Primitive prime divisors in the critical orbits of one-parameter families of rational polynomials

TL;DR

This paper investigates the occurrence of primitive prime divisors in the critical orbits of one-parameter families of rational polynomials, establishing a uniform bound on the size of the Zsigmondy set for critical points.

Contribution

It proves a uniform bound on the number of primitive prime divisors in the critical orbits of rational polynomial families when evaluated at rational critical points.

Findings

Existence of a uniform bound for the Zsigmondy set size

Bound depends only on the polynomial family, not on the parameter c

Critical points play a key role in primitive prime divisor distribution

Abstract

For a rational polynomial $f$ and rational numbers $c, u$, we put $f_c(x):=f(x)+c$, and consider the Zsigmondy set $\mathcal{Z}(f_c,u)$ associated to the sequence $\{f_c^n(u)-u\}_{n\geq 0}$, where $f_c^n$ is the $n$-st iteration of $f_c$. In this paper, we prove that if $u$ is a rational critical point of $f$, then there exists an $\mathbf M_f>0$ such that $\mathbf M_f\geq \max_{c\in \mathbb{Q}}\{\#\mathcal{Z}(f_c,u)\}$.

Equations

$${\text{SS}}_{f,u}:=\{c\in \mathbb{Q}\mid \{f_c^n(u)-u{\}}_{n\ge 1}\text{ is infinite}\},$$

$$\mathcal{Z}(f,u):=\{n\ge 1|f^n(u)-u\text{ has no primitive prime divisor}\}.$$

$$\#\mathcal{Z}(f_c,u)\le {\mathbf{M}}_f$$

$$g(x)=u_d{x}^d+\cdots +u_2{x}^2\in \mathbb{Q}[x].$$

$$\#\mathcal{Z}(g_c,0)\le {\mathbf{M}}_g$$

$$L_g:=1+\sum _{i=2}^{d-1}|u_i|/|u_d|.$$

$$g_c^{n+1}(0)=g_c^n(c):=\frac{A_n}{B_n},$$

$$\ln |A_n|>\sum _{p|n}\ln \left|A_{n/p}\right|.$$

$$\ln |A_n|\le \sum _{p|n}\ln \left|A_{n/p}\right|.$$

$$\ln |A_n|=\ln B_n+\ln |g_c^n(c)|.$$

$$|g_{c}^{n}(c)|\leq(2|u_{d}|)^{\frac{d^{n}-1}{d-1}}\Big{(}\max\left\{|c|,4L_{g}\right\}\Big{)}^{d^{n}}\quad\textrm{for all~{}}n\geq 0.$$

$$|g^{{n}+1}_{c}(c)|\leq|u_{d}||g_{c}^{n}(c)|^{d}+\sum_{i=2}^{d-1}|u_{i}||g_{c}^{n}(c)|^{i}+|c|\leq|u_{d}|(T_{n}+L_{g})T_{n}^{d-1}\\ \leq|u_{d}|\cdot 2T_{n}\cdot T_{n}^{d-1}=(2|u_{d}|)^{\frac{d^{n+1}-1}{d-1}}\Big{(}\max\left\{|c|,4L_{g}\right\}\Big{)}^{d^{n+1}}.$$

$$\frac{A_{n+1}}{B_{n+1}}=g_c^{n+1}(c)=g_c\left(\frac{A_n}{B_n}\right)=\sum _{i=2}^d{u}_i\frac{A_n^i}{B_n^i}+\frac{A_0}{B_0}.$$

$${\mathrm{val}}_p\left(\frac{u_d{A}_n^d}{B_n^d}\right)$$

$${\mathrm{val}}_p\left(\frac{u_i{A}_n^i}{B_n^i}\right)$$

$${\mathrm{val}}_p(\frac{A_0}{B_0})$$

$${\mathrm{val}}_p\left(\frac{u_d{A}_n^d}{B_n^d}\right)<(1-d){\mathrm{val}}_p(B_n)\le {\mathrm{val}}_p\left(\frac{u_i{A}_n^i}{B_n^i}\right)$$

$${\mathrm{val}}_p\left(\frac{A_1}{B_1}\right)={\mathrm{val}}_p\left(\frac{u_d{A}_0^d}{B_0^d}\right)={\mathrm{val}}_p(u_d)-d{\mathrm{val}}_p(B_0)<-{\mathrm{val}}_p(u_d)$$

