$, only on the set of places of $K$ ramifying in
$K_{\infty}(f,\alpha)$. This constant can be made explicit.

We next turn to the case of _post-critically infinite_ unicritical
polynomials, i.e. $f:=x^{d}-c$ for some $d\in\mathbb{Z}_{\geq 2}$ and $c$ in
$K$. In what follows we focus on the number field case, since for function
fields one has, in the post-critically infinite case, tools such as Vojta’s
conjecture.

######  Theorem 1.3.

Let $K$ be a number field, let $d$ be in $\mathbb{Z}_{\geq 2}$ and $c\in K$.
Suppose that $f:=x^{d}-c$ is post-critically infinite. Then for each $\alpha$
in $K$ there exists a positive constant $c(f,\alpha)$ such that

$\\#G_{N}(f,\alpha)\geq\textup{exp}(c(f,\alpha)\cdot N),$

for all $N$ in $\mathbb{Z}_{\geq 1}$.

The lower bound in Theorem 1.3 is unconditional, however, as explained at the
beginning of this introduction, it is far off from the conjectural growth of
$D_{N}(f,\alpha)$ for post-critically infinite $f$. Nevertheless, to the best
of our knowledge, this is the first general exponential lower bound for
$D_{N}(f,\alpha)$ proven unconditionally for all unicritical post-critically
infinite polynomials over number fields. With some extra work, one can provide
the constant $c(f,\alpha)$ explicitly, see the discussion immediately before
Proposition 3.2. We remark that the proof of Theorem 1.3 adapts very easily to
cover also all $x^{d}+c$ that are PCF and with periodical critical orbit.

As the reader can learn at the end of the proof of Theorem 1.3, the lower
bound we prove there is of the form $C\cdot\text{deg}(f)^{\frac{N}{n_{0}}}$,
where $C$ is a positive constant and $n_{0}$ is the smallest integer such that
there exists $v$ a non-archimedean place of $K$ satisfying $v(f^{n_{0}}(0))>0$
and $v(\alpha+c)\geq 0$. In particular if we take $\alpha:=0$ and $c$ to be
not the inverse of an element of $\mathcal{O}_{K}[\frac{1}{\text{deg}(f)}]$ we
get a lower bound of the form $C\cdot\text{deg}(f)^{N}$. As we next explain,
this special case is also a previously known instance of Theorem 1.3, which
brings us to the last part of this section, namely the relation between our
lower bounds and the notion of stability.

In case one shows that the pair $(f,\alpha)$ is eventually stable then it
follows an exponential lower bound of the form $\\#G_{N}(f,\alpha)\geq
C\cdot\text{deg}(f)^{N}$, for all $N$ in $\mathbb{Z}_{\geq 1}$, where $C$ is a
positive constant [17, Proposition 2.1]. Eventual stability had been
established in case $f=x^{d}+c,\alpha:=0$ and $v(c)>0$ for some non-
archimedean place $v$ of $K$ [13, Theorem 1.6] and for $K:=\mathbb{Q}$ and
$f:=x^{d}+c,\alpha:=0$ where $c$ is the inverse of an odd integer different
from $-1$ [17, Corollary 1.5]. From this perspective, Theorem 1.3 can be
viewed as a generalization 333Within the specialization from eventually stable
to exponential lower bounds. of [13, Theorem 1.6], where we get an exponential
lower bound whose quality is inversely proportional to the amount of time the
critical orbit is not divided by some prime where $\alpha$ is integral.
Indeed, the proof of Theorem 1.3 works more in general for all $f=x^{d}+c$
where such a valuation $v$ exists and not only in the post-critically infinite
case, where such a $v$ can be always found. For more details on how we exploit
also orbits of length larger than $1$ (the case of orbits of length $1$ being
in particular covered by [13, Theorem 1.6]) we refer to the next subsection,
where we give a quick summary of the key steps of the proofs.

