# Shared Dynamically-Small Points for Polynomials on Average

Yan Sheng Ang<EMAIL_ADDRESS>and  Jit Wu Yap<EMAIL_ADDRESS>
###### Abstract.

Given two rational maps $f,g:\mathbb{P}^{1}\to\mathbb{P}^{1}$ of degree $d$
over $\mathbb{C}$, DeMarco–Krieger–Ye [DKY22] has conjectured that there
should be a uniform bound $B=B(d)>0$ such that either they have at most $B$
common preperiodic points or they have the same set of preperiodic points. We
study their conjecture from a statistical perspective and prove that the
average number of shared preperiodic points is zero for monic polynomials of
degree $d\geq 6$ with rational coefficients. We also investigate the quantity
$\liminf_{x\in\overline{\mathbb{Q}}}\left(\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\right)$
for a generic pair of polynomials and prove both lower and upper bounds for
it.

##  1\. Introduction

Let $f:\mathbb{P}^{1}\to\mathbb{P}^{1}$ be a rational function of degree
$d\geq 2$ defined over a field $K$. The set of preperiodic points of $f$,
denoted by $\operatorname{Prep}(f)$, is the set of points
$x\in\mathbb{P}^{1}(\overline{K})$ such that $f^{m}(x)=f^{n}(x)$ for two
distinct integers $m>n\geq 0$, where $\overline{K}$ denotes an algebraic
closure of $K$. Our starting point is the following finiteness theorem for the
common preperiodic points of two rational maps of degree $d\geq 2$.

######  Theorem 1.1 ([Cor. 1.3]baker-demarco[Thm. 1.3]yuan-zhang).

For any pair of rational functions $f,g:\mathbb{P}^{1}\to\mathbb{P}^{1}$
defined over $\mathbb{C}$ of degree $d\geq 2$, we have

$|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|<\infty\quad\text{or}\quad\operatorname{Prep}(f)=\operatorname{Prep}(g).$

We remark that the case of $f,g$ being defined over $\overline{\mathbb{Q}}$
had been proven previously by Mimar [Mim97]. We now consider two possible
directions in which we can study extensions of this result. The first is to
ask if the number of common preperiodic points can be uniformly bounded in
families of pairs $(f,g)$:

######  Conjecture 1.2 ([DKY22, Conj. 1.4]).

Let $d\geq 2$. There exists a constant $B=B(d)$ so that for every pair of
rational functions $f,g:\mathbb{P}^{1}\to\mathbb{P}^{1}$ defined over
$\mathbb{C}$ of degree $d$, we have

$|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|\leq
B\quad\text{or}\quad\operatorname{Prep}(f)=\operatorname{Prep}(g).$

In this direction, DeMarco–Krieger–Ye [DKY22, Thm. 1.1] obtain an effective
uniform bound on the family of quadratic polynomials of the form
$f(z)=z^{2}+c_{1}$, $g(z)=z^{2}+c_{2}$ with $c_{1},c_{2}\in\mathbb{C}$.
Mavraki–Schmidt [MS22, Thm. 1.3] prove a uniform bound along any one-parameter
family of pairs $(f,g)$ defined over ${\overline{\mathbb{Q}}}$. More recently,
DeMarco–Mavraki [DM23] proves the existence of a Zariski open
$U\subseteq\operatorname{Rat}_{d}\times\operatorname{Rat}_{d}$, where
$\operatorname{Rat}_{d}$ is the space of all degree $d$ rational maps, for
which a uniform bound holds.

Following Le Boudec–Mavraki [LM21] and also Olechnowicz [Ole23], we will take
a statistical point of view and instead consider the average count of common
preperiodic points over pairs of monic polynomials over $\mathbb{Q}$. We
consider only monic polynomials because the exceptional locus
$\operatorname{Prep}(f)=\operatorname{Prep}(g)$ is easily described as
$\\{f=g\\}$. We let

$\displaystyle\mathcal{P}$
$\displaystyle=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\,|\,a_{i}\in\mathbb{Q}\\},$
$\displaystyle\mathcal{P}_{c}$
$\displaystyle=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\in\mathcal{P}\,|\,a_{d-1}=0\\},$

be the set of monic polynomials and _monic, centered polynomials_ respectively
with rational coefficients. For $X\geq 1$, define

$\displaystyle\mathcal{P}(X)$
$\displaystyle=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\,|\,a_{i}\in\mathbb{Q},\,H(a_{i})\leq
X\\},$ $\displaystyle\mathcal{P}_{c}(X)$
$\displaystyle=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\in\mathcal{P}(X)\,|\,a_{d-1}=0\\},$

the set of monic polynomials and monic, centered polynomials respectively with
rational coefficients of Weil height $\leq X$ (see Section 2). Note that any
pair of monic polynomials $(f,g)$ over $\mathbb{Q}$ can be simultaneously
conjugated by a translation to lie in $\mathcal{P}(X)\times\mathcal{P}_{c}(X)$
for some $X$.

In Section 4, we prove: theoremmainthmprep Let $d\geq 6$, and write
$\mathcal{S}(X)=\mathcal{P}_{c}(X)\times\mathcal{P}(X)\setminus\\{(f,f)\,|\,f\in\mathcal{P}_{c}(X)\\}$.
Then

$\lim_{X\to\infty}\frac{1}{|\mathcal{S}(X)|}\sum_{(f,g)\in\mathcal{S}(X)}|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|=0$

where $\operatorname{Prep}(f),\operatorname{Prep}(g)$ denotes all preperiodic
points of $f$ and $g$ in $\overline{\mathbb{Q}}$.

######  Remark.

The condition $d\geq 6$ is purely technical and we certainly expect the
average to be zero for all degrees $d\geq 2$. With more work, we believe it is
possible to improve our lower bound on $d$. Also, many of our lemmas adapt
straightforwardly to number fields $K$ instead of $\mathbb{Q}$ (see Section
3), but it is not clear if we can obtain a version of Theorem 1.2 where we
vary over polynomials over a fixed number field $K$ instead of $\mathbb{Q}$.

Our arguments also allow us to conclude the average remain zero even when
restricted to coordinate slices of $\cal{P}$ of large enough dimension. Let
$m$ be a positive integer and let $V$ be a linear subspace of $\cal{P}$ of
dimension $m$ that can be written as an intersection of some coordinate
hyperplanes $\\{a_{i}=0\\}$. Let $x=(x_{d-1},\ldots,x_{0})\in\mathbb{Q}^{d-1}$
and let $A$ be the affine subspace given by $x+V$. We shall assume that the
projection of $V$ onto the last coordinate $a_{0}$ is surjective. Choose $m$
coordinates $a_{i_{1}},\ldots,a_{i_{m}}$ that give rise to an isomorphism
$V\simeq\mathbb{Q}^{m}$ such that $i_{j}$ are decreasing and $i_{m}=0$. We
define

$\cal{P}_{A}(X)=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\mid(a_{d-1},\ldots,a_{0})\in
A\text{ and }H(a_{i_{j}})\leq X)\\},$
$\cal{P}_{c,A}(X)=\\{z^{d}+a_{d-2}z^{d-2}+\cdots+a_{0}\mid(0,a_{d-2},\ldots,a_{0})\in
A\text{ and }H(a_{i_{j}})\leq X\\},$
$\cal{S}_{A}(X)=\cal{P}_{c,A}(X)\times\cal{P}_{A}(X)\setminus\\{(f,f)\mid
f\in\cal{P}_{c,A}(X)\\}.$

theoremmainthmprepsubspace Assume that $\dim_{\mathbb{Q}}V\geq\frac{d}{2}+3$.
Then

$\lim_{X\to\infty}\frac{1}{|\cal{S}_{A}(X)|}\sum_{(f,g)\in\cal{S}_{A}(X)}|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|=0.$

In another direction, when $f,g$ are defined over ${\overline{\mathbb{Q}}}$,
we can reinterpret the common preperiodic points of $f,g$ as the set of common
zeroes of the Call–Silverman canonical heights of $f,g$ [CS93]:

$\operatorname{Prep}(f)\cap\operatorname{Prep}(g)=\\{x\in{\overline{\mathbb{Q}}}\,|\,\widehat{h}_{f}(x)+\widehat{h}_{g}(x)=0\\}.$

In this setting, Baker–DeMarco and Mimar prove the following stronger form of
Theorem 1.1:

######  Theorem 1.3 (mimar[Thm. 1.8]Mimar2).

For any pair of rational maps $f,g:\mathbb{P}^{1}\to\mathbb{P}^{1}$ of degree
$d\geq 2$ defined over $\overline{\mathbb{Q}}$, we have

$\liminf_{x\in{\overline{\mathbb{Q}}}}\left(\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\right)>0\quad\text{or}\quad\widehat{h}_{f}=\widehat{h}_{g}.$

Here $\liminf_{x\in{\overline{\mathbb{Q}}}}F(x)$ refers to the infimum of all
accumulation points in $\mathbb{R}$ of the set
$\\{F(x)\,|\,x\in{\overline{\mathbb{Q}}}\\}$; equivalently,

$\liminf_{x\in{\overline{\mathbb{Q}}}}F(x)=\sup_{\begin{subarray}{c}S\subseteq{\overline{\mathbb{Q}}}\\\
|S|<\infty\end{subarray}}\inf_{x\in{\overline{\mathbb{Q}}}\setminus S}F(x).$

In particular if $\widehat{h}_{f}\not=\widehat{h}_{g}$, then there exists some
$\varepsilon=\varepsilon(f,g)>0$ such that
$\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\leq\varepsilon$ has only finitely many
solutions $x\in{\overline{\mathbb{Q}}}$; i.e. there are only finitely many
points $x\in\overline{\mathbb{Q}}$ that are both small for both
$\widehat{h}_{f}$ and $\widehat{h}_{g}$. A natural question to ask is how
large can $\varepsilon$ be for a generic pair $(f,g)$ as the height of $f$ and
$g$ grows. For any fixed $\delta>0$, applying Zhang’s inequality [Zha95, Thm.
1.10] (see Section 6) gives us

$\left(\frac{1}{2}-\delta\right)(h(f)+h(g))\leq
d\liminf_{x\in\overline{\mathbb{Q}}}\left(\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\right)\leq(1+\delta)(h(f)+h(g))$

for a generic pair $(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$ and $X$
large enough, where $h(f)$ is the height of a polynomial $f$ (see Section 2).
Our next theorem improves this inequality.

theoremmainthmbogomolov Fix a $\delta>0$. We let
$\cal{S}=\cal{P}_{c}\times\cal{P}$ and
$\cal{S}(X)=\cal{P}_{c}(X)\times\cal{P}(X)$. Then for all sufficiently large
degrees $d$, there exists $a=a(d)>0,A=A(d)>0$ such that the following
statement holds: there exists a subset $\cal{T}\subseteq\cal{S}$ such that for
all $X\geq 1$,

$\frac{|T\cap\cal{S}(X)|}{|\cal{S}(X)|}\geq 1-\frac{A}{X^{a}}$

and

$\left(\ln 2-\delta\right)(h(f)+h(g))\leq
d\liminf_{x\in\overline{\mathbb{Q}}}\left(\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\right)\leq\left(\mathcal{C}+\delta\right)(h(f)+h(g)),$

for all $(f,g)\in\cal{T}$, where $\ln 2=0.69314\ldots$ and
$\mathcal{C}=\log\left(\frac{9+\sqrt{17}}{8}\right)+\frac{\sqrt{17}-1}{8}=0.88532\ldots$

It will be interesting to determine the exact asymptotic, if it exists at all.

###  1.1. Strategy of proof

Our main tool for the proof of Theorems 1.2 and 1 is the energy pairing of
Favre–Rivera-Letelier in [FR06], see also [PST12]. Given a polynomial
$f:\mathbb{P}^{1}\to\mathbb{P}^{1}$ defined over a number field $K$ of degree
$d\geq 2$, for each place $v\in M_{K}$ we can attach to it a corresponding
equilibrium measure $\mu_{f,v}$. For archimedean $v$, this is a probability
measure on $\mathbb{C}$ while for non-archimedean $v$, it is a probability
measure on the Berkovich affine line
${\mathbb{A}^{1}_{\operatorname{Berk},v}}$. The collection of measures
$\\{\mu_{f,v}\\}_{v\in M_{K}}$ gives us an adelic measure, where $M_{K}$ are
the places of $K$. For further details on
${\mathbb{A}^{1}_{\operatorname{Berk},v}}$, one may consult [BR10] or [Ben19].

Given two polynomials $f,g$, Favre–Rivera-Letelier first define a local energy
pairing between $\mu_{f,v}$ and $\mu_{g,v}$ for a place $v$, and then define a
global energy pairing by summing the local pairings over $v\in M_{K}$. This
gives us a quantitative way to measure a distance between $f$ and $g$. Using
the energy pairing and quantitative equidistribution, DeMarco–Krieger–Ye were
able to establish a uniform bound on the number of common preperiodic points
for quadratic unicritical polynomials.

We follow their approach in using the adelic energy pairing, proving bounds
that hold for all monic polynomials in general but are weaker as we do not
need a uniform bound. The hardest situation to handle is when both polynomials
are close to each other. For example when $f(z)$ has integer coefficients and
$g(z)=f(z)+1$, the Julia sets at every non-archimedean place are the
corresponding Gauss point $\zeta(0,1)$ and so their local energy pairing is
zero. But if $f$ and $g$ have large coefficients, then they are also
relatively close for the archimedean places. To obtain a usable bound for
Theorems 1.2 and 1, we use the fact that the moments of the equilibrium
measure $\mu_{f,v}$ are determined by the coefficients of $f$ when $f$ is
monic (see Section 3.4).

Now for Theorem 1.3, we introduce a notion of genericity for pairs of
polynomials in $\mathcal{P}_{c}(X)\times\mathcal{P}(X)$ which may be of
independent interest. For an integer $n$, we let $\operatorname{rad}(n)$ be
the product of all distinct primes that divides $n$.

######  Definition 1.4.

Given $X$ and $\varepsilon>0$, we will say that a pair
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$ is _$\varepsilon$ -ordinary_
if

$\displaystyle\operatorname{rad}(\operatorname{denom}(c))$ $\displaystyle\geq
X^{1-2\varepsilon},$
$\displaystyle\gcd(\operatorname{denom}(c),\operatorname{denom}(c^{\prime}))$
$\displaystyle\leq X^{2\varepsilon},$

for any distinct coefficients $c,c^{\prime}$ of $f$ or $g$.

In other words, a pair $(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$ is
$\varepsilon$-ordinary if the denominators of the coefficients for both $f$
and $g$ are large, and are mostly coprime and squarefree. Similarly, we can
make an analogous definition for a single polynomial $f\in\cal{P}(X)$ to be
$\varepsilon$-ordinary. As a consequence of the definitions, for most places
$v$ of bad reduction of an $\varepsilon$-ordinary pair, there is only one
coefficient of $f,g$ that satisfies $|c|_{v}>1$.

The dynamics of an $\varepsilon$-ordinary polynomial $f$ at such a place $v$
is simple enough to be analyzed thoroughly, but still more complicated than
that of a unicritical polynomial. For example when $v$ is non-archimedean,
elements $x$ of the Julia set of $f$ have three possible absolute values
$|x|_{v}$( see Proposition 4.3) whereas that for the Julia set of a polynomial
of the form $z^{d}+c$ has one absolute value when $|c|_{v}>1$. In Section 6,
we study the properties of the dynamical Green’s function $g_{f,v}$ and
equilibrium measure $\mu_{f,v}$.

For such pairs, we control some measures of arithmetic complexity and the
structure of the Julia sets in Section 5. The lower bound of Theorem 1.3 then
becomes a consequence of the structure and the product formula. The upper
bound is more involved as it requires us to construct Galois orbits whose
local heights are not too large. We do so by constructing an appropriate
adelic set and then applying the adelic Fekete–Szegő theorem due to Rumely
[Rum14].

###  1.2. Acknowledgements

The authors would like to thank our advisor Laura DeMarco for helpful
discussions and Niki Myrto Mavraki for suggesting the problem.

##  2\. Definitions and notations

Here we collect some notation used throughout this paper.

The cardinality of a set $S$ is written as $|S|$ or $\\#S$.

We write $f=O(g)$ or $f\lesssim g$ for $f\leq Cg$ where $C>0$ is an absolute
constant; $f=O_{\alpha,\beta,\ldots}(g)$ or $f\lesssim_{\alpha,\beta,\ldots}g$
for $f\leq C_{\alpha,\beta,\ldots}g$ where $C_{\alpha,\beta,\ldots}>0$ is a
constant depending on parameters $\alpha,\beta,\ldots$; and $f=o(g)$ for
$\frac{f}{g}\to 0$. In such asymptotic notation, the limit to be taken will
either be explicitly stated or clear from context.

The set of preperiodic points of $f$ is denoted by $\operatorname{Prep}(f)$.

The Weil height $H$ on $\mathbb{Q}$ is defined by
$H(\frac{a}{b})=\max(|a|,|b|)$ for $a,b\in\mathbb{Z}$ with $b\neq 0$ and
$\gcd(a,b)=1$.

The radical of a positive integer $n$, denoted by $\operatorname{rad}(n)$, is
defined as the product of all distinct primes that divide $n$.

For a number field $K$, the set of all places for $K$ is denoted by $M_{K}$.

For each finite place $v\mid p<\infty$, the non-archimedean norm $|\cdot|_{v}$
on $\mathbb{C}_{v}$ is normalized with $|p|_{v}=\frac{1}{p}$. For a polynomial
$f\in\mathbb{Q}[z]$ given by $f(z)=z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}$, the
height of $f$ is defined as

$h(f)=h_{\mathbb{P}^{d}}([1:a_{d-1}:\cdots:a_{0}])=\sum_{v\in
M_{\mathbb{Q}}}\log\max(1,|a_{d-1}|_{v},\ldots,|a_{0}|_{v}).$

We now recall some notions from non-archimedean dynamics; we refer the reader
to [Ben19] for the theory of dynamics on Berkovich space, and to [BR10] for
non-archimedean potential theory.

For each finite place $v\mid p<\infty$, write
${\mathbb{A}^{1}_{\operatorname{Berk},v}}$ for the Berkovich affine line. Now
$\mathbb{C}_{v}$ embeds in ${\mathbb{A}^{1}_{\operatorname{Berk},v}}$ as the
type I points, which form a dense set under the weak/Gel’fand topology. Write
$\delta(z,w)_{\infty}$ for the Hsia kernel on
${\mathbb{A}^{1}_{\operatorname{Berk},v}}$, the unique continuous extension of
the metric $|z-w|_{v}$ on Type I points.

Write $\mu_{f,v}$ for the invariant measure of a polynomial
$f\in\mathbb{C}_{v}[z]$, and $g_{f,v}$ for its potential function. When $f$ is
monic, this potential is also equal to the escape rate function for $f$, i.e.,

$g_{f,v}(\zeta)=\lim_{n\to\infty}d^{-n}\log^{+}[f^{n}]_{\zeta}.$

The local energy pairing between two signed measures
$\mu_{v},\mu_{v}^{\prime}$ on ${\mathbb{A}^{1}_{\operatorname{Berk},v}}$ is
defined as

$(\mu_{v},\mu_{v}^{\prime})_{v}=-\int_{{\mathbb{A}^{1}_{\operatorname{Berk},v}}\times{\mathbb{A}^{1}_{\operatorname{Berk},v}}\setminus\operatorname{Diag}}\\!\log\delta(z,w)_{\infty}\,d\mu_{v}(z)\otimes
d\mu^{\prime}_{v}(w).$

The energy of $\mu_{v}$ is defined as

$I(\mu_{v})=\int_{{\mathbb{A}^{1}_{\operatorname{Berk},v}}\times{\mathbb{A}^{1}_{\operatorname{Berk},v}}}\\!\log\delta(z,w)_{\infty}\,d\mu_{v}(z)\otimes
d\mu_{v}(w),$

which is also equal to $-(\mu_{v},\mu_{v})_{v}$ when $\mu_{v}$ has continuous
potential. By abuse of notation, for any compact set
$E_{v}\subseteq{\mathbb{A}^{1}_{\operatorname{Berk},v}}$ we define the
logarithmic capacity of $E_{v}$ as

$I(E_{v})=\sup_{\mu_{v}}I(\mu_{v}),$

where the supremum is taken over all probability measures $\mu_{v}$ supported
on $E_{v}$. When this is greater than $-\infty$, we have
$I(E_{v})=I(\mu_{E_{v}})$, where $\mu_{E_{v}}$ is the equilibrium measure of
$E_{v}$.