$${\mathrm{val}}_p(B_0)\le {\mathrm{val}}_p(u_d)<{\mathrm{val}}_p(B_n).$$

$${\mathrm{val}}_p\left(\frac{A_{n+1}}{B_{n+1}}\right)={\mathrm{val}}_p\left(\frac{u_d{A}_n^d}{B_n^d}\right)={\mathrm{val}}_p(u_d)-d{\mathrm{val}}_p(B_n)<-{\mathrm{val}}_p(u_d)$$

$${\mathrm{val}}_p(B_n)=d^n{\mathrm{val}}_p(B_0)-{\mathrm{val}}_p(u_d)\sum _{i=0}^{n-1}{d}^i\ge {\mathrm{val}}_p(B_0).$$

$$I(a):=\{p\in \Sigma \mid {\mathrm{val}}_p(a)>{\mathrm{val}}_p(u_d)\},$$

$$\hat{a}:=\prod _{p\in I(a)}{p}^{{\mathrm{val}}_p(a)}.$$

$$|a|\le |u_d\hat{a}|.$$

$$\ln B_n\ge \frac{d^{n-n^{\prime }}}{3}\ln \widehat{B_{n^{\prime }}}.$$

$${\mathrm{val}}_p(B_n)\ge \frac{d^{n-n^{\prime }}}{3}{\mathrm{val}}_p\left(B_{n^{\prime }}\right).$$

$${\mathrm{val}}_p\left(B_n\right)=d^{n-n^{\prime }}{\mathrm{val}}_p(B_{n^{\prime }})-{\mathrm{val}}_p(u_d)\sum _{i=0}^{n-n^{\prime }-1}{d}^i\ge \left(d^{n-n^{\prime }}-\sum _{i=0}^{n-n^{\prime }-1}{d}^i\right){\mathrm{val}}_p(B_{n^{\prime }}).$$

$$d^{n-n^{\prime }}-\sum _{i=0}^{n-n^{\prime }-1}{d}^i\ge d^{n-n^{\prime }}-2d^{n-n^{\prime }-1}\ge \frac{1}{3}{d}^{n-n^{\prime }},$$

$$|g_c^n(c)|\ge 2^{-\frac{d^{n-n^{\prime }}-1}{d-1}}\cdot |g_c^{n^{\prime }}(c){|}^{d^{n-n^{\prime }}}.$$

$$|g_c^n(c)|\ge 2^{-\frac{d^{n-n^{\prime }}-1}{d-1}}\cdot |g_c^{n^{\prime }}(c){|}^{d^{n-n^{\prime }}}\ge \max \{4L_g,|c|\}\cdot {\left|4L_g/2\right|}^{\frac{d^{n-n^{\prime }}-1}{d-1}}\ge \max \{4L_g,|c|\},$$

$$|g_{c}^{n+1}(c)|\geq|u_{d}||g_{c}^{n}(c)|^{d}-\sum_{i=2}^{d}|u_{i}||g_{c}^{n}(c)|^{i}-|c|\\ \geq|u_{d}|\Big{(}|g_{c}^{n}(c)|^{d}-L_{g}|g_{c}^{n}(c)|^{d-1}\Big{)}=|g_{c}^{n}(c)|^{d-1}|u_{d}|\Big{(}|g_{c}^{n}(c)|-L_{g}\Big{)}\\ \geq|g_{c}^{n}(c)|^{d}/2\geq 2^{-\frac{d^{n-n^{\prime}+1}-1}{d-1}}\cdot|g_{c}^{n^{\prime}}(c)|^{d^{n-n^{\prime}+1}}.\qed$$

Full text

1. 1 Introduction

2. 2 Introduction of Proposition 2.2 and some estimates

3. 2.1 Upper bounds for $\ln B_{n}$ and $\ln|g_{c}^{n}(c)|$

4. 2.2 A lower bound for $\ln B_{n}$

5. 2.3 The lower bound for $\ln|g_{c}^{n}(c)|$

6. 3 Proof of Proposition 2.2

7. 4 Theorem 1.4 implies Theorem 1.2

1 Introduction

For every polynomial $f(x)\in\mathbb{Q}[x]$ and $\alpha\in\mathbb{Q}$ we put $f_{\alpha}(x)\coloneqq f(x)+\alpha.$ Therefore, $f_{\alpha}$ can be considered as a one-parameter family of polynomials. For every $u\in\mathbb{Q}$ we write

[TABLE]

where $f_{c}^{n}$ is the $n$-st iteration of $f_{c}$. In particular, if $u=0$, we put $\SS_{f}\coloneqq\SS_{f,0}$.