###  1.1 Outline of the proofs:

In the proof of Theorem 1.2, the main feature of PCF polynomials that we
exploit is that the entire extension $K_{\infty}(f,\alpha)/K$ ramifies only at
finitely many places, and, even more, each of the polynomials $f^{N}-\alpha$
has discriminant supported at a fixed finite set of places $S(f,\alpha)$
independent of $N$ (Proposition 2.2). This forces the smallest splitting prime
of the extension $K_{N}(f,\alpha)/K$, outside of $S(f,\alpha)$, to have norm
at least $\text{deg}(f)^{N}$ (Proposition 2.4). Hence, using GRH (Proposition
2.3), one deduces that the logarithm of the absolute discriminant
$\text{log}(d_{K_{N}(f,\alpha)})$ must be exponentially large in $N$
(Proposition 2.5). Since $K_{N}(f,\alpha)$ is ramified only at the fixed
finite set $S(f,\alpha)$, this forces $[K_{N}(f,\alpha):K]$ to be
exponentially large in $N$ (Proposition 2.6). Putting everything together then
yields Theorem 1.2, as we articulate at the end of Section 2. We remark that
GRH has played a role in previous results in arithmetic dynamics, such as for
instance [21] and [24], in studying the density of prime divisors of linear
recurrence sequences. However this assumption here is used in a drastically
different manner and for entirely different reasons: to give a sense of the
difference we remark that in Theorem 1.2 the relevant recurrence sequences are
supported in finite sets (and the polynomials have degree at least $2$). We
also remark that it would be enough to have a real number $\sigma<1$ such that
$\text{Re}(s)\leq\sigma$ holds for any $s$ zero of any zeta function.
It would be interesting to adapt Theorem 1.2 to function fields, and obtain
unconditional bounds therein. For that one needs to take care of two issues.
One comes from constant field extensions, which has serious consequences on
the application of Chebotarev. If the constant field extension in
$K_{N}(f,\alpha)$ grows even only linearly in $N$, then the strategy of proof
used here does not work. A possible way out could be to establish a dichotomy,
which shows that either the constant field grows very slowly in $N$ or
exponentially fast, for all PCF polynomials $f$. However, at the moment, we do
not know whether this holds and how one could prove it. The other issue comes
from wild ramification. Contrarily to a local number field, a local function
field can have extensions of arbitrarily large (wild) ramification and given
degree, and this will create problems in establishing a good analogue of
Proposition 2.6. A way out for the second problem can be found by imposing
$\text{deg}(f)<p$, which imposes all the ramification to be tame. This
assumption seems particularly natural in the context of PCF polynomials over
function fields, see for instance [5, page 2350].
In the proof of Theorem 1.3 we explore a generalization of [1, Lemma 4.1] to
general unicritical PCF with _periodic_ critical orbit (Proposition 3.1): [1]
focused on the polynomial $x^{2}-1$, where the critical orbit has period $2$,
and showed, among other things, in that case how to explicitly construct
larger and larger $2$-power roots of unity by moving $2$ steps at the time up
on $T_{\infty}(f,\alpha)$. In Proposition 3.1 we show how to construct larger
and larger $d$-power roots of unity by moving $n_{0}$-steps at the time up on
$T_{\infty}(f,\alpha)$, where $n_{0}$ is the period of the critical orbit.
After having done this, we simply observe that a post-critically infinite
unicritical polynomial must be PCF with periodic critical orbit modulo some
prime that is sufficiently non-degenerate from the standpoint of Proposition
3.1 (this is done in Proposition 3.2). Hence we lower bound $D_{N}(f,\alpha)$
by the residue field degree at this suitably chosen prime. The conclusion then
follows easily. This proof is in line with [2, Remark 1.4], where one uses an
auxiliary prime to lower bound the size of global arboreal Galois groups by
the size of local arboreal Galois groups. It would be very interesting to
devise a way to use several auxiliary periodic primes at the same time,
keeping in mind tools such as [29] giving some control on the number of prime
divisors dividing the critical orbit. To this end [27] might provide useful
guidance.