An adelic measure is a collection of probability measures
$\\{\rho_{v}\\}_{v\in M_{K}}$ on ${\mathbb{A}^{1}_{\operatorname{Berk},v}}$
such that $\rho_{v}$ has continuous potentials and
$\rho_{v}=\delta_{\zeta(0,1)}$ for all but finitely many $v$. Following
Favre–Rivera-Letelier [FR06], we can define a global pairing between two
adelic measures $\rho,\rho^{\prime}$ by

$\langle\rho,\rho^{\prime}\rangle=\frac{1}{2}\sum_{v\in
M_{\mathbb{Q}}}(\rho_{v}-\rho^{\prime}_{v},\rho_{v}-\rho^{\prime}_{v})_{v}.$

For a finite
$\operatorname{Gal}({\overline{\mathbb{Q}}}/\mathbb{Q})$-invariant set
$F\subseteq{\overline{\mathbb{Q}}}$, write $[F]$ for the uniform probability
distribution on $F$, i.e.,

$[F]\coloneqq\frac{1}{|F|}\sum_{w\in F}\delta_{w}.$

Furthermore, for $\varepsilon>0$ we can define the regularized measure

$[F]_{\varepsilon}\coloneqq\frac{1}{|F|}\sum_{w\in F}\delta_{w,\varepsilon},$

where $\delta_{w,\varepsilon}$ is the uniform probability measure on $\partial
B(w,\varepsilon)$ for $v$ archimedean, or the delta mass on
$\zeta(w,\varepsilon)$ for $v$ non-archimedean.

For any polynomial $f\in\mathbb{Q}[z]$, we will generalize the definition of
the canonical height $\widehat{h}_{f}$ to finite
$\operatorname{Gal}({\overline{\mathbb{Q}}}/\mathbb{Q})$-invariant sets
$F\subseteq{\overline{\mathbb{Q}}}$ by

$\widehat{h}_{f}(F)\coloneqq\frac{1}{|F|}\sum_{w\in F}\widehat{h}_{f}(w).$

Note that since $\widehat{h}_{f}$ is constant on
$\operatorname{Gal}({\overline{\mathbb{Q}}}/\mathbb{Q})$-orbits, we have
$\widehat{h}_{f}(x)=\widehat{h}_{f}(F)$ when $F$ is the set of Galois
conjugates of $x\in{\overline{\mathbb{Q}}}$.

For a fixed degree $d$, we have the following sets

$\displaystyle\mathcal{P}$
$\displaystyle=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\,|\,a_{i}\in\mathbb{Q}\\},$
$\displaystyle\mathcal{P}_{c}$
$\displaystyle=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\in\mathcal{P}\,|\,a_{d-1}=0\\},$

of monic polynomials with rational coefficients and monic, centered
polynomials with rational coefficients respectively. For $X\geq 1$, we also
define

$\displaystyle\mathcal{P}(X)$
$\displaystyle=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\,|\,a_{i}\in\mathbb{Q},\,H(a_{i})\leq
X\\},$ $\displaystyle\mathcal{P}_{c}(X)$
$\displaystyle=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\in\mathcal{P}(X)\,|\,a_{d-1}=0\\},$

as subsets of $\mathcal{P}$ and $\mathcal{P}_{c}$ consisting of polynomials
with coefficients all having Weil height $\leq X$.

##  3\. Upper bound on energy pairing

Let $f,g$ be two monic polynomials of degree $d\geq 2$ defined over
$\mathbb{Q}$ and let $\mu_{f}=\\{\mu_{f,v}\\}_{v\in
M_{\mathbb{Q}}},\mu_{g}=\\{\mu_{g,v}\\}_{v\in M_{\mathbb{Q}}}$ be the adelic
measures associated to them. Here, $\mu_{f,v},\mu_{g,v}$ denotes the
equilibrium measures of $f$ and $g$ at the place $v$ respectively and they are
both probability measures supported on
${\mathbb{A}^{1}_{\operatorname{Berk},v}}$.

Recall from the proof outline for 1.2 that we want to bound the adelic energy
pairing

$\langle\mu_{f},\mu_{g}\rangle=\frac{1}{2}\sum_{v\in
M_{K}}\int_{{\mathbb{A}^{1}_{\operatorname{Berk},v}}\times{\mathbb{A}^{1}_{\operatorname{Berk},v}}\setminus\operatorname{Diag}}\\!\log\delta(z,w)_{\infty}\,d\mu_{v}(z)\otimes
d\mu^{\prime}_{v}(w)$

on both sides. In this section, we give an upper bound in terms of
$|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|$ and $h(f),h(g)$,
following the approach of DeMarco–Krieger–Ye [DKY22]. First, the energy
pairing $\langle\cdot,\cdot\rangle^{1/2}$ is a metric on adelic measures
(Lemma 3.17). Now by the Favre–Rivera-Letelier approach to quantitative
equidistribution [FR06], if
$F\subseteq\operatorname{Prep}(f)\cap\operatorname{Prep}(g)$ is Galois-
invariant and large, then for some suitable $\epsilon>0$, the adelic measure
$[F]_{\varepsilon}$ must be close to both $\mu_{f}$ and $\mu_{g}$ in this
metric; hence $\mu_{f}$ and $\mu_{g}$ must be close to each other, i.e.,
$\langle\mu_{f},\mu_{g}\rangle$ must be small.

As input to this argument, we prove technical estimates on the Hölder
continuity of the equilibrium potential $G_{f,v}$. We then finish the argument
using quantitative equidistribution.

###  3.1. Archimedean bounds

We will first handle the case where $v$ is an archimedean place. As all
computations here are local, we will omit $v$ from our notation.

We give a uniform Hölder bound on the equilibrium potential $G_{f}$
(Proposition 3.5), by an argument of Carleson–Gamelin [CG93, Thm. VIII.3.2].
We begin by bounding $G_{f}$ near the Julia set.

######  Proposition 3.1.

Let $\Omega=\\{z\in\mathbb{C}\,:\,\operatorname{dist}(z,K_{f})\leq 1\\}$.
Suppose $f$ has Lipschitz constant $A$ on $\Omega$, i.e.,

$\forall z,z^{\prime}\in\Omega\,[|f(z)-f(z^{\prime})|\leq A|z-z^{\prime}|].$

Also suppose that $G_{f}\leq M$ on $\Omega$ for some $M>0$. Then for all
$z\in\Omega$, we have

$G_{f}(z)\leq dM\operatorname{dist}(z,K_{f})^{\alpha},$

where $\alpha=\frac{\log d}{\log A}$.

###### Proof.

If $z\in K_{f}$, then both sides of the claimed inequality are 0.

Suppose $z\in\Omega\setminus K_{f}$. Then $z$ is in the attracting basin of
$\infty$, so there exists a minimal $n\geq 1$ for which
$f^{n}(z)\not\in\Omega$. Since $K_{f}$ is invariant under $f$, the Lipschitz
condition on $f$ gives

$\operatorname{dist}(f(z),K_{f})\leq A\operatorname{dist}(z,K_{f}).$

Now $f^{n}(z)\not\in\Omega$ implies $\operatorname{dist}(z,K_{f})\geq A^{-n}$,
and $f^{n-1}(z)\in\Omega$ implies

$M\geq G_{f}(f^{n-1}(z))=d^{n-1}G_{f}(z).$

Hence

$G_{f}(z)\leq dMd^{-n}=dM(A^{-n})^{\alpha}\leq
dM\operatorname{dist}(z,K_{f})^{\alpha}$

giving us our upper bound on $G_{f}(z)$. ∎

Now we give explicit values for the various parameters in Proposition 3.1.
First we bound the size of the filled Julia set $K_{f}$ in terms of the
coefficients of $f$.

######  Lemma 3.2.

Let $R_{f}\coloneqq 3\max(1,|a_{d-1}|,|a_{d-2}|^{1/2},\ldots,|a_{0}|^{1/d})$.
Then $K_{f}\subseteq B(0,R_{f})$.

###### Proof.

For $|z|\geq R_{f}$, we have for $i\geq 1$ that $|a_{d-i}z^{d-i}|\leq
3^{-i}|z|^{d}$. Hence

$|f(z)|\geq|z^{d}|\left(1-\frac{1}{3}-\frac{1}{3^{2}}-\cdots-\frac{1}{3^{d}}\right)\geq\frac{1}{2}|z^{d}|\geq\frac{3}{2}|z|.$

Hence $|f^{n}(z)|\geq\left(\frac{3}{2}\right)^{n}|z|$, so $z$ lies in the
attracting basin of $\infty$, i.e., $z\not\in K_{f}$. ∎

######  Proposition 3.3.

In Proposition 3.1, we may take

$M=\log(2R_{f}+1),\qquad A=\frac{3d}{2}(R_{f}+1)^{d-1},$

with $R_{f}$ as in Lemma 3.2.

###### Proof.

Note that $K_{f}\subseteq B(0,R_{f})$ implies $\Omega\subseteq B(0,R_{f}+1)$,
so $|z-w|\leq 2R_{f}+1$ for all $z\in\Omega$ and $w\in K_{f}$. Now
$\operatorname{supp}\mu_{f}=K_{f}$, so

$G_{f}(z)=\int\log|z-w|d\mu_{f}(w)\leq\log(2R_{f}+1),$

which proves the first claim.

Next, the Lipschitz constant of $f$ on $\Omega$ is at most
$\sup_{\Omega}|f^{\prime}|$. Now for $z\in\Omega\subseteq B(0,R_{f}+1)$, we
have

$\displaystyle|f^{\prime}(z)|$ $\displaystyle\leq
d|z^{d-1}|+(d-1)|a_{d-1}z^{d-2}|+\cdots+|a_{1}|$ $\displaystyle\leq
d\left(1+\frac{1}{3}+\frac{1}{3^{2}}\cdots\right)(R_{f}+1)^{d-1}$
$\displaystyle=\frac{3d}{2}(R_{f}+1)^{d-1},$

which proves the second claim. ∎

Since the equilibrium potential is asymptotic to $\log|\cdot|$ far from the
origin, the Hölder bound for $G_{f}$ near the Julia set (Proposition 3.1)
yields a global Hölder bound. In fact, we can take the same constants as in
Proposition 3.3:

######  Proposition 3.4.

Let $M,A$ be as in Proposition 3.3, and let $\alpha=\frac{\log d}{\log A}$.
For all $z\in\mathbb{C}$, we have

$G_{f}(z)\leq dM\operatorname{dist}(z,K_{f})^{\alpha}.$

###### Proof.

Let $x=\operatorname{dist}(z,K_{f})$. If $x\leq 1$, we are done by
Propositions 3.1 and 3.3.

Suppose that $x\geq 1$. We have $K_{f}\subseteq B(0,R_{f})$, so the diameter
of $K_{f}$ is at most $2R_{f}$. Hence for every $w\in K_{f}$ we have
$|z-w|\leq x+2R_{f}$, so

$G_{f}(z)=\int\log|z-w|d\mu_{f}(w)\leq\log(x+2R_{f}).$

Thus it suffices to show that $\log(x+2R_{f})\leq dMx^{\alpha}$.

_Case 1_ : $1\leq x\leq A$. Now $R_{f}\geq 3$, so

$e^{dM}=(2R_{f}+1)^{d}\geq 7(2R_{f}+1)^{d-1}\geq
6\left(\frac{7}{4}\right)^{d-1}(R_{f}+1)^{d-1}+2R_{f}\geq\frac{3d}{2}(R_{f}+1)^{d-1}+2R_{f}=A+2R_{f}.$

Hence

(1) $\log(x+2R_{f})\leq\log(A+2R_{f})\leq dM\leq dMx^{\alpha}.$

_Case 2_ : $x\geq A$. Write $x=A^{c}$ with $c\geq 1$; now $x\geq A\geq
2R_{f}\geq 2$ , so

$\log(x+2R_{f})\leq c\log A+\log 2\leq(c+1)\log A\leq d^{c}\log A\leq
dMd^{c}=dMx^{\alpha},$

where the last inequality follows from $\log A\leq dM$, by (1). ∎

We can now show uniform Hölder continuity for $G_{f}$.

######  Proposition 3.5.

Let $M,A$ be as in Proposition 3.3, and let $\alpha=\frac{\log d}{\log A}$.
For all $z_{1},z_{2}\in\mathbb{C}$, we have

$|G_{f}(z_{1})-G_{f}(z_{2})|\leq 3dM|z_{1}-z_{2}|^{\alpha}.$

###### Proof.

[CG93, Thm. VIII.3.2] Write $r_{i}=\operatorname{dist}(z_{i},K_{f})$; we may
assume that $r_{1}\geq r_{2}$. Note that $A=\frac{3d}{2}(R_{f}+1)^{d-1}\geq
d$, so that $\alpha\leq 1$.

_Case 1_ : $|z_{1}-z_{2}|\geq\frac{1}{3r_{1}}$. Now $0\leq G_{f}(z_{i})\leq
dMr_{i}^{\alpha}$ by Proposition 3.4, so

$|G_{f}(z_{1})-G_{f}(z_{2})|\leq dMr_{1}^{\alpha}\leq
3dM|z_{1}-z_{2}|^{\alpha}.$

_Case 2_ : $|z_{1}-z_{2}|=cr_{1}$, with $c\leq\frac{1}{3}$. Now $G_{f}$ is
harmonic and non-negative on $B(z_{1},r_{1})$, so by Harnack’s inequality we
have

$\frac{1-c}{1+c}G_{f}(z_{1})\leq G_{f}(z_{2})\leq\frac{1+c}{1-c}G_{f}(z_{1}).$

Therefore

$|G_{f}(z_{1})-G_{f}(z_{2})|\leq\frac{2c}{1-c}G_{f}(z_{1})\leq
3cdMr_{1}^{\alpha}\leq 3dM|z_{1}-z_{2}|^{\alpha}.$

as desired. ∎

######  Remark.

The arguments in Propositions 3.1 and 3.5 are essentially contained in [CG93,
Thm. VIII.3.2]; however, we note that their argument is incomplete, since they
do not control the equilibrium potential for large $z$ as in our proof of
Proposition 3.4.

The Hölder bound on $G_{f}$ allows us to control the local pairing of
$\mu_{f}$ with a regularized measure:

######  Corollary 3.6.

Let $M,A$ be as in Proposition 3.3, and let $\alpha=\frac{\log d}{\log A}$.
For any $\varepsilon>0$ and any nonempty finite set $F\subseteq\mathbb{C}$, we
have

$-(\mu_{f},[F]_{\varepsilon})\leq-(\mu_{f},[F])+3dM\varepsilon^{\alpha}.$

###### Proof.

Since $|w-\xi|=\varepsilon$ for all $\xi\in\partial B(w,\varepsilon)$, we have

$-(\mu_{f},[F]_{\varepsilon})=\frac{1}{|F|}\sum_{w\in
F}\int_{\mathbb{C}}G_{f}(\xi)d\delta_{w,\varepsilon}(\xi)\leq\frac{1}{|F|}\sum_{w\in
F}(G_{f}(w)+3dM\varepsilon^{\alpha})=-(\mu_{f},[F])+3dM\varepsilon^{\alpha}$

as desired. ∎

###  3.2. Non-archimedean bounds

Now we consider the non-archimedean case $v\mid p<\infty$ with $v\in M_{K}$.
By choosing an appropriate embedding $K\hookrightarrow\mathbb{C}_{p}$, we can
view $|\cdot|_{v}=|\cdot|_{p}$.

We first show that the Lipschitz constant of $f$ on a closed subset of
${\mathbb{A}^{1}_{\operatorname{Berk},v}}$ is the same as the Lipschitz
constant on its Type I points.

######  Lemma 3.7.

Let $\Omega\subseteq{\mathbb{A}^{1}_{\operatorname{Berk},v}}$ be a closed
subset. Suppose there is a constant $A>0$ such that for all
$z,z^{\prime}\in\Omega\cap\mathbb{C}_{v}$, we have

$|f(z)-f(z^{\prime})|_{v}\leq A|z-z^{\prime}|_{v}.$

Then for all $\zeta,\zeta^{\prime}\in\Omega$, we have

$\delta(f(\zeta),f(\zeta^{\prime}))_{\infty}\leq
A\delta(\zeta,\zeta^{\prime})_{\infty}.$

###### Proof.

_Case 1_ : $f(\zeta)\neq f(\zeta^{\prime})$. Take sequences
$(z_{n}),(z^{\prime}_{n})\subseteq\mathbb{C}_{v}$ such that $(z_{n})\to\zeta$
and $(z^{\prime}_{n})\to\zeta^{\prime}$. Since $\delta(\cdot,\cdot)_{\infty}$
is continuous off the diagonal, we have

$\displaystyle\delta(f(\zeta),f(\zeta^{\prime}))_{\infty}$
$\displaystyle=\lim_{n\to\infty}|f(z_{n})-f(z^{\prime}_{n})|_{v}$
$\displaystyle\leq A\lim_{n\to\infty}|z_{n}-z^{\prime}_{n}|_{v}$
$\displaystyle=A\delta(\zeta,\zeta^{\prime})_{\infty}.$

_Case 2_ : $f(\zeta)=f(\zeta^{\prime})$. Write $\zeta=\zeta(a,r)$ and
$f(\zeta)=\zeta(b,R)$; then [Ben19, Prop. 7.6] gives $f(B(a,r))=B(b,R)$, so

$\delta(f(\zeta),f(\zeta^{\prime}))_{\infty}=R\leq Ar\leq
A\delta(\zeta,\zeta^{\prime})_{\infty}.\qed$

From here on, the structure of the argument for the non-archimedean case is
analogous to the archimedean case in the previous section.

######  Proposition 3.8.

Let
$\Omega_{v}=\\{z\in\mathbb{C}_{v}\,:\,\operatorname{dist}_{v}(z,K_{f,v})\leq
1\\}$. Suppose $f$ has Lipschitz constant $A_{v}>1$ on $\Omega_{v}$, i.e.,

$\forall z,z^{\prime}\in\Omega_{v}\,[|f(z)-f(z^{\prime})|_{v}\leq
A_{v}|z-z^{\prime}|_{v}].$

Also suppose that $G_{f,v}\leq M_{v}$ on $\Omega_{v}$ for some $M_{v}>0$. Then
for all $z\in\Omega_{v}$, we have

$G_{f,v}(z)\leq dM_{v}\operatorname{dist}_{v}(z,K_{f,v})^{\alpha_{v}},$

where $\alpha_{v}=\frac{\log d}{\log A_{v}}$.

###### Proof.

In view of Lemma 3.7, we may replace $\Omega_{v}$ with its closure
$\overline{\Omega_{v}}$ in ${\mathbb{A}^{1}_{\operatorname{Berk},v}}$. The
proof is now completely analogous to the proof of Proposition 3.1. ∎

######  Proposition 3.9.

Let
$R_{f,v}:=\max(1,|a_{d-1}|_{v},|a_{d-2}|_{v}^{1/2},\ldots,|a_{0}|_{v}^{1/d})$.
Then $K_{f,v}\subseteq\overline{B}(0,R_{f,v})$.

###### Proof.

For $|z|_{v}>R_{f,v}$ and $i\geq 1$, we have
$|a_{d-i}z^{d-i}|_{v}<|z|_{v}^{d}$. Hence $|f(z)|_{v}=|z|_{v}^{d}$. By
iteration we see that $|f^{n}(z)|_{v}=|z|_{v}^{d^{n}}$, so $z$ lies in the
attracting basin of $\infty$, i.e., $z\not\in K_{f,v}$. ∎

######  Proposition 3.10.

Suppose that $f$ has bad reduction at $v$. In Proposition 3.8, we may take

$M_{v}=\log R_{f,v},\qquad A_{v}=R_{f,v}^{d-1},$

with $R_{f,v}$ as in Proposition 3.9.

###### Proof.

First, $f$ having bad reduction at $v$ implies $R_{f,v}>1$. Now
$K_{f,v}\subseteq\overline{B}(0,R_{f,v})$ implies that
$\Omega_{v}\subseteq\overline{B}(0,R_{f,v})$ as well, so $|z-w|_{v}\leq
R_{f,v}$ for all $z\in\Omega_{v}$ and $w\in K_{f,v}$. Now
$\operatorname{supp}\mu_{f,v}=K_{f,v}$, so

$G_{f,v}(z)=\int\log|z-w|d\mu_{f,v}(w)\leq\log R_{f,v},$

which proves the first claim.

Next, note that for all $z\in\Omega_{v}\subseteq\overline{B}(0,R_{f,v})$ we
have

$|f(z)|\leq\max_{i}|a_{d-i}z^{d-i}|_{v}\leq R_{f,v}^{d}.$

In other words, we have
$f(\overline{B}(0,R_{f,v}))\subseteq\overline{B}(0,R_{f,v}^{d})$. It now
follows from [Ben19, Prop. 3.20] that the Lipschitz constant for $f$ on
$B(0,R_{f,v})$ is bounded above by $R_{f,v}^{d-1}$. Thus the same also holds
on $\Omega_{v}$, which proves the second claim. ∎

######  Remark.

When $f$ has explicit good reduction at $v$, we have $R_{v}=1$, so $A_{v}=1$,
thus $\alpha_{v}=\frac{\log d}{\log A_{v}}$ is not well-defined, and
Proposition 3.8 cannot be applied. However, in this case we have
$J_{f,v}=\\{\zeta(0,1)\\}$, so
$\Omega_{v}=K_{f,v}\cap\mathbb{C}_{v}=\overline{B}(0,1)$, and thus
$G_{f,v}\equiv 0$ on $\Omega_{v}$.