We denote by $\mathrm{val}_{p}(-)$ the $p$-adic valuation of $\mathbb{Q}$ normalized by $\mathrm{val}_{p}(p)=1$. In keeping with the terminology of [5], for every polynomial $f(x)\in\mathbb{Q}[x]$, $u\in\mathbb{Q}$ and $n\geq 1$ we say that $p$ is a primitive prime divisor of $f^{n}(u)-u$ if $\mathrm{val}_{p}(f^{n}(u)-u)>0$ and $\mathrm{val}_{p}(f^{k}(u)-u)\leq 0$ for all $1\leq k<n$.

Definition 1.1**.**

The Zsigmondy set of the sequence $\{f^{n}(u)-u\}_{n\geq 1}$ is defined by

[TABLE]

The primary application of bounds on the Zsigmondy set is towards understanding arboreal Galois representations associated to iteration of rational maps over number fields. It is first studied by Bang [1] and Zsigmondy [9]. Since then, there have been quite a few research papers on characterizing/bounding Zsigmondy sets of various sequences in various settings, e.g., Carmichael [2], Schinzel [8], Rice [7], Ingram–Silverman [5], Doerksen–Haensch [3], Gratton–Nguyen–Tucker[4], Krieger [6], etc.

In this work, we are interested in the size of the Zsigmondy set of a sequence obtained from the critical orbit of polynomials with rational coefficients of degree $d\geq 2$. We denote by $\Sigma$ the set of finite primes of $\mathbb{Z}$ and reserve $p$ for a prime number.

We first state our main theorem.

Theorem 1.2**.**

For every polynomial $f\in\mathbb{Q}[x]$ of degree $d\geq 2$ with a critical point $u\in\mathbb{Q}$ there is a constant $\mathbf{M}_{f}>0$, depending only on $f$ (independent of $c\in\mathbb{Q}$), such that

[TABLE]

for every $c\in\SS_{f,u}$.

It is worth mentioning that Rice [7] was the first to prove the finiteness of $\mathcal{Z}(f,0)$ for each individual polynomial $f(x)\neq x^{d}$. In [3], Doerksen–Haensch prove Theorem 1.2 for the case that $f(x)=x^{d}$, $u=0$ and $c\in\mathbb{Z}$, which is generalized by Krieger in [6] to every $c\in\mathbb{Q}$, see [6, Theorem 1.1]. Our contribution is to prove Theorem 1.2 for general polynomials which are not necessary to be monic nor integer. As we consider polynomials $f$ that are more complicated than $x^{d}$, we did not aim to get the sharpest uniform bound $\mathbf{M}_{f}$.

Definition 1.3**.**

A polynomial $g(x)\in\mathbb{Q}[x]$ is called $x^{2}$-divisible if it has degree $d\geq 2$ and is of the form

[TABLE]

At the last section we will prove that the following theorem implies Theorem 1.2.

Theorem 1.4**.**

Given an $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ of degree $d\geq 3$ there is a constant $\mathbf{M}_{g}>0$, depending only on $g$, such that

[TABLE]

for every $c\in\SS_{g}$.

Note that one can give an explicit expression of the lower bound $\mathbf{M}_{g}$ when combining the decomposition of $\SS_{g}$ in Proposition 2.2 with Propositions 3.3, 3.4, 3.7 and 3.8.

This paper is inspired by Krieger’s work in [6]. We generalize her result from the special polynomial $f(x)=x^{d}$ to arbitrary polynomials in $\mathbb{Q}[x]$. We first address the difficulties on this generalization as follows.