## Acknowledgements

This project started as a correspondence with Jorge Mello, Alina Ostafe and
Igor Shparlinski, who drew my attention to this problem. I warmly thank them
for generously sharing with me their own ideas on this subject and for their
feedback on previous versions of this work. I am grateful to Andrea Ferraguti
for his comments on an earlier draft that improved the presentation, for
noticing a strenghtening of Theorem 1.2 and for insightful conversations. I
thank Rob Benedetto, Andrew Bridy, Rafe Jones, Valentijn Karemaker and Tom
Tucker for valuable feedback. I am grateful to Daniel El-Baz, Pieter Moree and
Marco Pagano for giving many suggestions that drastically improved the
presentation. I wish to thank the Max Planck Institute for Mathematics in Bonn
for its great working conditions and an inspiring atmosphere. I gratefully
acknowledge the financial support through EPSRC Fellowship EP/P019188/1,
“Cohen–Lenstra heuristics, Brauer relations, and low-dimensional manifolds”.

##  2 Proof of Theorem 1.2

Let $K$ be a number field and $d$ be in $\mathbb{Z}_{\geq 2}$. Until the final
paragraph of this section $f$ denotes a PCF polynomial over $K$ of degree $d$
and $\alpha$ denotes any element of $K$ outside of the critical orbits of $f$.
At the very end of the section we will reduce the proof of Theorem 1.2 to this
case. Let us begin with a standard fact.

######  Proposition 2.1.

Let $K,f,\alpha$ be as above. Then

$\\#f^{-N}(\alpha)=d^{N},$

for all $N$ in $\mathbb{Z}_{\geq 0}$.

###### Proof.

This follows immediately, for instance, by induction from [15, Lemma 2.6],
with $g(x):=x-\alpha$. We now provide a direct argument as well. The desired
conclusion is obviously true for $N=0$. Suppose by contradiction that the
conclusion does not hold for some positive value of $N$. Let $M$ be the
smallest positive integer with $\\#f^{-M}(\alpha)<d^{M}$. Then there must
necessarily be $\beta\in f^{-M+1}(\alpha)$ with $f^{-1}(\beta)$ possessing a
critical point, $x_{0}$, of $f$: otherwise each of the $d^{M-1}$ points of
$f^{-M+1}(\alpha)$ would have pairwise disjoint $d$-sets of preimages yielding
$\\#f^{-M}(\alpha)=d^{M}$, which cannot hold by definition of $M$. Hence
$\alpha$ is in the orbit of $x_{0}$, which means that $\alpha$ is in a
critical orbit, which has been excluded from the beginning of this section. ∎

We now recall how the assumption that $f$ is PCF, along with the assumption
that $\alpha$ is outside the critical orbits, forces the various discriminants
$\text{Disc}(f^{N}-\alpha)$ to be supported in a uniform finite set of primes.

######  Proposition 2.2.

Let $K,f,\alpha$ be as above. Then there exists a finite set $S(f,\alpha)$ of
places of $K$, such that $f$ is integral outside of $S(f,\alpha)$ and for any
finite prime $\mathfrak{p}$ of $K$ outside of $S(f,\alpha)$ and all positive
integers $N$, we have that $v_{\mathfrak{p}}(\textup{Disc}(f^{N}-\alpha))=0$.

###### Proof.

This is an immediate consequence of the formula for
$\text{Disc}(f^{N}-\alpha)$ given in [15, Lemma 2.6], plugging in
$g(x):=x-\alpha$ and keeping in mind that Proposition 2.1 forces each factor
in that formula to be non-zero. ∎

For the rest of this section we fix such a finite set of places $S(f,\alpha)$
provided by Proposition 2.2.

For a number field $L/\mathbb{Q}$, we denote by $d_{L}$ the absolute
discriminant of $L$. Fix now $\epsilon\in\mathbb{R}_{>0}$ any positive
constant. The following result is a consequence of the work of
Lagarias–Odlyzko [22].