From now on in this subsection, we will assume that $f$ has bad reduction at
$v$.

######  Proposition 3.11.

Let $v,M_{v},A_{v}$ be as in Proposition 3.10, and let $\alpha_{v}=\frac{\log
d}{\log A_{v}}$. For all $z\in\mathbb{C}_{v}$, we have

$G_{f,v}(z)\leq dM\operatorname{dist}(z,K_{f,v})^{\alpha_{v}}.$

###### Proof.

Let $x=\operatorname{dist}(z,K_{f,v})$. If $x\leq 1$, we are done by
Propositions 3.1 and 3.10.

Suppose that $x\geq 1$. We have $K_{f,v}\subseteq B(0,R_{f,v})$, so the
diameter of $K_{f,v}$ is at most $R_{f,v}$. Hence for every $w\in K_{f,v}$ we
have $\delta(z,w)_{\infty}\leq\max(x,R_{f,v})$, so

$G_{f,v}(z)=\int\log\delta(z,w)_{\infty}d\mu_{f,v}(w)\leq\log\max(x,R_{f,v}).$

Thus it suffices to show that $\log\max(x,R_{f,v})\leq dM_{v}x^{\alpha_{v}}$.

Now $\log R_{f,v}=M_{v}\leq dM_{v}x^{\alpha_{v}}$; also, writing $x=A_{v}^{c}$
for some $c\geq 0$, we have

$\log xc(d-1)\log R_{f,v}\leq d^{c+1}\log
R_{f,v}=dM_{v}d^{c}=dM_{v}x^{\alpha_{v}}.\qed$

######  Proposition 3.12.

Let $v,M_{v},A_{v}$ be as in Proposition 3.10, and let $\alpha_{v}=\frac{\log
d}{\log A_{v}}$. For all
$z_{1},z_{2}\in{\mathbb{A}^{1}_{\operatorname{Berk},v}}$, we have

$|G_{f,v}(z_{1})-G_{f,v}(z_{2})|\leq dM_{v}|z_{1}-z_{2}|_{v}^{\alpha_{v}}.$

###### Proof.

Write $r_{i}=\operatorname{dist}_{v}(z_{i},K_{f,v})$; we may assume that
$r_{1}\geq r_{2}$.

_Case 1_ : $|z_{1}-z_{2}|_{v}\geq r_{1}$. Now $0\leq G_{f,v}(z_{i})\leq
dM_{v}r_{i}^{\alpha_{v}}$ by Proposition 3.8, so

$|G_{f,v}(z_{1})-G_{f,v}(z_{2})|\leq dM_{v}r_{1}^{\alpha_{v}}\leq
dM_{v}|z_{1}-z_{2}|^{\alpha_{v}}.$

_Case 2_ : $|z_{1}-z_{2}|_{v}<r_{1}$. Then for all $w\in
K_{f,v}=\operatorname{supp}\mu_{f,v}$, we have

$\delta(z_{1},w)_{\infty}\geq r_{1}>|z_{1}-z_{2}|,$

so that $\delta(z_{1},w)_{\infty}=\delta(z_{2},w)_{\infty}$. Hence

$G_{f,v}(z_{1})=\int\log\delta(z_{1},w)_{\infty}d\mu_{f,v}(w)=\int\log\delta(z_{2},w)_{\infty}d\mu_{f,v}(w)=G_{f,v}(z_{2}).\qed$

######  Corollary 3.13.

Let $M_{v},A_{v}$ be as in Proposition 3.10, and let $\alpha_{v}=\frac{\log
d}{\log A_{v}}$. For any $\varepsilon>0$ and any nonempty finite set
$F\subseteq\mathbb{C}_{v}$, we have

$-(\mu_{f,v},[F]_{\varepsilon})_{v}\leq-(\mu_{f,v},[F])_{v}+dM_{v}\varepsilon^{\alpha_{v}}.$

###### Proof.

Since $\delta(\zeta(w,\varepsilon),w)_{\infty}=\varepsilon$, we have

$-(\mu_{f,v},[F]_{\varepsilon})_{v}=\frac{1}{|F|}\sum_{w\in
F}G_{f,v}(\zeta(w,\varepsilon))\leq\frac{1}{|F|}\sum_{w\in
F}(G_{f,v}(w)+dM_{v}\varepsilon^{\alpha_{v}})=-(\mu_{f,v},[F])_{v}+dM_{v}\varepsilon^{\alpha_{v}}.\qed$

###  3.3. Quantitative equidistribution

To summarize the discussion in Sections 3.1 and 3.2, we have shown:

######  Theorem 3.14.

Let $\varepsilon>0$. For any place $v\in M_{K}$ such that either $v\mid\infty$
or $f$ has bad reduction at $v$, and any nonempty finite set
$F\subseteq\mathbb{C}_{v}$, we have

$-(\mu_{f},[F]_{\varepsilon})_{v}\leq-(\mu_{f},[F])_{v}+3dM_{v}\varepsilon^{\alpha_{v}},$

with constants $\alpha_{v}=\frac{\log d}{\log A_{v}},$

$M_{v}=\begin{cases}\log(2R_{f,\infty}+1)\quad v\mid\infty,\\\ \log
R_{f,v},\quad
v\nmid\infty,\end{cases}A_{v}=\begin{cases}\frac{3d}{2}(R_{f,\infty}+1)^{d-1}\quad
v\mid\infty,\\\ R_{f,v}^{d-1}\quad v\nmid\infty,\end{cases}$
$R_{f,v}=\begin{cases}3\max(1,|a_{d-1}|_{\infty},|a_{d-2}|_{\infty}^{1/2},\ldots,|a_{0}|_{\infty}^{1/d})\quad
v\mid\infty,\\\
\max(1,|a_{d-1}|_{v},|a_{d-2}|_{v}^{1/2},\ldots,|a_{0}|_{v}^{1/d})\quad
v\nmid\infty.\end{cases}$

###### Proof.

This is immediate from Corollaries 3.6 and 3.13. ∎

To convert these Hölder bounds into an upper bound for the energy pairing
$\langle\mu_{f},\mu_{g}\rangle$, we show the following quantitative
equidistribution result: for any large Galois-invariant subset
$F\subseteq\operatorname{Prep}(f)$, there is some small adelic radius
$\varepsilon$ such that the regularized adelic measure $[F]_{\varepsilon}$ is
close to $\mu_{f}$.

######  Lemma 3.15 ([Lem. 4.11]FRL06[Lem. 12]Fil17).

For any place $v\in M_{K}$, any nonempty finite set
$F\subseteq\mathbb{C}_{v}$, and any $\varepsilon>0$, we have

$([F]_{\varepsilon},[F]_{\varepsilon})_{v}\leq([F],[F])_{v}+\frac{\log(1/\varepsilon)}{|F|}.$

######  Proposition 3.16.

With notation as in Theorem 3.14, let $F\subseteq K$ be a finite
$\operatorname{Gal}({\overline{\mathbb{Q}}}/\mathbb{Q})$-invariant set with
$|F|=N\geq 2$. Define the adelic radius
$\varepsilon=\\{\varepsilon_{v}\\}_{v\in M_{K}}$ as

$\varepsilon_{v}=\begin{cases}(dN)^{-1/\alpha_{v}}&\text{if
}v\mid\infty\text{, or }f\text{ has bad red.\ at }v,\\\ 1&\text{if }f\text{
has explicit good red.\ at }v.\end{cases}$

Then

$\langle\mu_{f},[F]_{\varepsilon}\rangle\leq\widehat{h}_{f}(F)+O\left(d\frac{\log
N}{N}(h(f)+1)\right).$

###### Proof.

We extend an argument of [DKY22, Lem. 9.2]. Let $\mathcal{B}\subseteq M_{K}$
be the set of places $v$ such that either $v\mid\infty$, or $f$ has bad
reduction at $v$.

First, by using $(\mu_{f},\mu_{f})_{v}=0$, we expand the left side of the
claim as

(2)
$\langle\mu_{f},[F]_{\varepsilon}\rangle=\frac{1}{2[K:\mathbb{Q}]}\sum_{v\in
M_{K}}\\!n_{v}([F]_{\varepsilon}-\mu_{f},[F]_{\varepsilon}-\mu_{f})_{v}$
$=\frac{1}{[K:\mathbb{Q}]}\sum_{v\in
M_{K}}\\!n_{v}\left(\frac{1}{2}([F]_{\varepsilon},[F]_{\varepsilon})_{v}-(\mu_{f},[F]_{\varepsilon})_{v}\right).$

We bound the first term in the sum of eq. 2 by Lemma 3.15 and the product
formula $\sum_{v}n_{v}([F],[F])_{v}=0$, to obtain

(3) $\frac{1}{2[K:\mathbb{Q}]}\sum_{v\in
M_{K}}\\!n_{v}([F]_{\varepsilon},[F]_{\varepsilon})_{v}\leq\frac{1}{[K:\mathbb{Q}]}\sum_{v\in
M_{K}}\\!n_{v}\frac{\log(1/\varepsilon_{v})}{2N}=\frac{1}{[K:\mathbb{Q}]}\sum_{v\in\mathcal{B}}n_{v}\frac{1+\frac{\log
N}{\log d}}{2N}\log A_{v}.$

Next, we bound the second term in the sum of (2). If $v\in\mathcal{B}$ then we
can use Theorem 3.14. Suppose $v\in M_{k}\setminus\mathcal{B}$, i.e., $f$ has
explicit good reduction at $v$. Then $\varepsilon_{v}=1$, and
$\mu_{f,v}=\delta_{\zeta(0,1)}$, so $G_{f,v}=\log^{+}|\cdot|_{v}$. Hence

$-(\mu_{f},[F]_{\varepsilon})_{v}=\frac{1}{|F|}\sum_{w\in
F}\log^{+}|\zeta(w,1)|_{v}=\frac{1}{|F|}\sum_{w\in
F}\log^{+}\max(1,|w|_{v})=\frac{1}{|F|}\sum_{w\in
F}\log^{+}|w|_{v}=-(\mu_{f},[F])_{v}.$

The above two cases yield

(4) $\frac{1}{[K:\mathbb{Q}]}\sum_{v\in
M_{K}}\\!n_{v}(-(\mu_{f},[F]_{\varepsilon})_{v})\leq\frac{1}{[K:\mathbb{Q}]}\left(\sum_{v\in
M_{K}}\\!n_{v}(-(\mu_{f},[F])_{v})+\sum_{v\in\mathcal{B}}n_{v}(3dM_{v}\varepsilon_{v}^{\alpha_{v}})\right)$
$=\widehat{h}_{f}(F)+\frac{1}{[K:\mathbb{Q}]}\sum_{v\in\mathcal{B}}n_{v}(3dM_{v}\varepsilon_{v}^{\alpha_{v}})=\widehat{h}_{f}(F)+\frac{1}{[K:\mathbb{Q}]}\sum_{v\in\mathcal{B}}n_{v}\frac{3M_{v}}{N}.$

Now (2), (3), (4), imply

(5)
$\langle\mu_{f},[F]_{\varepsilon}\rangle-\widehat{h}_{f}(F)\lesssim\frac{\log
N}{N}\frac{1}{[K:\mathbb{Q}]}\sum_{v\in\mathcal{B}}n_{v}(M_{v}+\log
A_{v})\leq\frac{\log N}{N}\sum_{v\in M_{\mathbb{Q}}}(M_{v}+\log A_{v}).$

We now bound the sum in (5). Note that

$\sum_{v\in M_{\mathbb{Q}}}\log R_{f,v}=\sum_{v\nmid\infty}\log R_{f,v}+\log
R_{f,\infty}\leq h(f)+O(1).$

For the first term $M_{v}$ in eq. 5, we have

(6) $\sum_{v\in M_{\mathbb{Q}}}M_{v}\leq\sum_{v\nmid\infty}\log
R_{f,v}+\log(2R_{f,\infty}+1)\leq\sum_{v\nmid\infty}\log R_{f,v}+\log
R_{f,\infty}+O(1)\leq h(f)+O(1).$

where the first inequality follows from Theorem 3.14. Similarly, for the
second term $\log A_{v}$ in (5), we deduce

(7) $\sum_{v\in M_{\mathbb{Q}}}\log A_{v}\leq\sum_{v\nmid\infty}(d-1)\log
R_{f,v}+(d-1)\log(R_{f,\infty}+1)+\log(\tfrac{3}{2}d)\leq(d-1)h(f)+O(d).$

Substituting (6),(7) back into (5), we deduce

$\langle\mu_{f},[F]_{\varepsilon}\rangle-\widehat{h}_{f}(F)\lesssim
d\frac{\log N}{N}(h(f)+1).\qed$

We can now show that $\mu_{f}$ and $\mu_{g}$ are close to each other, thereby
proving our upper bound for the adelic energy pairing.

######  Lemma 3.17 ([Fil17, Thm. 1]).

For any adelic measures $\rho,\sigma,\tau$, we have

$\langle\rho,\tau\rangle^{1/2}\leq\langle\rho,\sigma\rangle^{1/2}+\langle\sigma,\tau\rangle^{1/2}.$

######  Theorem 3.18.

Suppose $N=|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|\geq 2$. Then

$\langle\mu_{f},\mu_{g}\rangle\lesssim d\frac{\log N}{N}(\max(h(f),h(g))+1).$

###### Proof.

Take $F=\operatorname{Prep}(f)\cap\operatorname{Prep}(g)$, so that
$\widehat{h}_{f}(F)=\widehat{h}_{g}(F)=0$. Then by Lemmas 3.17 and 3.16, we
have

$\langle\mu_{f},\mu_{g}\rangle\leq\left(\langle\mu_{f},[F]_{\varepsilon}\rangle^{1/2}+\langle\mu_{g},[F]_{\varepsilon}\rangle^{1/2}\right)^{2}\lesssim
d\frac{\log N}{N}\left(\sqrt{h(f)+1}+\sqrt{h(g)+1}\right)^{2}$ $\lesssim
d\frac{\log N}{N}(\max(h(f),h(g))+1).\qed$

######  Remark.

The proof of Theorem 3.18 in fact carries over to any pair of monic $f,g\in
K[z]$, with the same implied constant independent of the field $K$.

We will also use the quantitative equidistribution bound of Proposition 3.16
to give a $O(\log X)$ bound in some cases where we know the measures differ
significantly in a non-archimedean place.

######  Lemma 3.19.

Let $v\in M_{K}$ be non-archimedean. Let
$\varphi:{\mathbb{A}^{1}_{\operatorname{Berk},v}}\to\mathbb{R}$ be continuous
with compact support, and let $\mu$ be a measure on
${\mathbb{A}^{1}_{\operatorname{Berk},v}}$ with total mass 0 and continuous
potential. Then

$\left|\int_{\mathbb{A}^{1}_{\operatorname{Berk},v}}\varphi\operatorname{d}\mu\right|\leq(\triangle\varphi,\triangle\varphi)_{v}^{1/2}(\mu,\mu)_{v}^{1/2}.$

###### Proof.

If the right side is $\infty$ then there is nothing to prove. Otherwise, let
$\mu=\triangle\psi$. By [BR10, Cor. 5.38], the expression
$[F,G]:=\int_{\mathbb{A}^{1}_{\operatorname{Berk},v}}F\triangle G$ defines a
positive semidefinite symmetric bilinear form on
$\\{F\in\operatorname{BDV}({\mathbb{A}^{1}_{\operatorname{Berk},v}})\,:\,[F,F]<\infty\\}$.
Now

$[\varphi,\psi]=\int_{\mathbb{A}^{1}_{\operatorname{Berk},v}}\varphi\operatorname{d}\mu,\qquad[\varphi,\varphi]=(\triangle\varphi,\triangle\varphi)_{v},\qquad[\psi,\psi]=(\mu,\mu)_{v},$

and we are done by the Cauchy–Schwarz inequality.

∎

######  Proposition 3.20.

Let $v\in M_{\mathbb{Q}}$ be non-archimedean, with $p$ its associated prime.
If

$\mu_{g,v}({\mathbb{A}^{1}_{\operatorname{Berk},v}}\setminus
D_{\operatorname{an}}(0,p^{1/d}))\geq 1/d,$

then there are at most $C_{d}(h(g)^{2}+1)\log^{2}p$ points
$x\in\operatorname{Prep}(g)$ such that $|\sigma(x)|_{v}\leq 1$ for all
$\sigma\in\operatorname{Gal}({\overline{\mathbb{Q}}}/\mathbb{Q})$.

###### Proof.

Consider the path segment $T$ that connects $\zeta(0,1)$ and
$\zeta(0,p^{1/d})$ in ${\mathbb{A}^{1}_{\operatorname{Berk},v}}$, namely
$T=\\{\zeta(0,p^{t/d})\,:\,t\in[0,1]\\}$. Let $\varphi$ be the continuous
function on ${\mathbb{A}^{1}_{\operatorname{Berk},v}}$ which is locally
constant on ${\mathbb{A}^{1}_{\operatorname{Berk},v}}\setminus T$, and takes
values $\varphi(\zeta(0,p^{t/d}))=t$ on $T$. We check that

$\triangle\varphi=\frac{d}{\log
p}\left(\delta_{\zeta(0,1)}-\delta_{\zeta(0,p^{1/d})}\right).$

Now if $F$ is a nonempty
$\operatorname{Gal}({\overline{\mathbb{Q}}}/\mathbb{Q})$-invariant set of
preperiodic points for $g$ contained in $\overline{B}(0,1)$, take the adelic
radius $\varepsilon$ as in Proposition 3.16. Since $0<\varepsilon_{v}\leq 1$,
we have $\zeta(w,\varepsilon_{v})\in\overline{D}_{\operatorname{an}}(0,1)$ for
all $w\in F$. Hence

$\int_{\mathbb{A}^{1}_{\operatorname{Berk},v}}\varphi
d[F]_{\varepsilon_{v}}=0,\qquad\int_{\mathbb{A}^{1}_{\operatorname{Berk},v}}\varphi\operatorname{d}\mu_{g,v}\geq\frac{1}{d}.$

Now Proposition 3.16 gives

$d\frac{\log
N}{N}(h(g)+1)\gtrsim\langle\mu_{g},[F]_{\varepsilon}\rangle=\frac{1}{2}\sum_{w\in
M_{\mathbb{Q}}}(\mu_{g}-[F]_{\varepsilon},\mu_{g}-[F]_{\varepsilon})_{w}\geq\frac{1}{2}(\mu_{g}-[F]_{\varepsilon},\mu_{g}-[F]_{\varepsilon})_{v},$

since the terms in the sum are non-negative. Then by Lemma 3.19,

$\frac{1}{d}\leq\left|\int\varphi\operatorname{d}[F]_{\varepsilon_{v}}-\int\varphi\operatorname{d}\mu_{g,v}\right|\leq(\triangle\varphi,\triangle\varphi)^{1/2}_{v}([F]_{\varepsilon_{v}}-\mu_{g,v},[F]_{\varepsilon_{v}}-\mu_{g,v})^{1/2}_{v}\lesssim\left(\frac{d}{\log
p}\right)^{1/2}\left(d\frac{\log N}{N}(h(g)+1)\right)^{1/2},$

which yields $\frac{N}{\log N}\lesssim d^{4}(h(g)+1)\log p$. Therefore
$N\lesssim_{d}(h(g)^{2}+1)\log^{2}p$. ∎

###  3.4. Lower bound on the archimedean energy pairing

We now prove the lower bound on $\langle\mu_{f},\mu_{g}\rangle$. In
particular, we will give a lower bound on the local energy pairing
$(\mu_{f}-\mu_{g},\mu_{f}-\mu_{g})_{\infty}$ at the archimedean place. As
usual, we will suppress the subscript $\infty$ in this section.

The simplest instance of our argument goes as follows: the centroid of
$\mu_{f}$ is a constant multiple of $a_{d-1}$, and similarly for $\mu_{g}$.
Thus if $a_{d-1}\neq b_{d-1}$, then $\mu_{f}$ and $\mu_{g}$ cannot be too
close to each other, so we should get a lower bound on their energy pairing.
In general, we will need to look at higher moments of $\mu_{f},\mu_{g}$,
depending on the first coefficient that $f,g$ differ in.