The first difficulty is from dealing with the non-monic case, in which the denominator $f_{c}^{n}(c)$ is no longer always equal to $d^{n}$’s power of the denominator of $c$. To conquer it, we introduce a factorization of an integer with respect to the leading term $u_{d}$ of $f$, see (2.9), which allows us to focus on the major factor of the denominator of $f_{c}^{n}(c)$.

The second difficulty is from the critical points of large multiplicities. Due to this reason, some arguments in [6] do not work for our case. For example, Krieger uses Mahler’s theorem to control $|f_{c}^{n}(c)|$ by $|f_{c}^{n-1}(c)-f_{c}^{-1}(0)|$. However, this estimation might not be enough when $f_{c}^{n-1}(c)$ is very close to a critical point with large multiplicity. It forces us to control $|f_{c}^{n}(c)|$ in Proposition 2.11 by $|f_{c}^{n-N}(c)-f_{c}^{-N}(0)|$ for some relatively large $N>1$.

Acknowledgment

The author would like to thank Tom Tucker and Shenhui Liu for their valuable discussions.

2 Introduction of Proposition 2.2 and some estimates

We split this section into two parts. In the first part, we introduce our main technical result Proposition 2.2 whose proof will be given in §3, and prove that it implies Theorem 1.4. In the second part, we focus on estimating $\ln|A_{n}|$ which appears in Proposition 2.2.

Let us first set conventions and introduce some notations.

1. (1).

We set $\mathbb{N}\coloneqq\{1,2,\dots\}$ to be the set of natural numbers and for every $n\in\mathbb{N}$ we denote by $[n]$ the finite set $\{1,2,\dots,n\}$. 2. (2).

We denote by $\Sigma$ the set of finite primes of $\mathbb{Z}$ and reserve $p$ for a prime number. For every $n\in\mathbb{Z}\backslash\{0\}$ the sum $\sum_{p|n}$ and the product $\prod_{p|n}$ are taken over all its distinct prime factors whose number is denoted by $\omega(n)$.

We will always write an $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ by $g(x)=u_{d}x^{d}+\cdots+u_{2}x^{2}\in\mathbb{Z}[x]$, and define its length by

[TABLE]

For every $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$, $c\in\mathbb{Q}$ and $n\geq 0$ we write the $(n+1)$-st iteration $g_{c}^{n+1}(0)$ as

[TABLE]

where $B_{n}>0,A_{n}$ are coprime and both depend on $g$ and $c$. Clearly, we have $c=g_{c}(0)=\frac{A_{0}}{B_{0}}$.

Definition 2.1**.**

For an $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ and a set $S$ in $\SS_{g}$ we call that $g$ has rapidly increasing numerators on $S$ if there exists an integer $N>0$ such that for every $c\in S$ there is a finite set $J_{c}$ with $\#J_{c}\leq N$ such that for every $n\notin J_{c}$ we have

[TABLE]

We now state our main proposition, which is followed by the proof of Theorem 1.4.

Proposition 2.2** (Main Proposition).**

Every $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ has rapidly increasing numerators on $\SS_{g}$.

Proof of Theorem 1.4 in assuming proposition 2.2.

By [6, Lemma 2.3 and Corollary 2.4], if $n\in\mathcal{Z}(g_{c},0)$, then $A_{n}\;|\;\prod_{p|n}A_{n/p}$ and hence

[TABLE]

Together with Proposition 2.2, this finishes the proof. ∎

The naive idea of proving Proposition 2.2 is to give a lower bound for $\ln|A_{n}|$ and an upper bound for $\prod_{p|n}|A_{n/p}|$ such that the lower bound is always greater than the upper bound when $n$ is large enough. Consider that

[TABLE]

It is sufficient for us to control $\ln B_{n}$ and $\ln|g_{c}^{n}(c)|$.

2.1 Upper bounds for $\ln B_{n}$ and $\ln|g_{c}^{n}(c)|$

Lemma 2.3**.**

Given any $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ and $c\in\mathbb{Q}$, for every $n\geq 0$ we have

1. (1).

$\ln B_{n}\leq{d^{n}}\ln B_{0};$** 2. (2).

\ln|g_{c}^{n}(c)|\leq d^{n}\ln\Big{(}2|u_{d}|\max\left\{|c|,4L_{g}\right\}\Big{)}.**

Proof.