######  Proposition 2.3.

Assume GRH. Then there exists a positive constant $c_{1}(f,\alpha)$ such that
for each $N$ in $\mathbb{Z}_{\geq 1}$ there exists a prime $\mathfrak{p}_{N}$
in $K$ outside of $S(f,\alpha)$ with the following two properties.
$(1)$ $\mathfrak{p}_{N}$ splits completely in $K_{N}(f,\alpha)$.
$(2)$ $\\#\mathcal{O}_{K}/\mathfrak{p}_{N}\leq
c_{1}(f,\alpha,\epsilon)\cdot\textup{log}(d_{K_{N}(f,\alpha)})^{2+\epsilon}$

###### Proof.

We apply [28, Theorem 4] with the same $K$, with $E:=K_{N}(f,\alpha)$, with

$x:=c_{1}(f,\alpha,\epsilon)\cdot\text{log}(d_{E})^{2+\epsilon},$

and with $C$ the conjugacy class of the identity. The constant
$c_{1}(f,\alpha)$ will be chosen sufficiently large, as we explain below. Then
[28, Theorem 4] combined with the negation of the proposition yields that
$\text{Li}(c_{1}(f,\alpha,\epsilon)\text{log}(d_{E})^{2+\epsilon})$ is at most

$c_{6}\text{log}(d_{E})^{1+\frac{\epsilon}{2}}\cdot(\text{log}(d_{E})+[E:\mathbb{Q}]\text{log}(c_{1}(f,\alpha,\epsilon)\text{log}(d_{E})^{2+\epsilon}))+\\#S(f,\alpha).$

Here $c_{6}$ is the positive absolute constant taken from the statement of
[28, Theorem 4].

Plugging in the Minkowski’s bound for $[E:\mathbb{Q}]$, we conclude in
particular that the quantity
$\text{Li}(c_{1}(f,\alpha,\epsilon)\text{log}(d_{E})^{2+\epsilon})$ is at most

$\text{log}(d_{E})^{1+\frac{\epsilon}{2}}\cdot(\text{log}(d_{E})+a_{\text{abs}}\cdot\text{log}(d_{E})\cdot\text{log}(c_{1}(f,\alpha,\epsilon)\text{log}(d_{E})^{2+\epsilon}))+\\#S(f,\alpha),$

for some absolute constant $a_{\text{abs}}$. Now we see that it is possible to
chose $c_{1}(f,\alpha,\epsilon)$ so large that the last inequality cannot hold
for any $E$. To avoid this contradiction, that arose from the negation of the
present statement, we obtain the desired conclusion. ∎

On the other hand we have the following elementary fact.

######  Proposition 2.4.

Let $\mathfrak{p}$ be a finite place of $K$ outside of $S(f,\alpha)$ and let
$N$ be in $\mathbb{Z}_{\geq 0}$. Suppose that $\mathfrak{p}$ splits completely
in $K_{N}(f,\alpha)/K$. Then

$\\#(\mathcal{O}_{K}/\mathfrak{p})\geq d^{N}.$

###### Proof.

Take $\mathfrak{p}$ as in the statement. Let
$\mathbb{F}_{\mathfrak{p}}:=\mathcal{O}_{K}/\mathfrak{p}$ be the residue field
of $\mathfrak{p}$. Choose any prime $\widetilde{\mathfrak{p}}$ above
$\mathfrak{p}$ in $K_{N}(f,\alpha)$. Since $\mathfrak{p}$ splits completely,
we have that the natural inclusion map induces an equality

$\mathcal{O}_{K_{N}(f,\alpha)}/\widetilde{\mathfrak{p}}=\mathbb{F}_{\mathfrak{p}}.$

Observe that, by definition of $S(f,\alpha)$, the set $f^{-N}(\alpha)$
consists of integral elements locally at $\widetilde{\mathfrak{p}}$ and thus
we can reduce each of its elements modulo $\widetilde{\mathfrak{p}}$.

When reduced modulo $\widetilde{\mathfrak{p}}$, the various elements of
$f^{-N}(\alpha)$ need to be distinct in view of Proposition 2.2. It follows
that

$\\#\mathbb{F}_{\mathfrak{p}}=\\#\mathcal{O}_{K_{N}(f,\alpha)}/\widetilde{\mathfrak{p}}\geq\\#f^{-N}(\alpha)=d^{N},$

where the last equality follows from Proposition 2.1. This is precisely the
desired inequality. ∎

Combining Proposition 2.3 and Proposition 2.4, we conclude the following.