######  Lemma 3.21.

If for some $k\leq d$ we have $a_{d-j}=b_{d-j}$ for all $1\leq j<k$, then

$\int z^{k}\operatorname{d}\mu_{f}(z)-\int
z^{k}\operatorname{d}\mu_{g}(z)=-\frac{k}{d}(a_{d-k}-b_{d-k}).$

###### Proof.

_Case 1_ : $k<d$. For any $c\in\mathbb{C}$, let $\rho_{1},\ldots,\rho_{d}$ be
the roots of $f-c$. By Newton’s identities for elementary symmetric
polynomials, the power sum $\rho_{1}^{k}+\cdots+\rho_{d}^{k}$ is a polynomial
in $a_{d-1},\ldots,a_{d-k}$, linear in $a_{d-k}$ with coefficient $-k$; in
other words, there is some polynomial $P_{k}$ such that

$\int
z^{k}\operatorname{d}(f^{*}\delta_{c})(z)=-ka_{d-k}+P_{k}(a_{d-1},\ldots,a_{d-k+1}).$

Now for any non-exceptional point $z_{0}\in\mathbb{C}$, we consider the
iterated pullback $\mu_{f}^{(n)}=d^{-n}(f^{n})^{*}\delta_{z_{0}}$. On one
hand, we have $\mu_{f}^{(n)}=\frac{1}{d}f^{*}\mu_{f}^{(n-1)}$, so

$\int z^{k}d\mu_{f}^{(n)}(z)=\frac{1}{d}\int
z^{k}d(f^{*}\mu_{f}^{(n-1)})(z)=\frac{1}{d}\int\left(\int
z^{k}d(f^{*}\delta_{c})(z)\right)d\mu_{f}^{(n-1)}(c)$
$=-\frac{k}{d}a_{d-k}+\frac{1}{d}P_{k}(a_{d-1},\ldots,a_{d-k+1}).$

On the other hand, the sequence $\mu_{f}^{(n)}$ has uniformly bounded support,
and converges weakly to $\mu_{f}$ by equidistribution. Hence we also have

$\int
z^{k}d\mu_{f}(z)=-\frac{k}{d}a_{d-k}+\frac{1}{d}P_{k}(a_{d-1},\ldots,a_{d-k+1}).$

By replacing $f$ with $g$ we get an analogous equation, and by taking the
difference we obtain the claim.

_Case 2_ : $k=d$. The constant coefficient of $f-c$ is $a_{0}-c$, so we now
have

$\int z^{d}d(f^{*}\delta_{c})(z)=-d(a_{0}-c)+P_{d}(a_{d-1},\ldots,a_{1})$

for some polynomial $P_{d}$. Arguing as before, we obtain

$\int z^{d}d\mu_{f}(z)=-a_{0}+\int
c\,d\mu_{f}(c)+\frac{1}{d}P_{d}(a_{d-1},\ldots,a_{1}).$

By replacing $f$ with $g$ and taking the difference, we get

$\int z^{d}d\mu_{f}(z)-\int z^{d}d\mu_{g}(z)=-(a_{0}-b_{0})+\left(\int
zd\mu_{f}(z)-\int zd\mu_{g}(z)\right)=-(a_{0}-b_{0}),$

by the $k=1$ case of this lemma. ∎

######  Proposition 3.22.

Let $(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$ with $f\neq g$. If
$a_{d-j}=b_{d-j}$ for all $1\leq j<k$ and $a_{d-k}\neq b_{d-k}$, then

$\langle\mu_{f},\mu_{g}\rangle\gtrsim_{d}\begin{cases}X^{-4}&\text{if }k=1,\\\
X^{-k-4}&\text{if }2\leq k\leq d.\end{cases}$

###### Proof.

Recall from Lemma 3.2 that $J_{f}=\operatorname{supp}\mu_{f}\subseteq
B(0,R_{f})$, where

$R_{f}=3\max(1,|a_{d-1}|,|a_{d-2}|^{1/2},\ldots,|a_{0}|^{1/d}).$

_Case 1_ : $k\geq 2$. Now $a_{d-1}=b_{d-1}=0$, so $R_{f},R_{g}\leq
3\sqrt{X}:=R$.

Fix a smooth function $\psi_{0}(z)$ which is zero outside of $B(0,2)$, and
equal to $z^{k}$ in $B(0,1)$. Then the function
$\psi(z)=R^{k}\psi_{0}(\frac{z}{R})$ satisfies $\psi(z)=z^{k}$ in $B(0,R)$,
and

$||\nabla\psi||_{L^{2}}=R^{k}||\nabla\psi_{0}||_{L^{2}}.$

Next, note that $a_{d-k},b_{d-k}$ are distinct rationals with numerator and
denominator $\leq X$, so $|a_{d-k}-b_{d-k}|\geq X^{-2}$. Hence by Lemmas 3.21
and 3.19, we have

$\frac{k}{d}X^{-2}\leq\frac{k}{d}|a_{d-k}-b_{d-k}|=\left|\int\psi
d\mu_{f}-\int\psi d\mu_{g}\right|$
$\leq||\nabla\psi||_{L^{2}}(\mu_{f}-\mu_{g},\mu_{f}-\mu_{g})_{\infty}^{1/2}\leq(3\sqrt{X})^{k}||\nabla\psi_{0}||_{L^{2}}\langle\mu_{f},\mu_{g}\rangle^{1/2}.$

Rearranging, we get

$\langle\mu_{f},\mu_{g}\rangle\geq\left(\frac{kX^{-k/2-2}}{3^{k}d||\nabla\psi_{0}||_{L^{2}}}\right)^{2}\geq\frac{X^{-k-4}}{(3^{d}d||\nabla\psi_{0}||_{L^{2}})^{2}}\gtrsim_{d}X^{-k-4}.$

_Case 2_ : $k=1$. In this case, we can only bound $R_{f},R_{g}$ from above by
$R\coloneqq 3X$; however, since $a_{d-1}=0\neq b_{d-1}$, we have the stronger
bound $|a_{d-1}-b_{d-1}|\geq X^{-1}$. Now the same argument with these
modifications yields

$\langle\mu_{f},\mu_{g}\rangle\geq\left(\frac{X^{-1-1}}{3d||\nabla\psi_{0}||_{L^{2}}}\right)^{2}\gtrsim_{d}X^{-4}.\qed$

######  Corollary 3.23.

Let $(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$ with $f\neq g$. If
$a_{d-j}=b_{d-j}$ for all $1\leq j<k$ and $a_{d-k}\neq b_{d-k}$, then
$|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|\lesssim_{\varepsilon,d}X^{k+4+\varepsilon}$.

###### Proof.

Write $N=|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|$. By Propositions
3.22 and 3.18,

$X^{-k-4}\lesssim_{d}\langle\mu_{f},\mu_{g}\rangle\lesssim d\frac{\log
N}{N}(\max(h(f),h(g))+1).$

Hence $\frac{N}{\log N}\lesssim_{d}X^{k+4}(\log
X+1)\lesssim_{\varepsilon}X^{k+4+\varepsilon/2}$, thus
$N\lesssim_{\varepsilon,d}X^{k+4+\varepsilon}$. ∎

##  4\. Average number of common preperiodic points

The main purpose of this section is to establish 1.2. Recall for $X\geq 1$, we
have the sets

$\displaystyle\mathcal{P}(X)$
$\displaystyle=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\,|\,a_{i}\in\mathbb{Q},\,H(a_{i})\leq
X\\},$ $\displaystyle\mathcal{P}_{c}(X)$
$\displaystyle=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\in\mathcal{P}(X)\,|\,a_{d-1}=0\\},$
$\displaystyle\mathcal{S}(X)$
$\displaystyle=\mathcal{P}_{c}(X)\times\mathcal{P}(X)\setminus\\{(f,f)\,|\,f\in\mathcal{P}_{c}(X)\\}.$

The strategy is to show that
$\left|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)\right|$ is small for
most pairs $(f,g)\in\mathcal{S}(X)$, and not very large for the remaining
pairs. In more detail:

  * •

For most pairs $(f,g)$, the coefficients of $f,g$ have a certain divisibility
structure (Proposition 4.1). Often this implies that the filled Julia sets of
$f,g$ are disjoint at some place (Corollary 4.4), so $f$ and $g$ cannot have
common preperiodic points. Otherwise, we can still use quantitative
equidistribution to show that $f$ and $g$ have few common preperiodic points
(Corollary 4.6).

  * •

For the remaining pairs $(f,g)$, we adapt the approach of DeMarco–Krieger–Ye
[DKY22] to prove

$X^{-k-4}\lesssim_{d}\langle\mu_{f},\mu_{g}\rangle\lesssim_{d}\frac{\log
N}{N}\log X,$

where $N=|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|$, and $f,g$ agree
in their first $k-1$ coefficients. The upper bound comes from Hölder bounds on
$G_{f,v}$ when $f$ has bad reduction at $v$ (Theorem 3.14), while the lower
bound comes from the moments of $\mu_{f,\infty}$ at the archimedean place
(Proposition 3.22). Together, these bounds yield
$N\lesssim_{\varepsilon,d}X^{k+4+\varepsilon}$ (Corollary 3.23).

###  4.1. Elementary properties for most pairs

We first formalize the notions of “most” and “divisibility structure” in the
proof outline: we will require that at some place, $f$ has all coefficients
with small norm, while $g$ has some coefficient with large norm.

######  Proposition 4.1.

Fix $f\in\mathcal{P}(X)$. Let $J\subseteq\\{0,1,\ldots,d-1\\}$ and choose
$b^{\prime}_{i}\in\mathbb{Q}$ for each $i\not\in J$ with
$h(b^{\prime}_{i})\leq\log X$. Then for any $\varepsilon>0$, among all
polynomials $g\in\mathcal{P}(X)$ with prescribed coefficients
$b_{i}=b^{\prime}_{i}$ for all $i\not\in J$, at most
$O_{\varepsilon,d}(X^{-|J|+\varepsilon})$ proportion satisfy the following
property: there does not exist a non-archimedean place $v\in M_{\mathbb{Q}}$
such that

$\forall i\,[|a_{i}|_{v}\leq 1],\quad\forall i\not\in J\,[|b_{i}|_{v}\leq
1],\quad\exists j\in J\,[|b_{j}|_{v}>1].$

(Recall that $a_{i},b_{i}$ are the coefficients of $f,g$ respectively.)

###### Proof.

Standard results from analytic number theory (e.g., [HW08, §18.5]) yield

(8) $\\#\\{x\in\mathbb{Q}\,:\,h(x)\leq\log
X\\}=\frac{2X^{2}}{\zeta(2)}+O(X\log X).$

Hence there are $O_{d}(X^{2|J|})$ many $g\in\cal{P}(X)$ with prescribed
coefficients $b_{i}=b^{\prime}_{i}$ for $i\not\in J$, so we need to show that
there are $O_{\varepsilon,d}(X^{|J|+\varepsilon})$ polynomials $g$ with the
given property.

Let $S=\\{q_{1},q_{2},\ldots,q_{m}\\}$ be the set of all prime factors of the
denominators of $a_{i}$ ($0\leq i\leq d-1$) and $b_{i}$ ($i\not\in J$). Then
the given property is equivalent to requiring that for each $j\in J$, every
prime factor of $\operatorname{denom}(b_{j})$ is in $S$. It now suffices to
show that there are at most $O_{\varepsilon,d}(X^{\varepsilon/|J|})$ integers
in $[1,X]$ with all prime divisors in $S$: then for each $j\in J$, there are
$O(X)$ possible numerators and $O_{\varepsilon,d}(X^{\varepsilon/|J|})$
possible denominators for $b_{j}$, so the total number of choices for
$(b_{j})_{j\in J}$ (and hence for $g$) is
$O_{\varepsilon,d}(X^{1+\varepsilon/|J|})^{|J|}=O_{\varepsilon,d}(X^{|J|+\varepsilon})$.

Let $2=p_{1}\leq p_{2}\leq\cdots$ denote the primes in increasing order; then
it suffices to show that there are $O_{\varepsilon,d}(X^{\varepsilon/|J|})$
integers in $[1,X]$ with all prime divisors in $\\{p_{1},\ldots,p_{m}\\}$,
i.e., which are $p_{m}$-smooth. But $\prod_{i}p_{i}\leq\prod_{i}q_{i}\leq
X^{2d-|J|}$, so [HW08, §22.2] gives $p_{m}\lesssim d\log X$. Then by [Gra08,
Eq. (1.19)], the number of $p_{m}$-smooth integers in $[1,X]$ is

$\lesssim\exp\left(C_{d}\frac{\log X}{\log\log
X}\right)\lesssim_{\varepsilon,d}X^{\varepsilon/|J|},$

and we are done. ∎

By Proposition 4.1, for most pairs
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$, there is some place $v$
where $f$ has explicit good reduction, while $g$ has some large coefficient.
We now prove some elementary properties of the filled Julia sets and
equilibrium measures of $f$ and $g$ in the above scenario, which will be used
to carry out the first step in the proof of 1.2.

######  Proposition 4.2.

Let $v\in M_{\mathbb{Q}}$ be non-archimedean. If $|a_{i}|_{v}\leq 1$ for all
$i$, then $|\zeta_{v}|\leq 1$ for all $\zeta\in K_{f,v}$.

###### Proof.

By the density of Type I points in $K_{f,v}$, it suffices to prove the
statement for $\zeta=z\in\mathbb{C}_{v}$. Now if $|z_{v}|>1$, then
$|f(z)|_{v}=|z_{v}|^{d}>1$; hence by induction we have
$|f^{n}(z)|_{v}=|z_{v}|^{d^{n}}\to\infty$, and thus $z\not\in K_{f,v}$. ∎

######  Proposition 4.3.

Let $v\in M_{\mathbb{Q}}$ be non-archimedean. If $|b_{i}|_{v}\leq 1$ for all
$i\neq 0$ but $|b_{0}|_{v}>1$, then $|\zeta|_{v}=|b_{0}|_{v}^{1/d}$ for all
$\zeta\in K_{g,v}$.

###### Proof.

By the density of Type I points in $K_{g,v}$, it suffices to prove the
statement for $\zeta=z\in\mathbb{C}_{v}$. If $|z_{v}|\neq|b_{0}|_{v}^{1/d}$,
then

$|g(z)|_{v}=\max(|z^{d}|_{v},|b_{0}|_{v})>|b_{0}|_{v}^{1/d};$

hence by induction we have $|g^{n}(z)|_{v}=|g(z)|_{v}^{d^{n-1}}\to\infty$, and
thus $z\not\in K_{g,v}$. ∎

Hence most pairs $(f,g)$ have no common preperiodic points:

######  Corollary 4.4.

Fix $f\in\mathcal{P}_{c}(X)$. For $1-O_{\varepsilon,d}(X^{-1+\varepsilon})$
proportion of $g\in\mathcal{P}(X)$, there exists a non-archimedean place $v\in
M_{\mathbb{Q}}$ such that

$\forall i\,[|a_{i}|_{v}\leq 1],\quad\forall j\neq 0\,[|b_{j}|_{v}\leq
1],\quad|b_{0}|_{v}>1.$

Furthermore, in this case we have
$\operatorname{Prep}(f)\cap\operatorname{Prep}(g)=\emptyset$.

###### Proof.

The first claim follows by applying Proposition 4.1 with $J=\\{0\\}$. Next, by
Propositions 4.2 and 4.3, we have

$\displaystyle z\in\operatorname{Prep}(f)\subseteq K_{f,v}$
$\displaystyle\implies|z|_{v}\leq 1,$ $\displaystyle
z\in\operatorname{Prep}(g)\subseteq K_{g,v}$
$\displaystyle\implies|z|_{v}=|b_{0}|_{v}^{1/d}>1,$

so $\operatorname{Prep}(f)\cap\operatorname{Prep}(g)=\emptyset$. ∎

When the large coefficient of $g$ is not the constant coefficient, we see
below that the distribution of mass for $\mu_{g,v}$ differs from
$\mu_{f,v}=\delta_{\zeta(0,1)}$ by a definite amount. Then Proposition 3.20
gives an upper bound on the number of common preperiodic points.

######  Proposition 4.5.

Let $v\in M_{\mathbb{Q}}$ be non-archimedean, with $p$ its associated prime.
If there exists $1\leq j\leq d-1$ such that $|b_{j}|_{v}>1$, then

$\mu_{g,v}({\mathbb{A}^{1}_{\operatorname{Berk},v}}\setminus
D_{\operatorname{an}}(0,p^{1/d}))\geq 1/d.$

###### Proof.

First, we claim that for any $c\in\mathbb{C}_{v}$, the polynomial $g-c$ has at
least one root $w$ with $|w|_{v}\geq|b_{j}|_{v}^{1/(d-j)}\geq p^{1/d}$. This
follows from considering the Newton polygon of $g-c$, which must have a side
with slope at least $\frac{1}{d-j}\log|b_{j}|_{v}$.

Now for any non-exceptional point $z\in\mathbb{C}_{v}$, we consider the
iterated pullback $d^{-n}(g^{n})^{\delta_{z}}$. On one hand, by the above
argument, this measure always assigns at least $1/d$ mass outside of
$D_{\operatorname{an}}(0,p^{1/d})$. On the other hand, by equidistribution,
this sequence of measures converges weakly to the equilibrium measure
$\mu_{g,v}$; hence $\mu_{g,v}$ must also assign at least $1/d$ mass outside of
$D_{\operatorname{an}}(0,p^{1/d})$. ∎

######  Corollary 4.6.

Let $(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$. If there exists a non-
archimedean place $v\in M_{\mathbb{Q}}$ such that

$\forall i\,[|a_{i}|_{v}\leq 1],\quad\exists j>0\,[|b_{j}|_{v}>1],$

then
$|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|\lesssim_{d}\log^{4}X$.

###### Proof.

Let $p$ be the prime associated to $v$. Then
$p\mid\operatorname{denom}(b_{j})$, so $p\leq X$.

Let $F=\operatorname{Prep}(f)\cap\operatorname{Prep}(g)$. Now $F$ is
$\operatorname{Gal}({\overline{\mathbb{Q}}}/\mathbb{Q})$-invariant, and
$F\subseteq\overline{B}(0,1)$ by Proposition 4.3. Hence by Propositions 4.5
and 3.20, we have

$|F|\lesssim_{d}(h(g)^{2}+1)\log^{2}p\lesssim\log^{4}X.\qed$

###  4.2. Proof of 1.2 and 1

We now restate and prove 1.2.

*

###### Proof.

Fix $f\in\mathcal{P}_{c}(X)$.

_Case 1_ : For some non-archimedean place $v\in M_{\mathbb{Q}}$, we have

$\forall i\,[|a_{i}|_{v}\leq 1],\quad\forall j\neq 0\,[|b_{j}|_{v}\leq
1],\quad|b_{0}|_{v}>1.$

By Corollary 4.4, $f$ and $g$ have no common preperiodic points.

_Case 2_ : Case 1 does not hold, and for some non-archimedean place $v\in
M_{\mathbb{Q}}$ we have

$\forall i\,[|a_{i}|_{v}\leq 1],\quad\exists j\,[|b_{j}|_{v}>1].$

By Corollary 4.4, this case occurs for at most
$O_{\varepsilon,d}(X^{-1+\varepsilon})$ proportion of $g\in\mathcal{P}(X)$,
and by Corollary 4.6,
$|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|\lesssim_{d}\log^{4}X$.

_Case 3_ : Neither Case 1 nor Case 2 holds. We will further split this case by
the value of $k$, $1\leq k\leq d$, for which

$\forall i<k\,[a_{d-i}=b_{d-i}],\quad a_{d-k}\neq b_{d-k}.$

By Proposition 4.1 with $J=\\{d-k,d-k-1,\ldots,0\\}$, the subcase for each
value of $k$ occurs for at most
$O_{\varepsilon,d}(X^{-2(k-1)}X^{-(d-k+1)+\varepsilon})=O_{\varepsilon,d}(X^{-(d+k-1)+\varepsilon})$
proportion of $g\in\mathcal{P}(X)$, and Corollary 3.23 gives
$|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|\lesssim_{d}X^{k+4+\varepsilon}$.