(1) Since $g_{c}^{n}(c)$ can be written as $A_{n}^{\prime}/B_{0}^{d^{n}}$ for some $A_{n}^{\prime}\in\mathbb{Z}$, we have $B_{n}|B_{0}^{d^{n}}$.

(2) It is enough to prove

[TABLE]

For $n=0$, we have $|g_{c}^{0}(c)|=|c|\leq\!\max\left\{|c|,4L_{g}\right\}.$

Assume that the desired inequality holds for some $n\geq 0$ and temporarily denote its right side by $T_{n}$. Then we have

[TABLE]

The proof follows by induction. ∎

2.2 A lower bound for $\ln B_{n}$

Consider that

[TABLE]

Lemma 2.4**.**

Given any $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ and $c\in\mathbb{Q}$, for every $n\geq 0$ if $p\in I(B_{n})$, then we have

$p\in I(B_{n+1})$* and* 2. 2.

$\mathrm{val}_{p}(B_{n+1})=d\mathrm{val}_{p}(B_{n})-\mathrm{val}_{p}(u_{d}).$**

Proof.

Note that for every $n\geq 0$ we have

[TABLE]

Therefore, if $p\in I(B_{n})$, then we have $\mathrm{val}_{p}(A_{n})=0$ and

[TABLE]

for every $2\leq i\leq d-1$. Note that the term on the right hand side of this inequality does not exist for the case $d=2$.

We now prove this lemma by induction.

For $n=0$ we have $\mathrm{val}_{p}(B_{0})>\mathrm{val}_{p}(u_{d})$. Combined with (2.3) and (2.7), this implies

[TABLE]

and hence $p\in I(B_{1})$.

Now we assume that this lemma holds for every $0\leq k\leq n$.

(1) If $p\not\in I(B_{0})$, we have

[TABLE]

Combined with (2.3) and (2.7), this implies

[TABLE]

and hence $p\in I(B_{n+1})$.

(2) If $p\in I(B_{0})$, then by induction, we have $p\in I(B_{n})$ and

[TABLE]

Combining (2.3) with (2.7), we also obtain (2.8). This completes the induction. ∎

For every $a\in\mathbb{Z}$ we denote

[TABLE]

and put

[TABLE]

When $I(a)$ is empty, we put $\widehat{a}\coloneqq 1$. Note that we always have

[TABLE]

Lemma 2.5**.**

Given any $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ of degree $d\geq 3$ and $c\in\mathbb{Q}$, for every $0\leq n^{\prime}\leq n$ we have

[TABLE]

Proof.

If $\widehat{B_{n^{\prime}}}=1$, it is trivial.

Now we assume that $\widehat{B_{n^{\prime}}}\geq 2$. It is enough to show that every prime $p\in I(B_{n^{\prime}})$ satisfies

[TABLE]

Using Lemma 2.4 inductively, we have

[TABLE]

From our assumption that $d\geq 3$, we have

[TABLE]

which completes the proof. ∎

2.3 The lower bound for $\ln|g_{c}^{n}(c)|$

Lemma 2.6**.**

Given any $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ and $c\in\mathbb{Q}$, if $|g_{c}^{n^{\prime}}(c)|\geq\max\left\{4L_{g},|c|\right\}$ for some $n^{\prime}\geq 0$, then for every $n\geq n^{\prime}$ we have

[TABLE]

Clearly, in this case $c$ is in the basin of infinity for $g_{c}$.

Proof.

The proof follows from induction. For $n=n^{\prime}$ this lemma is trivial.

Assume that this lemma holds for some $n\geq n^{\prime}$. Then we have

[TABLE]

and hence

[TABLE]

Corollary 2.7**.**

Given any $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ and $c\in\mathbb{Q}$, if $|g_{c}^{n^{\prime}}(c)|\geq\max\left\{4L_{g},|c|\right\}$ for some $n^{\prime}\geq 0$, then for every $n\geq n^{\prime}$ we have

[TABLE]

Proof.