######  Corollary 2.5.

Assume GRH. Then there exists a positive constant constant
$c_{2}(f,\alpha,\epsilon)$ such that for all $N$ in $\mathbb{Z}_{\geq 1}$, we
have that

$\textup{log}(d_{K_{N}(f,\alpha)})\geq
c_{2}(f,\alpha,\epsilon)\cdot(d^{\frac{1}{2+\epsilon}})^{N}.$

Our next step lower bounds the degree in terms of the discriminant for
extensions ramifying only above a fixed set of places.

######  Proposition 2.6.

Let $F$ be a number field, and let $S$ be a finite set of finite places in
$F$. Then there is a positive constant $c(S)$ such that for any extension
$E/F$ unramified at all finite places outside of $S$, we have that

$c(S)\cdot[E:\mathbb{Q}]\textup{log}([E:\mathbb{Q}])\geq\textup{log}(d_{E}).$

###### Proof.

We will show that this follows from [28, Proposition 4’]. We first explain the
correspondence between the notation therein and the notation of this
proposition. The $K$ in that proposition is our $F$, the notation for $E$
agrees. The set $P(E/K)$ in [28] consists for us of the set of rational primes
below a prime of $F$ that is in $S$. The number $n$ is $[E:F]$ and the number
$n_{E}$ is $[E:\mathbb{Q}]$. Hence the quantity
$(1-\frac{1}{n})\cdot\sum_{p\in P(E/K)}\text{log}(p)$ is a nonnegative
constant, which we denote by $c_{1}(S)$, and likewise $\\#P(E/K)$ and
$\text{log}(d_{K})$ are other nonnegative constants, which we denote by
$c_{2}(S)$ and $c_{3}(S)$, respectively. It now follows from [28, Proposition
4’] that

$\frac{\text{log}(d_{E})}{[E:\mathbb{Q}]}\leq
c_{3}(S)+c_{1}(S)+c_{2}(S)\text{log}([E:\mathbb{Q}]),$

which implies the desired conclusion.

∎

We are now ready to prove Theorem 1.2.
_Proof of Theorem 1.2:_ We keep assuming that $\alpha$ is outside of all the
critical orbits of $f$, as we did so far in this section, and we prove the
theorem under this assumption. We explain only at the end how to reduce to
this case when we only assume that $(f,\alpha)$ is not exceptional. Thanks to
Proposition 2.6 and Proposition 2.2 we know that for each $\epsilon>0$ we can
write

$[K_{N}(f,\alpha):K]\geq
c^{\prime}(f,\alpha,\epsilon)\text{log}(d_{K_{N}(f,\alpha)})^{\frac{1}{1+\epsilon}},$

for some positive constant $c^{\prime}(f,\alpha,\epsilon)$. By Corollary 2.5
we conclude that there exists a positive constant
$\widetilde{c}(\epsilon,f,\alpha)$ such that

$[K_{N}(f,\alpha):K]\geq\widetilde{c}(\epsilon,f,\alpha)\cdot(d^{\frac{1}{1+\epsilon}\cdot\frac{1}{2+\epsilon}})^{N},$

for all $N$ in $\mathbb{Z}_{\geq 1}$. Since $\epsilon>0$ is arbitrary, we can
rewrite this by saying that for each $\epsilon>0$ there exists a positive
constant $c(f,\alpha,\epsilon)$ such that

$[K_{N}(f,\alpha):K]\geq
c(f,\alpha,\epsilon)\cdot(d^{\frac{1}{2}-\epsilon})^{N},$

for all $N$ in $\mathbb{Z}_{\geq 1}$. This gives in particular the desired
conclusion under the assumption that $\alpha$ is outside of the critical orbit
of $f$. We now drop this assumption and we only assume that $(f,\alpha)$ is
not exceptional. This will conclude the proof in complete generality. Since
$(f,\alpha)$ is not exceptional we have that $T_{\infty}(f,\alpha)$ is
infinite, and thus, since $f$ is PCF, there must be $\beta$ in
$T_{\infty}(f,\alpha)$ such that $\beta$ is outside of all the critical orbits
of $f$. Then the above proof gives the conclusion for $D_{N}(f,\beta)$, where
the number field $K$ is replaced with the number field $K(\beta)$. This
clearly implies the desired lower bound for $D_{N}(f,\alpha)$.