Putting these three cases together, the average value of
$|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|$ over all
$g\in\mathcal{P}(X)\setminus\\{f\\}$ is at most

$O_{\varepsilon,d}\left(X^{-1+\varepsilon}\log^{4}X+\sum_{k=1}^{d-1}X^{-(d+k-1)+\varepsilon}X^{k+4+\varepsilon}\right)=O_{\varepsilon,d}(X^{-1+2\varepsilon}+X^{5-d+\varepsilon}).$

For $d\geq 6$ and $\varepsilon>0$ small, this goes to 0 as $X\to\infty$, and
we are done. ∎

We now prove Section 1 using the same argument. Let $V$ be a linear subspace
of $(a_{d-2},\ldots,a_{0})\simeq\mathbb{Q}^{d-1}$ of dimension $m$ that is
given by an intersection of coordinate hyperplanes $\\{a_{i}=0\\}$. Assume
that the projection map from $V$ to the last coordinate $a_{0}$ is surjective.
Let $x=(x_{d-1},\ldots,x_{0})$ be a point in $\mathbb{Q}^{d-1}$ and let $A$ be
the affine subspace $x+V$. Choose $m$ coordinates $a_{i_{1}},\ldots,a_{i_{m}}$
such that $i_{j}$ are decreasing, $i_{m}=0$, and they induce an isomorphism
$V\simeq\mathbb{Q}^{m}$. We define

$\cal{P}_{c,A}(X)=\\{z^{d}+a_{d-2}z^{d-2}+\cdots+a_{0}\mid(0,a_{d-2},\ldots,a_{0})\in
A\text{ and }H(a_{i_{j}})\leq X\\},$
$\cal{P}_{A}(X)=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\mid(a_{d-1},\ldots,a_{0})\in
A\text{ and }H(a_{i_{j}})\leq X\\},$
$\cal{S}_{V}(X)=\cal{P}_{c,A}(X)\times\cal{P}_{A}(X)\setminus\\{(f,f)\mid
f\in\cal{P}_{c,A}(X)\\}.$

*

###### Proof.

The same argument of 1.2 applies. Fix
$f(z)=z^{d}+a_{d-2}z^{d-2}+\cdots+a_{0}\in\cal{P}_{c,A}(X)$ and write
$g(z)=z^{d}+b_{d-1}z^{d-1}+\cdots+b_{0}\in\cal{P}_{A}(X)$. Then we keep Case 1
as the same as 1.2. We modify Case 2 to asking for a non-archimedean place $v$
such that $|b_{i_{k}}|_{v}>1$ for some $k$ and $|a_{i}|_{v}\leq 1$ for all
$i$. For Case 3, we split by the value of $1\leq k\leq m$ for which

$\forall j<k[a_{i_{j}}=b_{i_{j}}],a_{i_{k}}\not=b_{i_{k}}.$

As $V$ is given by an intersection of coordinate hyperplanes, it follows that
we can vary $b_{0}$ freely over rational numbers of height $\leq X$ while
keeping the rest of the coefficients fixed. Thus Proposition 4.1 again implies
Case 2 holds for $O(X^{-1+\epsilon})$ proportion of $g(z)$’s. The same bound
of $O(\log^{4}X)$ holds for this case as the height of each coefficient is
again $O(\log X)$. Similarly as we can vary each $b_{i_{k}}$ freely while
keeping the rest of the coefficients fixed, Proposition 4.1 implies that Case
3 occurs for at most

$O_{\epsilon,d}(X^{-2(k-1)}X^{-(m-k+1)+\epsilon})=O_{\epsilon,d}(X^{-(m+k-1)+\epsilon})$

proportion for all possible choices for $g$. Now by Corollary 3.23, since the
first index $j$ for which $a_{d-j}\not=b_{d-j}$ is at most $d-m+k$, we get

$|\operatorname{Prep}(f)\cap\operatorname{Prep}(g)|\lesssim_{d}X^{d-m+k+4+\epsilon}.$

Putting it together, we get an upper bound of

$O_{\epsilon,d}\left(X^{-1+\epsilon}\log^{2}(X)+\sum_{k=1}^{m}X^{-(m+k-1)+\epsilon}X^{d-m+k+4+\epsilon}\right)=O_{\epsilon,d}(X^{-1+\epsilon}+X^{5+d-2m+\epsilon}).$

Thus if $m\geq\frac{d}{2}+3$, this quantity goes to zero as $X\to\infty$ as
desired. ∎

##  5\. Generic polynomials and dynamics

In this section, we will define a notion of generic polynomials and study its
properties, mainly its dynamics. For a pair of polynomials $(f,g)$ with
rational coefficients, we first consider three notions of arithmetic
complexity, namely $h(f)+h(g)$, $\langle\mu_{f},\mu_{g}\rangle$, and $\log X$,
and show that these are comparable when the pair $(f,g)$ is generic. We then
make a detailed study of the equilibrium measures and potentials for generic
polynomials at most places. Recall again that for a fixed degree $d$ and
$X\geq 1$, we have the sets

$\displaystyle\mathcal{P}(X)$
$\displaystyle=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\,|\,a_{i}\in\mathbb{Q},\,H(a_{i})\leq
X\\},$ $\displaystyle\mathcal{P}_{c}(X)$
$\displaystyle=\\{z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}\in\mathcal{P}(X)\,|\,a_{d-1}=0\\},$

###  5.1. Complexity of generic pairs

The arithmetic complexity of a pair
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$ can be measured by the adelic
energy pairing $\langle\mu_{f},\mu_{g}\rangle$, by the heights of the
polynomials $h(f)+h(g)$, or simply by the height bound $\log X$. In this
section, we show that these three quantities are comparable in the generic
case. We first recall the notion of $\varepsilon$-ordinary. For an integer
$n$, we let $\operatorname{rad}(n)$ be the product of all distinct prime
factors of $n$.

######  Definition 5.1.

Given $X$ and $\varepsilon>0$, we will say that a pair
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$ is _$\varepsilon$ -ordinary_
if

$\operatorname{rad}(\operatorname{denom}(c))\geq X^{1-2\varepsilon}\text{ and
}\gcd(\operatorname{denom}(c),\operatorname{denom}(c^{\prime}))\leq
X^{2\varepsilon},$

for any distinct coefficients $c,c^{\prime}$ of $f$ or $g$.

######  Theorem 5.2.

For any $(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$, we have

$\langle\mu_{f},\mu_{g}\rangle-2\leq\frac{h(f)+h(g)}{d}\leq 2\log X.$

Moreover, if $(f,g)$ is $\varepsilon$-ordinary, then

$\langle\mu_{f},\mu_{g}\rangle+O(d\varepsilon\log
X)\geq\frac{h(f)+h(g)}{d}\geq\left(2-O(d\varepsilon+\tfrac{1}{d})\right)\log
X.$

We prove these bounds in Propositions 5.4, 5.11, 5.18 and 5.21. For
convenience, we make the following definition.

######  Definition 5.3.

For $v\in M_{K}$, write
$\cal{M}_{f,v}:=\max(1,|a_{d-1}|_{v},\ldots,|a_{0}|_{v})$.

######  Remark.

Note that $\log\cal{M}_{f,v}$ is the local contribution to the height of $f$
at $v$, i.e., $h(f)=\frac{1}{[K:\mathbb{Q}]}\sum_{v\in
M_{K}}\\!n_{v}\log\cal{M}_{f,v}$. Also, for any place $v^{\prime}$ of a finite
extension $K^{\prime}/K$ with $v^{\prime}\mid v$, we have
$\cal{M}_{f,v^{\prime}}=\cal{M}_{f,v}$.

We begin with an easy bound for $h(f)$ in terms of $\log X$.

######  Proposition 5.4.

For any $f\in\cal{P}(X)$, we have $h(f)\leq d\log X$.

###### Proof.

For any $v\in M_{\mathbb{Q}}$,

$\cal{M}_{f,v}=\max(1,|a_{d-1}|_{v},\ldots,|a_{0}|_{v})\leq\max(1,|a_{d-1}|_{v})\cdots\max(1,|a_{0}|_{v}).$

Hence

$h(f)=\sum_{v\in M_{\mathbb{Q}}}\\!\log\cal{M}_{f,v}\leq\sum_{v\in
M_{\mathbb{Q}}}\\!\left(\log^{+}|a_{d-1}|_{v}+\cdots+\log^{+}|a_{0}|_{v}\right)=h(a_{d-1})+\cdots+h(a_{0})\leq
d\log X.\qed$

Now we will relate $h(f)+h(g)$ to $\langle\mu_{f},\mu_{g}\rangle$. We first
note that $\log\cal{M}_{f,v}$ can be read off from the equilibrium potential
$G_{f,v}$ at the Gauss point:

######  Lemma 5.5.

Let $v\in M_{K}$ be non-archimedean. Then

$G_{f,v}(\zeta(0,1))=\frac{1}{d}\log\cal{M}_{f,v}.$

###### Proof.

It is routine to check that $f(\zeta(0,1))=\zeta(0,\cal{M}_{f,v})$, and
$f(\zeta(0,R))=\zeta(0,R^{d})$ for all $R\geq\cal{M}_{f,v}$. Hence for all
$n\geq 1$ we have

$f^{n}(\zeta(0,1))=\zeta(0,\cal{M}_{f,v}^{d^{n-1}}),$

and the result follows. ∎

On the other hand, we can bound $G_{f,v}$ in terms of its value at the Gauss
point, which in turn yields a bound on the local pairing
$-(\mu_{f},\mu_{g})_{v}$.

######  Lemma 5.6.

Let $v\in M_{K}$ be non-archimedean. For any
$\xi\in{\mathbb{A}^{1}_{\operatorname{Berk},v}}$, we have

$G_{f,v}(\xi)\leq\log^{+}|\xi|_{v}+G_{f,v}(\zeta(0,1)).$

###### Proof.

The left side of the claim is subharmonic on
${\mathbb{A}^{1}_{\operatorname{Berk},v}}$, while the right side is harmonic
on ${\mathbb{A}^{1}_{\operatorname{Berk},v}}\setminus\\{\zeta(0,1)\\}$. Hence
it suffices to check the inequality for $\xi=\zeta(0,1)$, where it is clear,
and for $|\xi|_{v}\to\infty$. But $G_{f,v}(\xi)=\log^{+}|\xi|_{v}$ for all
$\xi$ with $|\xi|_{v}$ sufficiently large, so we are done by observing that

$G_{f,v}(\zeta(0,1))\geq\inf G_{f,v}=I(\mu_{f,v})=0.\qed$

######  Corollary 5.7.

Let $v\in M_{K}$ be non-archimedean. Then

$-(\mu_{f},\mu_{g})_{v}\leq\frac{1}{d}(\log\cal{M}_{f,v}+\log\cal{M}_{g,v}).$

###### Proof.

We have by Lemmas 5.6 and 5.5,

$-(\mu_{f},\mu_{g})_{v}=\int
G_{f,v}(\xi)d\mu_{g,v}(\xi)\leq\int\log^{+}|\xi|_{v}d\mu_{g,v}(\xi)+G_{f,v}(\zeta(0,1))$
$=G_{g,v}(\zeta(0,1))+G_{f,v}(\zeta(0,1))=\frac{1}{d}(\log\cal{M}_{f,v}+\log\cal{M}_{g,v}).\qed$

The situation at archimedean places $v\mid\infty$ is analogous. As usual, we
fix an embedding ${\overline{\mathbb{Q}}}\hookrightarrow\mathbb{C}$, and write
$|\cdot|_{v}=|\cdot|_{\infty}$.

######  Lemma 5.8.

We have

$\max_{|w|_{\infty}=1}G_{f,\infty}(w)\leq\frac{1}{d}\log\cal{M}_{f,\infty}+1.$

###### Proof.

Write $|\cdot|=|\cdot|_{\infty}$ and $R=\max(2,\cal{M}_{f,\infty})$. For
$|z|\geq R$, we have

$|f(z)|\leq|z|^{d}+R(|z|^{d-1}+\cdots+|z|+1)\leq(|z|+R)^{d}-R,$

by comparing terms of the binomial expansion. By induction, we have
$|f^{n}(z)|\leq(|z|+R)^{d^{n}}-R$, so that by the maximum principle we have

$G_{f,\infty}(z)\leq\log(\max(|z|,R)+R)$

for all $z\in\mathbb{C}$.

Now for $|w|=1$, we have $|f(w)|\leq dR+1$. Hence

$d\cdot
G_{f,\infty}(w)=G_{f,\infty}(f(w))\leq\log(\max(|f(w)|,R)+R)\leq\log((d+1)R+1).$

If $1\leq\cal{M}_{f,\infty}\leq 2$ then $R=2$, so

$d\cdot G_{f,\infty}(w)\leq\log(2d+3)\leq d\leq\log\cal{M}_{f,\infty}+d.$

Otherwise, we have $\cal{M}_{f,\infty}>2$ and $R=\cal{M}_{f,\infty}$, so

$d\cdot
G_{f,\infty}(w)\leq\log\cal{M}_{f,\infty}+\log(d+2)\leq\log\cal{M}_{f,\infty}+d.\qed$

######  Lemma 5.9.

For any $z\in\mathbb{C}$, we have

$G_{f,\infty}(z)\leq\log^{+}|z|_{\infty}+\max_{|w|_{\infty}=1}G_{f,\infty}(w).$

###### Proof.

Write $|\cdot|=|\cdot|_{\infty}$. The left side of the claim is subharmonic on
$\mathbb{C}$, while the right side is harmonic on $\mathbb{C}\setminus\partial
B(0,1)$. Hence it suffices to check the inequality on $|z|=1$, where it is
clear, and for $|z|\to\infty$. But $G_{f,\infty}(z)-\log^{+}|z|\to 0$ as
$|z|\to\infty$, so we are done by observing that

$\max_{|w|=1}G_{f,\infty}(w)\geq\inf G_{f,\infty}=I(\mu_{f,\infty})=0.\qed$

######  Corollary 5.10.

We have

$-(\mu_{f},\mu_{g})_{\infty}\leq\frac{1}{d}(\log\cal{M}_{f,\infty}+\log\cal{M}_{g,\infty})+2.$

###### Proof.

We have by Lemmas 5.9 and 5.8

$-(\mu_{f},\mu_{g})_{\infty}=\int
G_{f,\infty}(z)d\mu_{g,\infty}(z)\leq\max_{|w|_{\infty}=1}G_{f,\infty}(w)+\int\log^{+}|z|_{\infty}d\mu_{g,\infty}(z)=\max_{|w|_{\infty}=1}G_{f,\infty}(w)-(\mu_{S^{1}},\mu_{g,\infty})_{\infty}$
$=\max_{|w|_{\infty}=1}G_{f,\infty}(w)+\int
G_{g,\infty}(e^{i\theta})\,\frac{d\theta}{2\pi}\leq\max_{|w|_{\infty}=1}G_{f,\infty}(w)+\max_{|w|_{\infty}=1}G_{g,\infty}(w)\leq\frac{1}{d}(\log\cal{M}_{f,\infty}+\log\cal{M}_{g,\infty})+2.\qed$

Combining the above results, we obtain the following bound on the adelic
energy pairing $\langle\mu_{f},\mu_{g}\rangle$.

######  Proposition 5.11.

We have

$\langle\mu_{f},\mu_{g}\rangle\leq\frac{h(f)+h(g)}{d}+2.$

###### Proof.

Since $(\mu_{f},\mu_{f})_{v}=(\mu_{g},\mu_{g})_{v}=0$, we have

$\langle\mu_{f},\mu_{g}\rangle=\frac{1}{2}\sum_{v\in
M_{\mathbb{Q}}}\\!(\mu_{f}-\mu_{g},\mu_{f}-\mu_{g})_{v}=\sum_{v\in
M_{\mathbb{Q}}}\\!-(\mu_{f},\mu_{g})_{v}.$

Now each term in the sum is bounded by Corollaries 5.7 and 5.10; summing over
all places, we obtain the desired bound. ∎

Our next objective is to show that most pairs
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$ are $\varepsilon$-ordinary,
which requires some preparatory estimates.

######  Proposition 5.12 ([LM21, Lem. 2.9]).

Fix $\varepsilon>0$. Then for all large enough $X$,

$\sum_{n\leq X}\frac{1}{\operatorname{rad}(n)}\leq X^{\varepsilon}.$

######  Corollary 5.13.

There are at most $O_{\varepsilon}(X^{2-\varepsilon})$ rationals $x$ such that
$h(x)\leq\log X$ and $\operatorname{rad}(\operatorname{denom}(x))\leq
X^{1-2\varepsilon}$.

###### Proof.

By Proposition 5.12, the number of possible values of
$\operatorname{denom}(x)$ is

$\\#\\{n\leq X\,:\,\operatorname{rad}(n)\leq
X^{1-2\varepsilon}\\}\lesssim_{\varepsilon}X^{\varepsilon}X^{1-2\varepsilon}=X^{1-\varepsilon},$

and for each such denominator there are at most $2X+1$ possibilities for $x$.
∎

######  Lemma 5.14.

Let $x_{1},x_{2}$ be chosen uniformly at random independently from the finite
set $\\{x\in\mathbb{Q}\,:\,h(x)\leq\log X\\}$, and write
$x_{i}=\frac{\alpha_{i}}{\beta_{i}}$ for some $|\alpha_{i}|\leq X$,
$1\leq\beta_{i}\leq X$, $\gcd(\alpha_{i},\beta_{i})=1$. Then

$\Pr(\gcd(\beta_{1},\beta_{2})\geq
X^{2\varepsilon})\lesssim_{\varepsilon}X^{-\varepsilon}.$

###### Proof.

Fix $x_{1}$. If $\gcd(\beta_{1},\beta_{2})\geq X^{2\varepsilon}$, then
$\beta_{2}=km$ for some $k,m$ satisfying

$m\mid\beta_{1},\quad m\geq X^{2\varepsilon},\quad km\leq X.$

Since $\beta_{1}\leq X$, the number of divisors of $\beta_{1}$ is at most
$O_{\varepsilon}(X^{\varepsilon})$, by the divisor bound. Hence there are at
most $O_{\varepsilon}(X^{\varepsilon})$ choices for $m$, and for each such $m$
there are at most $X^{1-2\varepsilon}$ choices for $k$. Also, for each
$\beta_{2}$ there are at most $X$ choices for $\alpha_{2}$. Thus there are
$\lesssim_{\varepsilon}X^{\varepsilon}\cdot X^{1-2\varepsilon}\cdot
X=X^{2-\varepsilon}$ choices for $x_{2}$, out of $\sim X^{2}$ possibilities
from $\\{x\in\mathbb{Q}\,:\,h(x)\leq\log X\\}$ by eq. 8. Hence

$\Pr(\gcd(\beta_{1},\beta_{2})\geq X^{2\varepsilon}\,|\,\beta_{1}\text{ is
fixed})\lesssim_{\varepsilon}\frac{X^{2-\varepsilon}}{X^{2}}=X^{-\varepsilon},$

which implies the desired statement. ∎

######  Proposition 5.15.

The proportion of pairs $(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$ which
are $\varepsilon$-ordinary is $1-O_{\varepsilon}(d^{2}X^{-\varepsilon})$.

###### Proof.

By Corollary 5.13, there is at most an $O_{\varepsilon}(dX^{-\varepsilon})$
proportion of polynomials $f\in\mathcal{P}_{c}(X)$ where
$\operatorname{rad}(\operatorname{denom}(a_{i}))\leq X^{1-2\varepsilon}$ for
some $i$; the analogous bound holds for $g$. Also, by Lemma 5.14, there is at
most an $O_{\varepsilon}(d^{2}X^{-\varepsilon})$ proportion of pairs
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$ where
$\gcd(\operatorname{denom}(a_{i}),\operatorname{denom}(b_{j}))>X^{2\varepsilon}$
for some $i,j$.

Hence at most $O_{\varepsilon}(d^{2}X^{-\varepsilon})$ proportion of pairs
$(f,g)$ satisfies one of the above conditions; this is equivalent to not being
$\varepsilon$-ordinary, and we are done. ∎

We now show that for $\varepsilon$-ordinary pairs, the reverse inequalities to
Propositions 5.4 and 5.11 hold up to a small multiplicative error.

######  Definition 5.16.

Given a pair $(f,g)$, and $0\leq j\leq d-1$, we say that a non-archimedean
place $v\in M_{K}$ is associated to $a_{j}$ if

$\forall i\neq j\,[|a_{i}|_{v}\leq 1],\quad\forall i\,[|b_{i}|_{v}\leq
1],\quad|a_{j}|_{v}>1.$

We define places associated to $b_{j}$ similarly.

We call a non-archimedean place $v\in M_{K}$ a good place for the pair $(f,g)$
if $v$ is associated to some coefficient $a_{j}$ or $b_{j}$, or a bad place
for $(f,g)$ otherwise. (We will always treat archimedean places separately.)

######  Remark.

Each $v\in M_{K}$ is associated to at most one coefficient of $f$ or $g$. If
$v$ is associated to $a_{j}$, then $\cal{M}_{f,v}=|a_{j}|_{v}>1$ and
$\cal{M}_{g,v}=1$, and similarly for $b_{j}$.

We first show the reverse bound to Proposition 5.4 for $h(f)+h(g)$ and $\log
X$.

######  Lemma 5.17.

For any $\varepsilon$-ordinary pair
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$, we have

$\frac{1}{[K:\mathbb{Q}]}\sum_{\begin{subarray}{c}v\in M_{K}\\\ v\text{
assoc.\
}a_{j}\end{subarray}}\\!\\!n_{v}\log|a_{j}|_{v}\geq(1-O(d\varepsilon))\log X,$

and similarly for $b_{j}$.