##  3 Proof of Theorem 1.3

Our main tool will be a construction of roots of unity in the arboreal fields
of a PCF unicritical polynomial $f:=x^{d}-c$ over a field $F$ of
characteristic coprime with $d$, under the assumption that the critical orbit,
which equals to the orbit of $0$, is periodic under the action of $f$. This
construction is inspired by [1, Lemma 4.1], which dealt with the special case
$f:=x^{2}-1$, where the length of the period is $2$. We have the following.

######  Proposition 3.1.

Let $d$ be an integer, let $F$ be a field of characteristic coprime to $d$.
Let $c$ be in $F$. Suppose there is a positive integer $n_{0}$ such that
$f^{n_{0}}(0)=0$, where $f:=x^{d}-c$. Let $\alpha$ be an element of $K$
different from $-c$.

Then, for each positive integer $N$, the field $F_{N}(f,\alpha)$ contains an
element of multiplicative order equal to

$d^{\textup{max}(\lfloor\frac{N}{n_{0}}\rfloor-1,0)+1}$

###### Proof.

Since $\alpha$ is different from $-c$ and since $d$ is coprime to the
characteristic of $F$, we know that there are $d$ distinct non-zero elements
in $f^{-1}(\alpha)$, that form a free transitive set under the action of the
cyclic group of order $d$ given by

$\mu_{d}(F_{1}(f,\alpha)):=\\{\zeta\in F_{1}(f,\alpha):\zeta^{d}=1\\}.$

Take $\zeta_{d}$ a generator of $\mu_{d}(F_{1}(f,\alpha))$, pick an element
$\beta$ in $f^{-1}(\alpha)$, and denote by $\beta^{\prime}$ the element
$\zeta_{d}\cdot\beta$.

Observe that the polynomial $f^{n_{0}}-\beta$ is a polynomial in $x^{d}$ of
degree $d^{n_{0}}$. It follows that we can pick a vector
$(\gamma_{1}(1),\ldots,\gamma_{d^{n_{0}-1}}(1))$ in
$(f^{-n_{0}}(\beta))^{d^{n_{0}-1}}$ such that

$f^{n_{0}}-\beta=(x^{d}-\gamma_{1}(1)^{d})\ldots(x^{d}-\gamma_{d^{n_{0}-1}}(1)^{d}).$

We now evaluate in $0$ and take advantage of the fact that $f^{n_{0}}(0)=0$ to
obtain that

$-\beta=(-1)^{d^{n_{0}-1}}\cdot(\gamma_{1}(1)\ldots\gamma_{d^{n_{0}}-1}(1))^{d}.$

We can do the same for $\beta^{\prime}$ and obtain that

$-\beta^{\prime}=(-1)^{d^{n_{0}-1}}\cdot(\gamma_{1}(1)^{\prime}\ldots\gamma_{d^{n_{0}}-1}(1)^{\prime})^{d},$

with $(\gamma_{1}(1)^{\prime},\ldots,\gamma_{d^{n_{0}-1}}(1)^{\prime})$ in
$(f^{-n_{0}}(\beta^{\prime}))^{d^{n_{0}-1}}$.

Now we can iterate the procedure above and find a function $g:\mathbb{Z}_{\geq
1}\to\mathbb{F}_{2}$ such that for each $i$ in $\mathbb{Z}_{\geq 1}$ there are
two vectors $(\gamma_{1}(i),\ldots,\gamma_{d^{i\cdot
n_{0}-i}}(i))\in(f^{-n_{0}\cdot i}(\beta))^{d^{i\cdot n_{0}-i}}$ with

$-\beta=(-1)^{g(i)}\cdot(\gamma_{1}(i)\ldots\gamma_{d^{i\cdot
n_{0}-i}}(i))^{d^{i}}$

and

$-\beta^{\prime}=(-1)^{g(i)}\cdot(\gamma_{1}(i)^{\prime}\ldots\gamma_{d^{i\cdot
n_{0}-i}}(i)^{\prime})^{d^{i}}$

Recalling that the elements $\beta$ and $\beta^{\prime}$ are non-zero and
their ratio equals $\zeta_{d}$, we obtain that, for each $i$ in
$\mathbb{Z}_{\geq 1}$, $\zeta_{d}$ is a $d^{i}$-th power in $K_{i\cdot
n_{0}+1}(f,\alpha)$. That means that for each $i$ in $\mathbb{Z}_{\geq 0}$,
one has inside the field $K_{i\cdot n_{0}+1}(f,\alpha)$ an element of
multiplicative order precisely equal to $d^{i+1}$. This is the desired
conclusion.

∎

The next proposition will offer us a useful prime of periodic reduction in the
post-critically infinite case as an immediate consequence of a result in [20],
which, as explained therein, is a consequence of Siegel’s theorem on integral
points. As an alternative, we remark that in this special case one could
easily adapt the arguments given in [10, Thm 2.6]. We remark that [29, T.6]
obtains a much stronger conclusion for polynomials over $\mathbb{Q}$.

######  Proposition 3.2.

Let $K$ be a number field, $d$ in $\mathbb{Z}_{\geq 2}$ and $c\in K$ such that
$x^{d}-c$ is post-critically infinite. Let $\alpha$ be in $K$ different from
$-c$. Then there exists a positive integer $n_{0}$ and a prime $\mathfrak{p}$,
coprime to $d$, with $v_{\mathfrak{p}}(\alpha+c)=0$ and such that

$v_{\mathfrak{p}}(f^{n_{0}}(0))>0.$

###### Proof.

We will show that this is an immediate consequence of [20, Proposition 1.6,
(a)], which we can apply since $c\neq 0$. The orbit of $0$ takes negative
valuation only at those places where $c$ does, so finitely many. Since
$\alpha\neq c$ there are only finitely many valuation where $\alpha+c$ is non-
zero, and the places above prime divisors of $d$ are finite. Call $S$ the
finite set obtained collecting all these places listed so far. By [20,
Proposition 1.6, (a)], we know that there exists $n_{0}$ in $\mathbb{Z}_{\geq
1}$ such that $f^{n_{0}}(0)$ is not an $S$-unit. This means that there exists
a place $\mathfrak{p}$ outside of $S$ such that
$v_{\mathfrak{p}}(f^{n_{0}}(0))\neq 0$. Since $\mathfrak{p}$ is not in $S$,
the only possibility is that $v_{\mathfrak{p}}(f^{n_{0}}(0))>0$. Furthermore
since $\mathfrak{p}$ is not in $S$, we have that $\mathfrak{p}$ is coprime to
$d$ and $v_{\mathfrak{p}}(\alpha+c)=0$. This is precisely the desired
conclusion. ∎

To turn the combination of Proposition 3.1 and Proposition 3.2 into a lower
bound we need the following basic fact.

######  Proposition 3.3.

Let $\mathbb{F}$ be a finite field of cardinality coprime to an integer $d$.
Then there is a positive constant $c(d,\mathbb{F})$ such that the following
holds. If $\mathbb{L}/\mathbb{F}$ is a finite extension such that $\mathbb{L}$
has an element of multiplicative order equal to $d^{s}$, then

$[\mathbb{L}:\mathbb{F}]\geq c(d,\mathbb{F})\cdot d^{s}.$

###### Proof.

Denote by $q:=\\#\mathbb{F}$. Denote by $\Gamma$ the closure of the group
generated by $q$ in $\prod_{l|d}\mathbb{Z}_{l}^{*}$. Observe that $\Gamma$ is
open in $\prod_{l|d}\mathbb{Z}_{l}^{*}$. Define

$U(s):=\\{x\in\prod_{l|d}\mathbb{Z}_{l}^{*}:x\equiv 1\ \text{mod}\ d^{s}\\}.$

In order to prove the statement we need to show that there exists a positive
constant $c(d,\mathbb{F})$ such that for each positive integer $s$ we have
that