###### Proof.

We may assume that $K=\mathbb{Q}$. Let $A_{i},B_{i}$ denote the denominators
of $a_{i},b_{i}$ respectively. Now $v\in M_{\mathbb{Q}}$ is associated to
$a_{j}$ if and only if $v$ divides $A_{j}$, but no other $A_{i}$ or $B_{i}$;
hence

$\sum_{v\text{ assoc.\
}a_{j}}\\!\\!\log|a_{j}|_{v}\geq\log\operatorname{rad}(A_{j})-\sum_{i\neq
j}\log\gcd(A_{i},A_{j})-\sum_{i}\log\gcd(B_{i},A_{j})$
$\geq(1-2\varepsilon)\log X-(2d-1)2\varepsilon\log X\geq(1-4d\varepsilon)\log
X.$

The argument for $b_{j}$ is analogous. ∎

######  Proposition 5.18.

For any $\varepsilon$-ordinary pair
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$, we have

$\frac{h(f)+h(g)}{d}\geq\left(2-O(d\varepsilon+\tfrac{1}{d})\right)\log X.$

###### Proof.

Using Lemma 5.17, we have

$h(f)+h(g)=\sum_{v\in
M_{\mathbb{Q}}}\\!(\log\mathcal{M}_{f,v}+\log\mathcal{M}_{g,v})\geq\sum_{j}\sum_{v\text{
assoc.\ }a_{j}}\\!\\!\log|a_{j}|_{v}+\sum_{j}\sum_{v\text{ assoc.\
}b_{j}}\\!\\!\log|b_{j}|_{v}$ $\geq(2d-1)(1-4d\varepsilon)\log X\geq
2d\left(1-O(d\varepsilon+\tfrac{1}{d})\right)\log X,$

and we are done after dividing by $d$. ∎

we now proceed to show the reverse bound to Proposition 5.11 for $h(f)+h(g)$
and $\langle\mu_{f},\mu_{g}\rangle$. We begin by observing an important
equality case for Corollary 5.7.

######  Proposition 5.19.

Let $v\in M_{K}$ be non-archimedean. If $v$ is good for $(f,g)$, then

$-(\mu_{f},\mu_{g})_{v}=\frac{1}{d}(\log\mathcal{M}_{f,v}+\log\mathcal{M}_{g,v}).$

###### Proof.

Assume that $g$ has explicit good reduction at $v$; in particular,
$\mathcal{M}_{g,v}=1$ and $\mu_{g,v}=\delta_{\zeta(0,1)}$. Now by Lemma 5.5,

$-(\mu_{f},\mu_{g})_{v}=\int
G_{f,v}\operatorname{d}\mu_{g,v}=G_{f,v}(\zeta(0,1))=\frac{1}{d}\log\mathcal{M}_{f,v}=\frac{1}{d}(\log\mathcal{M}_{f,v}+\log\mathcal{M}_{g,v}).$

The argument when $f$ has explicit good reduction is analogous. ∎

Next, for an $\varepsilon$-ordinary pair $(f,g)$, we show that the local
contribution to $h(f)+h(g)$ from bad places is small.

######  Proposition 5.20.

For any $\varepsilon$-ordinary pair
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$, we have

$\frac{1}{[K:\mathbb{Q}]}\sum_{\begin{subarray}{c}v\in M_{K}\\\ v\text{
bad}\end{subarray}}\\!n_{v}(\log\cal{M}_{f,v}+\log\cal{M}_{g,v})\lesssim
d^{2}\varepsilon\log X.$

###### Proof.

We may assume that $K=\mathbb{Q}$. Write $A_{i},B_{i}$ for the denominator of
$a_{i},b_{i}$ respectively. Let $p_{v}$ denote the prime associated to a place
$v$. Now

(9) $\sum_{v\text{ bad}}\log\cal{M}_{f,v}\leq\sum_{v\text{ bad}}\log
p_{v}+\sum_{v\text{ bad}}\max_{i}(0,\log|a_{i}|_{v}-\log p_{v}).$

Note that $v$ is a bad place if and only if $p_{v}$ divides more than one of
the $A_{i}$ or $B_{i}$. Hence the first sum on the right in (9) is

$\sum_{v\text{ bad}}\log
p_{v}\leq\sum_{i,j}\log\gcd(A_{i},B_{j})+\sum_{i<j}\log\gcd(A_{i},A_{j})+\sum_{i<j}\log\gcd(B_{i},B_{j})\leq
4d^{2}\varepsilon\log X.$

Also, the second term on the right in (9) is

$\sum_{v\text{ bad}}\max_{i}(0,\log|a_{i}|_{v}-\log
p_{v})\leq\sum_{i}\sum_{v\nmid\infty}\max(0,\log|a_{i}|_{v}-\log
p_{v})=\sum_{i}(\log A_{i}-\log\operatorname{rad}(A_{i}))$ $\leq\sum_{i}(\log
X-\log X^{1-2\varepsilon})\leq 2d\varepsilon\log X.$

Substituting back into (9), we get $\sum_{v\text{
bad}}\log\cal{M}_{f,v}\lesssim d^{2}\varepsilon\log X$. The same argument
works for $g$. ∎

######  Proposition 5.21.

For any $\varepsilon$-ordinary pair
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$, we have

$\langle\mu_{f},\mu_{g}\rangle\geq\frac{h(f)+h(g)}{d}-O((d\varepsilon+\tfrac{1}{d})\log
X).$

###### Proof.

We may assume that $K=\mathbb{Q}$. Note that
$(\mu_{f},\mu_{f})_{v}=(\mu_{g},\mu_{g})_{v}=0$ for any $v\in M_{\mathbb{Q}}$,
so $0\leq(\mu_{f}-\mu_{g},\mu_{f}-\mu_{g})_{v}=-2(\mu_{f},\mu_{g})_{v}$. Hence
by Proposition 5.19

$\langle\mu_{f},\mu_{g}\rangle\geq\sum_{v\text{
good}}-(\mu_{f},\mu_{g})_{v}=\frac{1}{d}\sum_{v\text{
good}}(\log\mathcal{M}_{f,v}+\log\mathcal{M}_{g,v})=\frac{h(f)+h(g)}{d}-\frac{1}{d}\sum_{v\text{
bad}}(\log\mathcal{M}_{f,v}+\log\mathcal{M}_{g,v}).$

Using Proposition 5.20, we get the above expression is

$=\frac{h(f)+h(g)}{d}-\frac{1}{d}(\log\mathcal{M}_{f,\infty}+\log\mathcal{M}_{g,\infty})\geq\frac{h(f)+h(g)}{d}-O(d\varepsilon\log
X)-O(\tfrac{1}{d}\log X).\qed$

###  5.2. Dynamics on generic polynomials

For $\varepsilon$-ordinary pairs $(f,g)$, since the denominators of the
coefficients are almost pairwise coprime, most primes which divide one such
denominator would not divide any other denominator. Hence for most places $v$
where $f,g$ do not both have explicit good reduction, it will be the case that
one of them has explicit good reduction, while the other has exactly one large
coefficient.

We know in the former that the equilibrium measure and potential are
$\delta_{\zeta(0,1)}$ and $\log^{+}|\cdot|_{v}$ respectively; in this section,
we will study the equilibrium measure and potential for the latter case.

First, we show that the equilibrium measure lies in three distinct strata (cf.
Proposition 4.3, which corresponds to the case $j=0$).

######  Proposition 5.22.

Let $v\in M_{\mathbb{Q}}$ be non-archimedean with $|a_{0}|_{v}=1$. If there
exists $1\leq j\leq d-1$ such that

$\forall i\neq j\,[|a_{i}|_{v}\leq 1],\quad|a_{j}|_{v}>1,$

then
$|\zeta|_{v}\in\\{|a_{j}|_{v}^{\frac{1}{d-j}},|a_{j}|_{v}^{\frac{1}{j}(\frac{1}{d-j}-1)},|a_{j}|_{v}^{-\frac{1}{j}}\\}$
for all $\zeta\in K_{f,v}$.

###### Proof.

By the density of Type I points in $K_{f,v}$, it suffices to prove the
statement for $\zeta=z\in\mathbb{C}_{v}$. We will prove the contrapositive
statement, so suppose that $|z|_{v}$ is not one of the listed values.

_Case 1a_ : $|z|_{v}\geq 1$ and $j<d-1$. Then
$|z^{d}|_{v}\not=|a_{j}z^{j}|_{v}$ by assumption, so

$|f(z)|_{v}=\max(|z^{d}|_{v},|a_{j}z^{j}|_{v})>|a_{j}|_{v}^{\frac{1}{d-j}}.$

By induction, we have $|f^{n}(z)|_{v}=|f(z)|_{v}^{d^{n-1}}$; in particular,
$f^{n}(z)\to\infty$, so $z\not\in K_{f,v}$.

_Case 1b_ : $|z|_{v}\geq 1$ and $j=d-1$. Note that
$|z|_{v}\not=|a_{j}|_{v}^{\frac{1}{j}(\frac{1}{d-j}-1)}=1$, so $|z|_{v}>1$,
and now the argument of Case 1a works verbatim.

_Case 2_ : $|z|_{v}<1$. Then $|a_{j}z^{j}|_{v}\not=1$ and $|a_{0}|_{v}=1$ by
assumption, so

$|f(z)|_{v}=\max(|a_{j}z^{j}|_{v},1)\geq 1.$

Furthermore, $|a_{j}z^{j}|_{v}\not=|a_{j}|_{v}^{\frac{1}{d-j}}$ by assumption,
so $f(z)\not\in K_{f,v}$ by Cases 1a and 1b. Thus $z\not\in K_{f,v}$ as well.
∎

The next result shows that if some subset of
${\mathbb{A}^{1}_{\operatorname{Berk},v}}$ decomposes into finitely many
strata, then the equilibrium measure restricted to each stratum is a multiple
of the equilibrium measure on that stratum.

######  Lemma 5.23.

Let $r_{1}>r_{2}>\cdots>r_{k}>0$, and let
$K_{i}\subseteq\overline{D}_{\operatorname{an}}(0,r_{i})\setminus
D_{\operatorname{an}}(0,r_{i})$ be compact sets in
${\mathbb{A}^{1}_{\operatorname{Berk},v}}$ with equilibrium measures
$\mu_{i}$. Then the equilibrium measure $\mu$ on $K=\bigcup_{i=1}^{k}K_{i}$ is
a convex combination of the $\mu_{i}$.

###### Proof.

Write $\mu=\sum_{i=1}^{k}\alpha_{i}\nu_{i}$ for some $\alpha_{i}\in[0,1]$ and
$\nu_{i}$ probability measures supported on
$\overline{D}_{\operatorname{an}}(0,r_{i})\setminus
D_{\operatorname{an}}(0,r_{i})$. Now for all $i<j$ and all $\zeta_{i}\in
K_{i}$, $\zeta_{j}\in K_{j}$, we have
$\delta(\zeta_{i},\zeta_{j})_{\infty}=\log r_{i}$, so

(10) $I(\mu)=\sum_{i=1}^{k}\alpha_{i}^{2}I(\nu_{i})+\sum_{\mathclap{1\leq
i<j\leq k}}2\alpha_{i}\alpha_{j}\log r_{i}.$

Since $\mu$ is the equilibrium measure on $K$, the above expression is minimal
over all choices of $\nu_{i}$ supported on $K_{i}$. But the minimizer of
$I(\nu_{i})$ for $\nu_{i}$ supported on $K_{i}$ is precisely $\mu_{i}$; hence
$\nu_{i}=\mu_{i}$ when $\alpha_{i}\neq 0$, and so
$\mu=\sum_{i=1}^{k}\alpha_{i}\mu_{i}$.

∎

######  Remark.

The above result is known in more general contexts, cf. [Prop.
4.1.27]rumely1989[Prop. A.12]rumely2013.

######  Theorem 5.24.

Let $f,v,j$ be as in Proposition 5.22, and write

$r_{1}=|a_{j}|_{v}^{\frac{1}{d-j}},\qquad
r_{2}=|a_{j}|_{v}^{\frac{1}{j}(\frac{1}{d-j}-1)},\qquad
r_{3}=|a_{j}|_{v}^{-\frac{1}{j}}.$

Let $\mu_{f,v}=\alpha_{1}\mu_{1}+\alpha_{2}\mu_{2}+\alpha_{3}\mu_{3}$ for some
$\alpha_{i}\in[0,1]$ and $\mu_{i}$ probability measures supported on the
strata $S_{i}:=\overline{D}_{\operatorname{an}}(0,r_{i})\setminus
D_{\operatorname{an}}(0,r_{i})$. Then

$\displaystyle\alpha_{1}$ $\displaystyle=\frac{d-j}{d},$ $\displaystyle
I(\mu_{1})$ $\displaystyle=-\frac{j}{(d-j)^{2}}\log|a_{j}|_{v},$
$\displaystyle\alpha_{2}$ $\displaystyle=\frac{j(d-j)}{d^{2}},$ $\displaystyle
I(\mu_{2})$
$\displaystyle=-\left(\frac{1}{j}+\frac{1}{(d-j)^{2}}\right)\log|a_{j}|_{v},$
$\displaystyle\alpha_{3}$ $\displaystyle=\frac{j^{2}}{d^{2}},$ $\displaystyle
I(\mu_{3})$
$\displaystyle=-\left(\frac{1}{j}+\frac{1}{j^{2}}\right)\log|a_{j}|_{v}.$

###### Proof.

For $c\in K_{f,v}\cap\mathbb{C}_{v}$, we consider the Newton polygon for the
polynomial $f-c$. This is the same as the Newton polygon for $f$ except when
$|c|_{v}=r_{1}>1$, where the constant coefficient then has absolute value
$r_{1}$. The two possibilities are shown in fig. 1.

$\log|\cdot|_{v}$$0$$j$$d$$\frac{1}{d-j}\log|a_{j}|_{v}$$\log|a_{j}|_{v}$
Figure 1. Newton polygon for $f-c$ for $c\in K_{f,v}\cap\mathbb{C}_{v}$, when
$\lvert c\rvert_{v}=r_{1}$ (below), or $\lvert c\rvert_{v}=r_{2}$ or $r_{3}$
(above).

We conclude that:

  * •

If $|c|_{v}=r_{1}$, then $f-c$ has $d-j$ roots with absolute value $r_{1}$,
and $j$ roots with absolute value $r_{2}$;

  * •

If $|c|_{v}=r_{2}$ or $r_{3}$, then $f-c$ has $d-j$ roots with absolute value
$r_{1}$, and $j$ roots with absolute value $r_{3}$.

Hence for any measure $\rho$ with finite support in the Type I points within
the three strata $S_{1}\cup S_{2}\cup S_{3}$, we have

$\begin{pmatrix}(f^{*}\rho)(S_{1})\\\ (f^{*}\rho)(S_{2})\\\
(f^{*}\rho)(S_{3})\end{pmatrix}=\begin{pmatrix}d-j&d-j&d-j\\\ j&0&0\\\
0&j&j\end{pmatrix}\begin{pmatrix}\rho(S_{1})\\\ \rho(S_{2})\\\
\rho(S_{3})\end{pmatrix}.$

Now let $\rho=d^{-n}(f^{n})^{*}\delta_{z}$ for any non-exceptional
$z\in\mathbb{C}_{v}$ with (say) $|z|_{v}=r_{1}$. Taking $n\to\infty$ and using
equidistribution, we see that the above identity holds for $\rho=\mu_{f,v}$ as
well. But $f^{*}\mu_{f,v}=d\cdot\mu_{f,v}$, so we have

$d\begin{pmatrix}\alpha_{1}\\\ \alpha_{2}\\\
\alpha_{3}\end{pmatrix}=\begin{pmatrix}d-j&d-j&d-j\\\ j&0&0\\\
0&j&j\end{pmatrix}\begin{pmatrix}\alpha_{1}\\\ \alpha_{2}\\\
\alpha_{3}\end{pmatrix}.$

Combined with $\alpha_{1}+\alpha_{2}+\alpha_{3}=1$, we find that indeed

$\begin{pmatrix}\alpha_{1}\\\ \alpha_{2}\\\
\alpha_{3}\end{pmatrix}=\frac{1}{d^{2}}\begin{pmatrix}d(d-j)\\\ j(d-j)\\\
j^{2}\end{pmatrix}.$

We now compute the energies $I(\mu_{i})$. By eq. 10,

$0=I(\mu_{f,v})=I(\alpha_{1}\mu_{1}+\alpha_{2}\mu_{2}+\alpha_{3}\mu_{3})=\alpha_{1}^{2}I(\mu_{1})+\alpha_{2}^{2}I(\mu_{2})+\alpha_{3}^{2}I(\mu_{3})$
$+2\alpha_{1}\alpha_{2}\log(r_{1})+2\alpha_{1}\alpha_{3}\log(r_{1})+2\alpha_{2}\alpha_{3}\log(r_{2}).$

But taking the sum of $\alpha_{i}$ times the $i$-th expression above, we
recover the expression for $I(\mu_{f,v})$, which has value $0$; hence each of
the above expressions is $0$. Now we get the claimed values of $I(\mu_{i})$ by
isolating them within each expression.

∎

######  Corollary 5.25.

In the setting of Theorem 5.24, we have

$G_{f,v}(\xi)\leq\begin{cases}\log|\xi|_{v}&\text{if
}|\xi|_{v}\in[r_{1},\infty)\\\
\frac{j}{d}\log|\xi|_{v}+\frac{1}{d}\log|a_{j}|_{v}&\text{if
}|\xi|_{v}\in[r_{2},r_{1}]\\\
\frac{j^{2}}{d^{2}}\log|\xi|_{v}+\frac{j+1}{d^{2}}\log|a_{j}|_{v}&\text{if
}|\xi|_{v}\in[r_{3},r_{2}]\\\ \frac{1}{d^{2}}\log|a_{j}|_{v}&\text{if
}|\xi|_{v}\in[0,r_{3}],\end{cases}$

with equality when $|\xi|_{v}\not\in\\{r_{1},r_{2},r_{3}\\}$.

###### Proof.

We have
$G_{f,v}=\alpha_{1}G_{\mu_{1}}+\alpha_{2}G_{\mu_{2}}+\alpha_{3}G_{\mu_{3}}$,
where $G_{\mu_{i}}$ is the equilibrium potential for $\mu_{i}$. Since
$|w|_{v}=r_{i}$ for all $w\in\operatorname{supp}\mu_{i}$, we have

$G_{\mu_{i}}(\xi)=\int\delta(\xi,w)_{\infty}d\mu_{i}(w)\leq\log\max(|\xi|_{v},r_{i}),$

with equality when $|\xi|_{v}\neq r_{i}$. The claim follows by routine
computation. ∎

We conclude this section by computing the capacities and potentials on some
sets we will use in Section 6.

######  Lemma 5.26.

Let $s_{1}>s_{2}>0$, and let
$K_{i}\subseteq\overline{D}_{\operatorname{an}}(0,s_{i})\setminus
D_{\operatorname{an}}(0,s_{i})$ be compact sets with equilibrium measures
$\nu_{i}$. Then the equilibrium measure $\nu$ on $K=K_{1}\cup K_{2}$ is given
by $\nu=\alpha_{1}\nu_{1}+\alpha_{2}\nu_{2}$ for some
$\alpha_{1},\alpha_{2}\in[0,1]$.

Moreover, we have

$I(\nu)=\frac{\log^{2}s_{1}-I(\nu_{1})I(\nu_{2})}{2\log
s_{1}-I(\nu_{1})-I(\nu_{2})}.$

###### Proof.

By Lemma 5.23, $\nu$ has the given form. Now

(11) $I(\nu)=a\alpha_{1}^{2}+b\alpha_{2}^{2}+2c\alpha_{1}\alpha_{2},\ $

where $a=I(\nu_{1})$, $b=I(\nu_{2})$, and $c=\log s_{1}$. It is easy to verify
(e.g., by Lagrange multipliers) that this quadratic form is minimized on
$\alpha_{1}+\alpha_{2}=1$ at

$(\alpha_{1},\alpha_{2})=\left(\frac{c-b}{2c-a-b},\frac{c-a}{2c-a-b}\right).$

Since $K_{1},K_{2}\subseteq\overline{D}_{\operatorname{an}}(0,s_{1})$, we have
$a,b\leq V(\overline{D}_{\operatorname{an}}(0,s_{1}))=c$, so
$\alpha_{1},\alpha_{2}\in[0,1]$. Finally, substituting the values of
$\alpha_{1},\alpha_{2}$ into eq. 11 yields $I(\nu)=\frac{c^{2}-ab}{2c-a-b}$. ∎

######  Proposition 5.27.

In the setting of Theorem 5.24, define
$\mathcal{R}=|a_{j}|_{v}^{j/(d^{2}-j^{2})}$,

$S_{f,v}^{\cup}=\\{\zeta(0,\mathcal{R})\\}\cup\operatorname{supp}\mu_{1},\qquad
S_{f,v}^{\cap}=\operatorname{supp}\mu_{2}\cup\operatorname{supp}\mu_{3}.$

Then

$\displaystyle\sup_{S_{f,v}^{\cup}}(G_{f,v}+\log^{+}|\cdot|_{v})$
$\displaystyle=\frac{1}{d-j}\log|a_{j}|_{v},$ $\displaystyle
V(S_{f,v}^{\cup})$ $\displaystyle=\frac{1}{2(d-j)}\log|a_{j}|_{v},$
$\displaystyle\sup_{\mathclap{\overline{D}_{\operatorname{an}}(0,1)}}(G_{f,v}+\log^{+}|\cdot|_{v})$
$\displaystyle=\frac{1}{d}\log|a_{j}|_{v},$ $\displaystyle
V(\overline{D}_{\operatorname{an}}(0,1))$ $\displaystyle=0,$
$\displaystyle\sup_{S_{f,v}^{\cap}}(G_{f,v}+\log^{+}|\cdot|_{v})$
$\displaystyle=0,$ $\displaystyle V(S_{f,v}^{\cap})$
$\displaystyle=-\frac{1}{j}\log|a_{j}|_{v}.$

###### Proof.

Write $\nu_{f,v}^{\cup},\nu_{f,v}^{\cap}$ for the equilibrium measures on
$S_{f,v}^{\cup},S_{f,v}^{\cap}$ respectively. Then

$\nu_{f,v}^{\cup}=\alpha\mu_{1}+\beta\delta_{\zeta(0,\mathcal{R})},\qquad\nu_{f,v}^{\cap}=\alpha^{\prime}\mu_{2}+\beta^{\prime}\mu_{3}.$

By Corollary 5.25, the values of $G_{f,v}$ and $\log^{+}|\cdot|_{v}$ on each
of the pieces are given as follows. These yield the identities in the left
column of the claim.

$\begin{array}[]{rcc}&G_{f,v}&\log^{+}|\cdot|_{v}\\\
\hline\cr\zeta(0,\mathcal{R})&\frac{d}{d^{2}-j^{2}}\log|a_{j}|_{v}&\frac{j}{d^{2}-j^{2}}\log|a_{j}|_{v}\\\
\operatorname{supp}\mu_{1}&0&\frac{1}{d-j}\log|a_{j}|_{v}\\\
\overline{D}_{\operatorname{an}}(0,1)&\leq\frac{1}{d}\log|a_{j}|_{v}&0\\\
\operatorname{supp}\mu_{2}&0&0\\\ \operatorname{supp}\mu_{3}&0&0\end{array}$

Next, we have $V(\overline{D}_{\operatorname{an}}(0,1))=\log 1=0$. Also, by
Lemma 5.26,

$I(\nu_{f,v}^{\cup})=\frac{\log^{2}r_{1}-I(\mu_{1})\log\mathcal{R}}{2\log
r_{1}-I(\mu_{1})-\log\mathcal{R}},\quad
I(\nu_{f,v}^{\cap})=\frac{\log^{2}r_{2}-I(\mu_{2})I(\mu_{3})}{2\log
r_{2}-I(\mu_{2})-I(\mu_{3})}.$

By Theorem 5.24, these expressions simplify to the right column of the claim.
∎

##  6\. Shared Small Points on average

Let $f,g$ be polynomials of degree $d\geq 2$ over $\mathbb{Q}$. We will now
prove Theorem 1.3 on the essential minimum of
$\widehat{h}_{f}+\widehat{h}_{g}$ for generic pairs $(f,g)$. We first start
off with some results that apply to all pairs $(f,g)$, before specializing to
the generic case.

Let $\overline{L}_{f}$ be the canonical adelic line bundle associated to our
polynomial $f\in\mathbb{Q}[z]$ of degree $d\geq 2$. For background on adelic
line bundles, readers may refer to [PST12]. Then the height associated to
$\overline{L}_{f}$ is the canonical height $h_{f}$, and so applying Zhang’s
inequality [Zha95, Thm. 1.10] to the line bundle $\overline{L}_{f\otimes
g}=\overline{L}_{f}\otimes\overline{L}_{g}$ gives us

$\liminf_{\mathbb{P}^{1}(\overline{\mathbb{Q}})}\left(\widehat{h}_{f}+\widehat{h}_{g}\right)+\inf_{\mathbb{P}^{1}(\overline{\mathbb{Q}})}\left(\widehat{h}_{f}+\widehat{h}_{g}\right)\leq\langle\mu_{f},\mu_{g}\rangle\leq
2\liminf_{\mathbb{P}^{1}(\overline{\mathbb{Q}})}\left(\widehat{h}_{f}+\widehat{h}_{g}\right).$

Since $f$ and $g$ are both polynomials, $\infty$ is a common preperiodic point
and thus we have $\inf(\widehat{f}+\widehat{g})=0$. We then get

(12)
$\frac{1}{2}\langle\mu_{f},\mu_{g}\rangle\leq\liminf_{\mathbb{P}^{1}(\overline{\mathbb{Q}})}\left(\widehat{h}_{f}+\widehat{h}_{g}\right)\leq\langle\mu_{f},\mu_{g}\rangle.$

When $\langle\mu_{f},\mu_{g}\rangle=0$, this implies that
$\liminf(\widehat{h}_{f}+\widehat{h}_{g})=0$, but in general only gives a
possible range. We now give a different proof of (12) using equidistribubtion
and the energy pairing. We first prove the upper bound, which is an easy
consequence of the Arakelov–Zhang pairing in [PST12].

######  Proposition 6.1.

$\displaystyle\liminf_{x\in{\overline{\mathbb{Q}}}}\left(\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\right)\leq\langle\mu_{f},\mu_{g}\rangle$.

###### Proof.

Let $(x_{n})$ be a sequence of pairwise distinct points in
${\overline{\mathbb{Q}}}$ with $\widehat{h}_{f}(x_{n})\to 0$ (for instance
from $\operatorname{Prep}(f)$, which is infinite). By [PST12, Thm. 1], we have
$\widehat{h}_{g}(x_{n})\to\langle\mu_{f},\mu_{g}\rangle$, so
$\langle\mu_{f},\mu_{g}\rangle$ is a limit point for the values of
$\widehat{h}_{f}+\widehat{h}_{g}$ on ${\overline{\mathbb{Q}}}$. ∎

We next do the lower bound.

######  Proposition 6.2.

$\displaystyle\liminf_{x\in\mathbb{\overline{Q}}}\left(\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\right)\geq\frac{1}{2}\langle\mu_{f},\mu_{g}\rangle$.

###### Proof.

Given $x\in\mathbb{\overline{Q}}$, write $F\subseteq\mathbb{\overline{Q}}$ for
its $\operatorname{Gal}({\overline{\mathbb{Q}}}/\mathbb{Q})$-orbit, and let
$N=|F|$. Let $K$ be a number field containing $F$, and let
$\varepsilon=\\{\varepsilon_{v}\\}_{v\in M_{K}}$ be the adelic radius given in
the proof of Theorem 3.18.

Recall the parallelogram law for bilinear forms $(\cdot,\cdot)$:

$(a+b,a+b)+(a-b,a-b)=2(a,a)+2(b,b).$

Applying this for $a=\frac{\mu_{f}+\mu_{g}}{2}-[F]_{\varepsilon}$,
$b=\frac{\mu_{f}-\mu_{g}}{2}$, $(\cdot,\cdot)=(\cdot,\cdot)_{v}$, we have

(13)
$(\mu_{f}-[F]_{\varepsilon},\mu_{f}-[F]_{\varepsilon})_{v}+(\mu_{g}-[F]_{\varepsilon},\mu_{g}-[F]_{\varepsilon})_{v}$
$=2\left(\frac{\mu_{f}+\mu_{g}}{2}-[F]_{\varepsilon},\frac{\mu_{f}+\mu_{g}}{2}-[F]_{\varepsilon}\right)_{v}+\frac{1}{2}(\mu_{f}-\mu_{g},\mu_{f}-\mu_{g})_{v}\geq\frac{1}{2}(\mu_{f}-\mu_{g},\mu_{f}-\mu_{g})_{v}.$

Summing over $v\in M_{K}$ and using Proposition 3.16, we obtain

$\frac{1}{2}\langle\mu_{f},\mu_{g}\rangle\leq\langle\mu_{f},[F]_{\varepsilon}\rangle+\langle\mu_{g},[F]_{\varepsilon}\rangle\leq\widehat{h}_{f}(x)+\widehat{h}_{g}(x)+O\left(d\frac{\log
N}{N}(h(f)+h(g)+1)\right)\leq\widehat{h}_{f}(x)+\widehat{h}_{g}(x)+C_{d,f,g}\frac{\log
N}{N}.$

Now for any $\delta>0$, choose $M$ sufficiently large such that $\frac{\log
m}{m}\leq\frac{\delta}{C_{d,f,g}}$ for all $m\geq M$. Then we have the
implication

$\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\leq\frac{1}{2}\langle\mu_{f},\mu_{g}\rangle-\delta\implies
N=[\mathbb{Q}(x):\mathbb{Q}]<M.$

By Northcott’s theorem, there are only finitely many
$x\in{\overline{\mathbb{Q}}}$ satisfying both the hypothesis and our degree
bound. Hence

$\liminf_{x\in{\overline{\mathbb{Q}}}}\left(\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\right)\geq\frac{1}{2}\langle\mu_{f},\mu_{g}\rangle-\delta,$

and we are done since $\delta>0$ is arbitrary. ∎

The proof of Proposition 6.2 also explains why it is not an equality in
general. For (13) to be an equality, we must have
$[F]\approx\frac{1}{2}(\mu_{f}+\mu_{g})$ at every place $v$, but then
$\sum_{v\in M_{K}}N_{v}([F],{F})_{v}\approx\sum_{v\in
M_{K}}N_{v}\langle\mu_{f},\mu_{g}\rangle_{v}$ which is positive whenever
$\mu_{f}\not=\mu_{g}$, contradicting the product formula.

We now move onto the case of a generic pair. We start by controlling the
archimedean place. Recall from Lemma 3.2 that $K_{f,\infty}\subseteq
B(0,R_{f,\infty})$, where
$R_{f,\infty}=3\max(1,|a_{d-1}|_{\infty},|a_{d-2}|_{\infty}^{1/2},\ldots,|a_{0}|_{\infty}^{1/d})$.

######  Lemma 6.3.

For any $\varepsilon$-ordinary pair
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$, we have $R_{f,\infty}\leq
3X^{\varepsilon}$ and $R_{g,\infty}\leq 3X^{2\varepsilon}$.

###### Proof.

Since $f\in\mathcal{P}(X)$, the numerator of $a_{j}$ has absolute value at
most $X$. Also, since $(f,g)$ is $\varepsilon$-ordinary, we have
$\operatorname{denom}(a_{j})\geq X^{1-2\varepsilon}$. Hence
$|a_{j}|_{\infty}\leq X^{2\varepsilon}$, and now $a_{d-1}=0$ implies
$R_{f,\infty}\leq 3X^{\varepsilon}$. The argument for $g$ is analogous. ∎

The key step in the lower bound for
$\liminf_{\overline{\mathbb{Q}}}(\widehat{h}_{f}+\widehat{h}_{g})$ is the
following estimate on the sum of local heights, with an extra term
$-\tfrac{1}{2}\log|x|_{v}$ which vanishes by the product formula when we sum
over all places.

######  Lemma 6.4.

For any $\varepsilon$-ordinary pair
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$, and any $v\in M_{K}$
associated to $a_{j}$, we have

$\inf_{x\in{\mathbb{A}^{1}_{\operatorname{Berk},v}}}\left(G_{f,v}(x)+G_{g,v}(x)-\tfrac{1}{2}\log|x|_{v}\right)=\begin{cases}\frac{1}{2(d-j)}\log|a_{j}|_{v}&j\leq\frac{d-1}{2},\\\
\frac{d-j-1}{2j(d-j)}\log|a_{j}|_{v}&j\geq\frac{d-1}{2}.\end{cases}$

The same statement holds with $a_{j}$ replaced by $b_{j}$.

###### Proof.

Let
$\psi_{v}(\zeta)=G_{f,v}(\zeta)+G_{g,v}(\zeta)-\frac{1}{2}\log|\zeta|_{v}$.
Since $\psi_{v}(\zeta)\to\infty$ as $|\zeta|_{v}\to\infty$, the minimum value
of $\psi_{v}$ occurs on the support of the positive part of
$\triangle\psi_{v}$, which is $J_{f,v}\cup\\{\zeta(0,1)\\}$. Now $v$ is
associated to $a_{j}$, so $g$ has explicit good reduction at $v$, and thus
$G_{g,v}=\log^{+}|\cdot|_{v}$. Hence we can compute the value of $\psi_{v}$ at
each of the three strata of $J_{f,v}$ (Theorem 5.24) and at $\zeta(0,1)$
(Corollary 5.25), as follows:

$\begin{array}[]{*5c}|\zeta|_{v}&G_{f,v}(\zeta)&G_{g,v}(\zeta)&\log|\zeta|_{v}&\psi_{v}(\zeta)\\\
\hline\cr
r_{1}&0&\frac{1}{d-j}\log|a_{j}|_{v}&\frac{1}{d-j}\log|a_{j}|_{v}&\frac{1}{2(d-j)}\log|a_{j}|_{v}\\\
1&\frac{1}{d}\log|a_{j}|_{v}&0&0&\frac{1}{d}\log|a_{j}|_{v}\\\
r_{2}&0&0&\frac{1}{j}(\frac{1}{d-j}-1)\log|a_{j}|_{v}&\frac{1}{2j}(1-\frac{1}{d-j})\log|a_{j}|_{v}\\\
r_{3}&0&0&-\frac{1}{j}\log|a_{j}|_{v}&\frac{1}{2j}\log|a_{j}|_{v}\end{array}$

Since $\frac{2}{d}\geq\min(\frac{1}{j},\frac{1}{d-j})$, and
$\frac{1}{j}>\frac{1}{j}(1-\frac{1}{d-j})$, we have

$\inf_{\mathbb{A}^{1}_{\operatorname{Berk},v}}\psi_{v}=\min\left(\tfrac{1}{2j}\left(1-\tfrac{1}{d-j}\right),\tfrac{1}{2(d-j)}\right)\log|a_{j}|_{v},$

which is equivalent to the claim. The argument for $b_{j}$ is analogous. ∎

######  Theorem 6.5.

For any $\varepsilon$-ordinary pair
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$, we have

$\inf_{x\in{\overline{\mathbb{Q}}}\setminus\\{0\\}}\left(\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\right)\geq
2\left(\log 2-O(d^{2}\varepsilon+\tfrac{\log d}{d})\right)\log X-O(1).$

###### Proof.

Let $K$ be a number field with $x\in K\setminus\\{0\\}$. For any $v\in M_{K}$,
write $\psi_{v}(x)=G_{f,v}(x)+G_{g,v}(x)-\frac{1}{2}\log|x|_{v}$. Then by the
product formula $\sum_{v\in M_{K}}n_{v}\log|x|_{v}=0$, we have

$\widehat{h}_{f}(x)+\widehat{h}_{g}(x)=\frac{1}{[K:\mathbb{Q}]}\sum_{v\in
M_{K}}\\!n_{v}\psi_{v}(x),$

We split the sum over good, bad, and archimedean places. For good places, by
Lemma 6.4 we have

$\frac{1}{[K:\mathbb{Q}]}\sum_{v\in M_{K}\text{
good}}\\!\\!n_{v}\psi_{v}(x)\geq\frac{1}{[K:\mathbb{Q}]}\left(\sum_{\begin{subarray}{c}v\text{
assoc.\ }a_{j}\\\
0<j\leq\frac{d-1}{2}\end{subarray}}\\!\\!n_{v}\frac{1}{2(d-j)}\log|a_{j}|_{v}+\\!\\!\sum_{\begin{subarray}{c}v\text{
assoc.\ }b_{j}\\\
0<j\leq\frac{d-1}{2}\end{subarray}}\\!\\!n_{v}\frac{1}{2(d-j)}\log|b_{j}|_{v}\right.$
$+\left.\\!\\!\sum_{\begin{subarray}{c}v\text{ assoc.\ }a_{j}\\\
\frac{d-1}{2}<j<d\end{subarray}}\\!\\!n_{v}\frac{d-j-1}{2j(d-j)}\log|a_{j}|_{v}+\\!\\!\sum_{\begin{subarray}{c}v\text{
assoc.\ }b_{j}\\\
\frac{d-1}{2}<j<d\end{subarray}}\\!\\!n_{v}\frac{d-j-1}{2j(d-j)}\log|b_{j}|_{v}\right).$

By Lemma 5.17, this is

(14) $\geq
2(1-O(d\varepsilon))\left(\sum_{0<j\leq\frac{d-1}{2}}\frac{1}{2(d-j)}+\sum_{\frac{d-1}{2}<j<d}\frac{d-j-1}{2j(d-j)}\right)\log
X.$

We recognize the first sum in (14) as a Riemann sum:

(15)
$\sum_{0<j\leq\frac{d-1}{2}}\frac{1}{2(d-j)}\geq\int_{0}^{1/2}\frac{1}{2(1-t)}dt-O\left(\frac{1}{d}\right)=\frac{\log
2}{2}-O\left(\frac{1}{d}\right).$

The summand in the second sum of (14) splits into
$\frac{1}{2j}-\frac{1}{2j(d-j)}$. As before,

(16)
$\sum_{\frac{d-1}{2}<j<d}\frac{1}{2j}\geq\int_{1/2}^{1}\frac{1}{2t}dt-O\left(\frac{1}{d}\right)=\frac{\log
2}{2}-O\left(\frac{1}{d}\right),$

while

(17)
$\sum_{\frac{d-1}{2}<j<d}\frac{1}{2j(d-j)}\leq\sum_{\frac{d-1}{2}<j<d}\frac{1}{j(d-j)}\leq
O\left(\frac{\log d}{d}\right).$

Substituting (15), (16), (17) back into (14), we have

(18) $\frac{1}{[K:\mathbb{Q}]}\sum_{v\in M_{K}\text{
good}}\\!\\!n_{v}\psi_{v}(x)\geq 2(1-O(d\varepsilon))\left(\log
2-O(\tfrac{\log d}{d})\right)\log X$ $\geq 2\left(\log
2-O(d\varepsilon+\tfrac{\log d}{d})\right)\log X.$

For bad places, we have
$K_{f,v}\subseteq\overline{D}_{\operatorname{an}}(0,R_{f,v})\subseteq\overline{D}_{\operatorname{an}}(0,\mathcal{M}_{f,v})$
by Proposition 3.9, and similarly for $g$. Hence
$G_{f,v}(x)=G_{g,v}(x)=\log^{+}|x|_{v}$ for all $x$ such that
$|x|_{v}>\max(\mathcal{M}_{f,v},\mathcal{M}_{g,v})$. Therefore

$\psi_{v}(x)\geq\begin{cases}\tfrac{3}{2}\log|x|_{v}&\text{if
}|x|_{v}>\max(\mathcal{M}_{f,v},\mathcal{M}_{g,v})\\\
-\frac{1}{2}\log\max(\mathcal{M}_{f,v},\mathcal{M}_{g,v})&\text{if
}|x|_{v}\leq\max(\mathcal{M}_{f,v},\mathcal{M}_{g,v}).\end{cases}$

Hence using Proposition 5.20,

(19) $\frac{1}{[K:\mathbb{Q}]}\sum_{v\in M_{K}\text{
bad}}\\!\\!n_{v}\psi_{v}(x)\geq-\frac{1}{2[K:\mathbb{Q}]}\sum_{v\in
M_{K}\text{
bad}}\\!\\!n_{v}\left(\log\mathcal{M}_{f,v}+\log\mathcal{M}_{g,v}\right)\geq-O(d^{2}\varepsilon)\log
X.$

For archimedean places $v\mid\infty$ we split into two cases. If $|x|_{v}\leq
R_{f,\infty}+1$ then

$\psi_{v}(x)\geq-\tfrac{1}{2}\log|x|_{v}\geq-\tfrac{1}{2}\log(R_{f,\infty}+1).$

If $|x|_{v}\geq R_{f,\infty}+1$, then for all $w\in K_{f,v}$ we have
$|w/x|_{v}\leq R_{f,\infty}/(R_{f,\infty}+1)$, so

$G_{f,v}(x)-\log|x|_{v}\geq\int\log|\frac{x-w}{x}|_{v}d\mu_{f,v}(w)\geq-\log(R_{f,\infty}+1),$

and thus

$\psi_{v}(x)\geq\frac{1}{2}(G_{f,v}(x)-\log|x|_{v})\geq-\frac{1}{2}\log(R_{f,\infty}+1).$

In either case, by Lemma 6.3, we have

(20)
$\frac{1}{[K:\mathbb{Q}]}\sum_{v\mid\infty}n_{v}\psi_{v}(x)\geq-\frac{1}{2}\log(R_{f,\infty}+1)=-O(\varepsilon\log
X+1).$

Adding (18), (19) and (20) yields the desired bound. ∎

We now prove our upper bound of
$\liminf\widehat{h}_{f}(x)+\widehat{h}_{g}(x)$, which is improves Proposition
6.1 by a constant. As in the proof of Proposition 6.1, we need to construct
infinitely many $x\in{\overline{\mathbb{Q}}}$ such that
$\widehat{h}_{f}(x)+\widehat{h}_{g}(x)$ is small. Our main tool will be
Theorem 6.7, which is a special case of the adelic Fekete–Szegő theorem
[Rum13, Thm. 0.4].

######  Definition 6.6.

A compact Berkovich adelic set $\mathbb{E}=(E_{v})_{v\in M_{\mathbb{Q}}}$ is a
collection of compact affinoids
$E_{v}\subseteq{\mathbb{A}^{1}_{\operatorname{Berk},v}}$ for each place $v\in
M_{\mathbb{Q}}$, with $E_{v}=\overline{D}_{\operatorname{an}}(0,1)$ for all
but finitely many $v$. The Robin constant $V(\mathbb{E})$ of $\mathbb{E}$ is
$V(\mathbb{E}):=\sum_{v\in M_{\mathbb{Q}}}V(E_{v})$.