$[\Gamma:\Gamma\cap U(s)]\geq c(d,\mathbb{F})\cdot d^{s}.$

Indeed for $\mathbb{L}$ as in the statement we have

$[\mathbb{L}:\mathbb{F}]\geq[\Gamma:\Gamma\cap U(s)].$

Observe that there exists an isomorphism of profinite group, $\phi_{d}$,
sending

$\phi_{d}:U(2)\to\prod_{l|d}\mathbb{Z}_{l},$

such that $\phi_{d}(U(s))=d^{s-2}\cdot\prod_{l|d}\mathbb{Z}_{l}$ for each $s$
at least $2$. Finally observe that

$\phi_{d}(\Gamma\cap U(2))=\prod_{l|d}l^{f_{l}(\mathbb{F})}\mathbb{Z}_{l},$

for suitable positive integers $f_{l}(\mathbb{F})$. This is an immediate
consequence of the fact that $\Gamma$ is open in
$\prod_{l|d}\mathbb{Z}_{l}^{*}$. Define $m_{0}(\mathbb{F})$ to be the maximum
of the various $f_{l}(\mathbb{F})$, with $l|d$. We now see that

$[\Gamma\cap U(2):\Gamma\cap U(s)]\geq d^{s-m_{0}(\mathbb{F})},$

for each $s\geq m_{0}(\mathbb{F})$, which gives the desired conclusion.

∎

We are now ready to prove Theorem 1.3.
_Proof of Theorem 1.3:_ Assume first that $\alpha$ is different from $-c$.
Thanks to Proposition 3.2, we can find a prime $\mathfrak{p}$ such that $f$
reduces modulo $\mathfrak{p}$ to a polynomial of the form $x^{d}-c$ with the
orbit of $0$ periodic of period $n_{0}>0$, with $v_{\mathfrak{p}}(\alpha+c)=0$
and with $d$ coprime to $\mathfrak{p}$. Let us fix such a place
$\mathfrak{p}$. Let now $N$ be a positive integer and take
$\widetilde{\mathfrak{p}}_{N}$ a place above $\mathfrak{p}$ in
$K_{N}(f,\alpha)$. Let us denote by
$\mathbb{F}_{\widetilde{\mathfrak{p}}_{N}}$ and $\mathbb{F}_{\mathfrak{p}}$
the corresponding residue fields. Observe that $c$ is integral locally at
$\mathfrak{p}$, since $v_{\mathfrak{p}}(f^{n_{0}}(0))>0$, while
$v_{\mathfrak{p}}(c)<0$ certainly implies $v_{\mathfrak{p}}(f^{n_{0}}(0))<0$.
Likewise, we know that $\alpha$ is integral locally at $\mathfrak{p}$, since,
if not, we could not have $v_{\mathfrak{p}}(\alpha+c)=0$, since
$v_{\mathfrak{p}}(c)\geq 0$. Therefore we can reduce the entire set
$f^{-N}(\alpha)$ modulo $\widetilde{\mathfrak{p}_{N}}$ and apply Proposition
3.1. Hence we deduce that there is in
$\mathbb{F}_{\widetilde{\mathfrak{p}}_{N}}$ an element of multiplicative order
larger than $d^{\frac{N}{n_{0}}-1}$. Hence, via Proposition 3.3, we deduce
that

$[\mathbb{F}_{\widetilde{\mathfrak{p}}_{N}}:\mathbb{F}_{\mathfrak{p}}]\geq
c(d,\mathfrak{p})\cdot d^{\frac{N}{n_{0}}},$

where $c(d,\mathfrak{p})$ is a positive constant depending only on $d$ and the
place $\mathfrak{p}$. Recalling that
$[\mathbb{F}_{\widetilde{\mathfrak{p}}_{N}}:\mathbb{F}_{\mathfrak{p}}]$
divides $[K_{N}(f,\alpha):K],$ we obtain the desired conclusion. We are now
left with the case that $\alpha=-c$, which is equivalent to
$f^{-1}(\alpha)=\\{0\\}$. We can redo exactly the same argument for
$\alpha^{\prime}:=0$, which is certainly different from $-c$ since $f$ is
post-critically infinite. This provides a satisfactory bound for
$[K_{N}(f,\alpha^{\prime}):K]=[K_{N+1}(f,\alpha):K]$ and thus for
$D_{N}(f,\alpha)$ as desired.

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