######  Theorem 6.7.

Let $\mathbb{E}=(E_{v})_{v\in M_{\mathbb{Q}}}$ be a compact Berkovich adelic
set. If $V(\mathbb{E})\geq 0$, then there exists infinitely many
$x\in\overline{\mathbb{Q}}$ such that the
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-orbit of $x$ lies
inside $E_{v}$ at every place $v$.

Our strategy is to construct, given any $\varepsilon$-ordinary $(f,g)$, a
compact Berkovich adelic set $\mathbb{E}$ with $V(\mathbb{E})>0$ and
$\sum_{v}n_{v}\sup_{x\in E_{v}}\left(G_{f}(x)+G_{g}(x)\right)$ small.

######  Theorem 6.8.

For any $\varepsilon$-ordinary pair
$(f,g)\in\mathcal{P}_{c}(X)\times\mathcal{P}(X)$, we have

$\liminf_{x\in{\overline{\mathbb{Q}}}}\left(\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\right)\leq
2\left(\mathcal{C}+O(d\varepsilon+\tfrac{1}{d})\right)\log X+O(1),$

where
$\mathcal{C}=\log\left(\frac{9+\sqrt{17}}{8}\right)+\frac{\sqrt{17}-1}{8}=0.88532\ldots$

###### Proof.

Let $\alpha:=\frac{\sqrt{17}-1}{8}$, and let $c>0$ be a small parameter to be
chosen later. We consider the adelic set $\mathbb{E}=(E_{v})_{v\in
M_{\mathbb{Q}}}$ given by

$E_{v}\coloneqq\begin{cases}S_{f,v}^{\cup}&v\text{ assoc.\ }a_{j}\text{ with
}0<\frac{j}{d}<\alpha+c,\\\ S_{g,v}^{\cup}&v\text{ assoc.\ }b_{j}\text{ with
}0<\frac{j}{d}<\alpha+c,\\\ S_{f,v}^{\cap}&v\text{ assoc.\ }a_{j}\text{ with
}2\alpha<\frac{j}{d}<1,\\\ S_{g,v}^{\cap}&v\text{ assoc.\ }b_{j}\text{ with
}2\alpha<\frac{j}{d}<1,\\\
\overline{D}_{\operatorname{an}}(0,1)&\text{otherwise},\end{cases}$

with $S_{f,v}^{\cup},S_{f,v}^{\cap}$ as defined in Proposition 5.27. Using
Proposition 5.27 and Lemma 5.17, we bound $V(\mathbb{E})$ from below:

$V(\mathbb{E})=\sum_{v\in
M_{\mathbb{Q}}}\\!V(E_{v})=\sum_{\begin{subarray}{c}v\text{ assoc.\ }a_{j}\\\
0<\frac{j}{d}<\alpha+c\end{subarray}}\frac{1}{2(d-j)}\log|a_{j}|_{v}+\\!\\!\sum_{\begin{subarray}{c}v\text{
assoc.\ }b_{j}\\\
0<\frac{j}{d}<\alpha+c\end{subarray}}\frac{1}{2(d-j)}\log|b_{j}|_{v}$
$-\\!\\!\sum_{\begin{subarray}{c}v\text{ assoc.\ }a_{j}\\\
2\alpha<\frac{j}{d}<1\end{subarray}}\frac{1}{j}\log|a_{j}|_{v}-\\!\\!\sum_{\begin{subarray}{c}v\text{
assoc.\ }b_{j}\\\
2\alpha<\frac{j}{d}<1\end{subarray}}\frac{1}{j}\log|b_{j}|_{v}\geq\left(2-O(d\varepsilon)\right)\log
X\\!\\!\sum_{0<\frac{j}{d}<\alpha+c}\frac{1}{2(d-j)}-2\log
X\\!\\!\sum_{2\alpha<\frac{j}{d}<1}\frac{1}{j}.$

We recognize the sums above as Riemann sums:

$\sum_{0<\frac{j}{d}<\alpha+c}\frac{1}{2(d-j)}\geq\int_{0}^{\alpha+c}\frac{1}{2(1-t)}dt-O\left(\frac{1}{d}\right)=\frac{\log(1-\alpha-c)}{2}-O\left(\frac{1}{d}\right),$
$\sum_{2\alpha<\frac{j}{d}<1}\frac{1}{j}\geq\int_{2\alpha}^{1}\frac{1}{t}dt-O\left(\frac{1}{d}\right)=\log(2\alpha)-O\left(\frac{1}{d}\right).$

Hence

$V(\mathbb{E})\geq\left(-\left(1-O(d\varepsilon)\right)\log(1-\alpha-c)-2\log(2\alpha)-O(\tfrac{1}{d})\right)\log
X\geq\left(\Theta(c)-O(d\varepsilon+\tfrac{1}{d})\right)\log X,$

where we have used the identity $-\frac{1}{2}\log(1-\alpha)=\log(2\alpha)$.
Hence we may choose $c=O(d\varepsilon+\frac{1}{d})$ so that $V(\mathbb{E})\geq
0$.

We want to apply the adelic Fekete–Szegő theorem to $\mathbb{E}$, but the
issue is that $S_{f,v}^{\cup},S_{f,v}^{\cap}$ are not Berkovich affinoids. To
fix this, we construct a new adelic set
$\widetilde{\mathbb{E}}=(\widetilde{E}_{v})_{v\in M_{\mathbb{Q}}}$ by
replacing the sets $S_{f,v}^{\cup},S_{f,v}^{\cap}$ in the definition of
$\mathbb{E}$ by compact affinoids $\widetilde{S}_{f,v}^{\cup}\supseteq
S_{f,v}^{\cup}$ and $\widetilde{S}_{f,v}^{\cap}\supseteq S_{f,v}^{\cap}$ such
that

(21)
$\sup_{\widetilde{S}_{f,v}^{\cup}}(G_{f,v}+\log^{+}|\cdot|_{v})\leq\frac{1}{d-j}\log|a_{j}|_{v}+\eta,\qquad\sup_{\widetilde{S}_{f,v}^{\cap}}(G_{f,v}+\log^{+}|\cdot|_{v})\leq\eta,$

where $\eta>0$ is a small parameter to be chosen later. (This is possible by
Proposition 5.27, and the fact that $G_{f,v}+\log^{+}|\cdot|_{v}$ is
continuous on ${\mathbb{A}^{1}_{\operatorname{Berk},v}}$.) Similarly, we also
replace $S_{g,v}^{\cup},S_{g,v}^{\cap}$ with closed neighborhoods
$\widetilde{S}_{g,v}^{\cup},\widetilde{S}_{g,v}^{\cap}$.

Now $\widetilde{\mathbb{E}}$ is a compact Berkovich adelic set and
$V(\widetilde{\mathbb{E}})\geq V(\mathbb{E})\geq 0$, so by Theorem 6.7 we can
find infinitely many distinct $x_{n}\in{\overline{\mathbb{Q}}}$ whose
$\operatorname{Gal}({\overline{\mathbb{Q}}}/\mathbb{Q})$-orbit $F_{n}$ lies
entirely in $\widetilde{\mathbb{E}}$.

Let $K_{n}$ be a number field containing $F_{n}$. The local contribution to
$\widehat{h}_{f}(x_{n})$ at $v\in M_{\mathbb{Q}}$ is

$\frac{1}{[K_{n}:\mathbb{Q}]}\sum_{\begin{subarray}{c}v^{\prime}\in
M_{K_{n}}\\\ v^{\prime}\mid
v\end{subarray}}\\!\\!n_{v^{\prime}}G_{f,v^{\prime}}(x_{n})=\frac{1}{|F_{n}|}\sum_{x\in
F_{n}}G_{f,v}(x)\leq\sup_{\widetilde{E}_{v}}G_{f,v},$

and similarly for $g$. Hence we are left to bound
$\sup_{\widetilde{E}_{v}}(G_{f,v}+G_{g,v})$ from above at good, bad, and
archimedean places $v\in M_{\mathbb{Q}}$.

For good places $v\in M_{\mathbb{Q}}$, if $v$ is associated to $a_{j}$ then
$g$ has good reduction at $v$, so
$G_{f,v}+G_{g,v}=G_{f,v}+\log^{+}|\cdot|_{v}$, and we are in the setting of
Proposition 5.27 and (21). If $v$ is associated to $b_{j}$, we use the
analogous statements with $f,g$ swapped. Working through the possible cases in
terms of $j$, we have

$\sup_{\widetilde{E}_{v}}(G_{f,v}+G_{g,v})\leq\begin{cases}\frac{1}{d-j}\log|a_{j}|_{v}+\eta&v\text{
assoc.\ }a_{j}\text{ with }0<\frac{j}{d}<\alpha+c,\\\
\frac{1}{d-j}\log|b_{j}|_{v}+\eta&v\text{ assoc.\ }b_{j}\text{ with
}0<\frac{j}{d}<\alpha+c,\\\ \frac{1}{d}\log|a_{j}|_{v}&v\text{ assoc.\
}a_{j}\text{ with }\alpha+c<\frac{j}{d}<2\alpha,\\\
\frac{1}{d}\log|b_{j}|_{v}&v\text{ assoc.\ }b_{j}\text{ with
}\alpha+c<\frac{j}{d}<2\alpha,\\\ \eta&v\text{ assoc.\ }a_{j}\text{ or
}b_{j}\text{ with }2\alpha<\frac{j}{d}<1.\end{cases}$

or bad places $v\in M_{\mathbb{Q}}$, we have by Lemma 5.6 and Lemma 5.5,

$\sup_{\overline{D}_{\operatorname{an}}(0,1)}(G_{f,v}+G_{g,v})\leq
G_{f,v}(\zeta(0,1))+G_{g,v}(\zeta(0,1))=\frac{1}{d}(\log\mathcal{M}_{f,v}+\log\mathcal{M}_{g,v}).$

Finally, for the archimedean place, we have $\widetilde{E}_{\infty}=\partial
B(0,1)$, so for any $z\in\widetilde{E}_{\infty}$ and $w\in K_{f,\infty}$ we
have $|z-w|\leq R_{f,\infty}+1$, with $R_{f,\infty}$ as in Lemma 3.2. Thus

$G_{f,\infty}(z)=\int\log|z-w|_{\infty}d\mu_{f,\infty}(w)\leq\log(R_{f,\infty}+1),$

and similarly for $g$. Hence by Lemma 6.3 we have

$\sup_{\partial
B(0,1)}(G_{f,\infty}+G_{g,\infty})\leq\log(R_{f,\infty}+1)+\log(R_{g,\infty}+1)\leq
O(\varepsilon\log X+1).$

Now we choose $\eta\leq\\#\\{\text{good places }v\in M_{\mathbb{Q}}\\}^{-1}$,
so

$\widehat{h}_{f}(x_{n})+\widehat{h}_{g}(x_{n})\leq\sum_{\begin{subarray}{c}v\text{
assoc.\ }a_{j}\\\
0<\frac{j}{d}<\alpha+c\end{subarray}}\frac{1}{d-j}\log|a_{j}|_{v}+\\!\\!\sum_{\begin{subarray}{c}v\text{
assoc.\ }b_{j}\\\
0<\frac{j}{d}<\alpha+c\end{subarray}}\frac{1}{d-j}\log|b_{j}|_{v}$
$+\\!\\!\sum_{\begin{subarray}{c}v\text{ assoc.\ }a_{j}\\\
\alpha+c<\frac{j}{d}<2\alpha\end{subarray}}\frac{1}{d}\log|a_{j}|_{v}+\\!\\!\sum_{\begin{subarray}{c}v\text{
assoc.\ }b_{j}\\\
\alpha+c<\frac{j}{d}<2\alpha\end{subarray}}\frac{1}{d}\log|b_{j}|_{v}+\frac{1}{d}\sum_{v\text{
bad}}(\log\mathcal{M}_{f,v}+\log\mathcal{M}_{g,v})+O(\varepsilon\log X+1).$

By Proposition 5.20, this is

$\leq\left(2\\!\\!\sum_{0<\frac{j}{d}<\alpha+c}\\!\\!\frac{1}{d-j}+2\\!\\!\sum_{\alpha+c<\frac{j}{d}<2\alpha}\\!\\!\frac{1}{d}+O(d\varepsilon)\right)\log
X+O(1).$

The first sum is a Riemann sum as before, and the second sum is easy to bound:

$\widehat{h}_{f}(x_{n})+\widehat{h}_{g}(x_{n})\leq\left(-2\log(1-\alpha-c)+2(\alpha-c)+O(d\varepsilon+\tfrac{1}{d})\right)\log
X+O(1)$
$=2\left(\mathcal{C}+O\left(c+d\varepsilon+\tfrac{1}{d}\right)\right)\log
X+O(1),$

where $\mathcal{C}=-\log(1-\alpha)+\alpha$, and we are done since
$c=O(d\varepsilon+\frac{1}{d})$. ∎

Putting together Theorems 6.5 and 6.8, we obtain Theorem 1.3. Recall that we
define $\cal{P}$ and $\cal{P}_{c}$ to be the space of monic polynomials and
monic centered polynomials with rational coefficients respectively.

*

###### Proof.

Let $\delta>0$ be given. Then for all large enough $d$ depending on $\delta$,
we can choose $\epsilon$ small enough depending on $d$ such that the terms
$O(d\epsilon+\frac{1}{d}),O(d^{2}\epsilon+\frac{\log d}{d})$ and
$O(d\epsilon+\frac{1}{d})$ occurring in Proposition 5.18, Theorem 6.5 and
Theorem 6.8 respectively are bounded above by $\frac{\delta}{3}$.

Let $X$ be sufficiently large such that for the terms $O(1)$’s occuring in
Theorem 6.5 and Theorem 6.8, we have $\frac{\delta}{3}\log X\geq O(1)$. Then
by Theorem 6.5 and Theorem 6.8, we know that for any $\varepsilon$-ordinary
pair $(f,g)\in\cal{S}(X)$ we have

$2\left(\ln 2-\frac{2}{3}\delta\right)\log
X\leq\liminf_{x\in\overline{\mathbb{Q}}}\left(\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\right)\leq
2\left(\cal{C}+\frac{2}{3}\delta\right)\log X.$

Now Proposition 5.18 implies that for an $\varepsilon$-ordinary pair of
$\cal{S}(X)$, we have

$h(f)+h(g)\geq d\left(2-\frac{1}{3}\delta\right)\log X.$

We also have the trivial upper bound

$h(f)+h(g)\leq 2d\log X$

since each coefficient of $f$ and $g$ have height bounded by $\log X$ and we
have at most $2d$ different coefficients. We now assume $\delta$ is small
enough so that

$\left(2-\frac{1}{3}\delta\right)(\cal{C}+\delta)\geq
2\left(\cal{C}+\frac{2}{3}\delta\right).$

This is possible since $\cal{C}<1$. Then substituting in our bounds of
$h(f)+h(g)$ against $\log X$, we obtain

$(\ln 2-\delta)(h(f)+h(g))\leq
d\liminf_{x\in\overline{\mathbb{Q}}}\left(\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\right)\leq(\cal{C}+\delta)(h(f)+h(g))$

for all $\varepsilon$-ordinary pairs $(f,g)\in\cal{S}(X)$, if $X$ is
sufficiently large. By Proposition 5.15, the proportion of
$\varepsilon$-ordinary pairs in $\cal{S}(X)$ is at least
$1-O_{\epsilon}(d^{2}X^{-\epsilon})$. Clearly we can take
$\epsilon=\frac{c}{d^{2}}$ for some constant $c>0$ that depends on $\delta$
and so the proportion of $\varepsilon$-ordinary pairs will be
$1-O_{d}(d^{2}X^{-c/d^{2}})$. Finally, we set $\cal{T}$ to be the all pairs
$(f,g)$ which are $\varepsilon$-ordinary for some $X$ which is sufficiently
large for our previous bounds to hold. Then

$\frac{|\cal{T}\cap\cal{S}(X)|}{|\cal{S}(X)|}\geq 1-\frac{A}{X^{a}}$

for some constants $a=a(d)>0,A=A(d)>0$ that depend on the degree $d$, and

$(\ln 2-\delta)(h(f)+h(g))\leq
d\liminf_{x\in\overline{\mathbb{Q}}}\left(\widehat{h}_{f}(x)+\widehat{h}_{g}(x)\right)\leq(\cal{C}+\delta)(h(f)+h(g))$

for all $(f,g)\in\cal{T}$ as desired.

∎

## References

  * [BD11] Matthew Baker and Laura Demarco  “Preperiodic points and unlikely intersections”  In _Duke Math. J._ 159.1 Duke University Press, 2011  DOI: 10.1215/00127094-1384773
  * [Ben19] Robert Benedetto  “Dynamics in One Non-Archimedean Variable”  American Mathematical Society, 2019
  * [BR10] Matthew Baker and Robert Rumely  “Potential Theory and Dynamics on the Berkovich Projective Line (Mathematical Surveys and Monographs)”  American Mathematical Society, 2010
  * [CG93] Lennart Carleson and Theodore W. Gamelin  “Complex Dynamics”  Springer New York, 1993
  * [CS93] Gregory S. Call and Joseph H. Silverman  “Canonical heights on varieties with morphisms”  In _Compos. Math._ 89.2 Kluwer Academic Publishers, 1993, pp. 163–205  URL: http://www.numdam.org/item/CM_1993__89_2_163_0/
  * [DKY22] Laura DeMarco, Holly Krieger and Hexi Ye  “Common preperiodic points for quadratic polynomials”  In _J. Mod. Dyn._ 18 American Institute of Mathematical Sciences (AIMS), 2022, pp. 363–413  DOI: 10.3934/jmd.2022012
  * [DM23] Laura DeMarco and Niki Myrto Mavraki  “Dynamics on $\mathbb{P}^{1}$: preperiodic points and pairwise stability”, 2023  arXiv:2212.13215 [math.DS]
  * [Fil17] Paul Fili  “A metric of mutual energy and unlikely intersections for dynamical systems”  arXiv, 2017  DOI: 10.48550/arXiv.1708.08403
  * [FR06] Charles Favre and Juan Rivera-Letelier  “Equidistribution quantitative des points de petite hauteur sur la droite projective”  In _Math. Ann._ 335.2 Springer ScienceBusiness Media LLC, 2006, pp. 311–361  DOI: 10.1007/s00208-006-0751-x
  * [Gra08] Andrew Granville  “Smooth numbers: computational number theory and beyond”  In _Algorithmic number theory: lattices, number fields, curves and cryptography_ 44, 2008, pp. 267–323
  * [HW08] G.. Hardy and E.. Wright  “An Introduction to the Theory of Numbers”  Oxford University Press, 2008
  * [LM21] Pierre Le Boudec and Niki Myrto Mavraki  “Arithmetic dynamics of random polynomials”  arXiv, 2021  DOI: 10.48550/arXiv.2112.12005
  * [Mim13] Arman Mimar  “On the preperiodic points of an endomorphism of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ which lie on a curve”  In _Trans. Amer. Math. Soc._ 365.1, 2013, pp. 161–193  DOI: 10.1090/S0002-9947-2012-05557-7
  * [Mim97] Arman Mimar  “On the preperiodic points of an endomorphism of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ which lie on a curve”, 1997
  * [MS22] Niki Myrto Mavraki and Harry Schmidt  “On the dynamical Bogomolov conjecture for families of split rational maps”  arXiv, 2022  DOI: 10.48550/arXiv.2201.10455
  * [Ole23] Matt Olechnowicz  “Distribution of preperiodic points in one-parameter families of rational maps”, 2023  arXiv:2306.00933 [math.NT]
  * [PST12] Clayon Petsche, Lucien Szpiro and Thomas J. Tucker  “A dynamical pairing between two rational maps”  In _Trans. Amer. Math. Soc._ , 2012
  * [Rum13] Robert Rumely  “Capacity theory with local rationality: The strong Fekete–Szegő theorem on curves”, Mathematical Surveys and Monographs  Providence, RI: American Mathematical Society, 2013  DOI: 10.1090/surv/193
  * [Rum14] Robert Rumely  “Capacity theory with local rationality: The strong Fekete–Szegö theorem on curves”, Mathematical Surveys and Monographs  Providence, RI: American Mathematical Society, 2014
  * [Rum89] Robert Rumely  “Capacity theory on algebraic curves”, Lecture Notes in Mathematics  Berlin, Germany: Springer, 1989
  * [YZ21] Xinyi Yuan and Shou-Wu Zhang  “The arithmetic Hodge index theorem for adelic line bundles II”, 2021  arXiv:1304.3539 [math.NT]
  * [Zha95] Shou-Wu Zhang  “Small points and adelic metrics”  In _J. Algebraic Geom._ 4.2, 1995, pp. 281–300