It follows directly from Lemma 2.6. ∎

Given an algebraic number $\gamma\in\mathbb{C}$ of degree $\ell$ with conjugates $\gamma_{1}\coloneqq\gamma,\gamma_{2},\dots,\gamma_{\ell}$ over $\mathbb{Q}$, let $a_{0}$ be an integer such that the coefficients of the polynomial $g(X)=a_{0}\prod_{i=1}^{\ell}(X-\gamma_{i})$ are integers of $\gcd\ 1$, then we define the Mahler measure of $\gamma$ by

[TABLE]

Notation 2.8**.**

For every $r\geq 1$ and $\delta>0$ we put

[TABLE]

Theorem 2.9** ([10], Theorem 1).**

Let $0<\delta<1.$ Then for every algebraic number $\gamma$ of degree $r\geq 1$, there are at most $W(r,\delta)$ solutions $a/b\in\mathbb{Q}$ to

[TABLE]

with $M(a/b)\geq\max\{4^{2/\delta},M(\gamma)\}.$

Theorem 2.9 implies the following result.

Corollary 2.10**.**

Given any $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$, for every $L>0$, $N\in\mathbb{N}$, $c\in[-L,L]\cap\mathbb{Q}$ and $\gamma\in\mathbb{C}$ such that $g^{N}_{c}(\gamma)=0$, there is an integer $D>0$, independent of $c$, such that

[TABLE]

has at most $W(d^{N},1/10)$ rational solutions $a/b$ with $b\geq\max\left\{4^{20},(|u_{d}|B_{0}D)^{d^{N}}\right\}.$

Proof.

Let $\gamma_{c,1},\dots,\gamma_{c,d^{N}}$ be the roots of $g_{c}^{N}(x)=0$ in $\mathbb{C}$ which are not necessary to be distinct. Since $g_{c}^{N}({x})$ is continuous as a function of $x$ and $c$, there exists an integer $D>1$ such that for every $c\in[-L,L]\cap\mathbb{Q}$ and $1\leq i\leq d^{N}$ we have

[TABLE]

Without loss of generality, we put $\gamma\coloneqq\gamma_{c,1}$ and $h(x)\coloneqq a_{0}\prod_{i=1}^{\ell}(x-\gamma_{c,i})$ to be the minimal polynomial of $\gamma$ with integer coefficients of $\gcd\ 1$.

Since $B_{0}^{d^{N}}g^{N}_{c}(\gamma)=0$ and $B_{0}^{d^{N}}g^{N}_{c}(x)$ is a polynomial with integer coefficients, we have $h(x)|B_{0}^{d^{N}}g^{N}_{c}(x)$. Combined with Gauss’s lemma, this implies

[TABLE]

Combining (2.15) with (2.16), we have

[TABLE]

On the other hand, for every rational number $a/b$ in the lowest terms such that $|a/b|\leq 2D$ we have

[TABLE]

Note that Theorem 2.9 still holds when we do the following modifications.

1. (1).

Restricting this theorem to a set of algebraic numbers and changing $M(\gamma)$ to a function of $\gamma$ which is larger than $M(\gamma)$ for every $\gamma$ in this set. 2. (2).

Changing the right hand side of (2.13) to a function of $a/b$ which is less than $M(a/b)^{-2-\delta}$ for every rational number $a/b$. 3. (3).

Changing the second $M(a/b)$ in Theorem 2.9 to a function of $a/b$ which is less than $M(a/b)$ for every rational number $a/b$.

Therefore, combined with (2.17) and (2.18), Theorem 2.9 implies that there are at most $W(\ell,\delta)$ rational solutions $a/b$ to

[TABLE]

such that $|a/b|\leq 2D$ and $b\geq\max\{4^{2/\delta},(|u_{d}|B_{0})^{d^{N}}D^{\ell}\}.$

For rational number $a/b$ such that $|a/b|\geq 2D$ we have

[TABLE]

Together with (2.19), this shows that there at most $W(\ell,\delta)$ rational solutions $a/b\in\mathbb{Q}$ to

[TABLE]

with $b\geq\max\{4^{2/\delta},(|u_{d}|B_{0})^{d^{N}}D^{\ell}\}.$

Take $\delta\coloneqq 1/10$. Combining $W(\ell,1/10)\leq W(d^{N},1/10)$ with the modification(3) above, we can replace $\ell$ by $d^{N}$ and $-2-1/10$ by $-3$, which completes the proof. ∎

Now we consider $|c|\leq 4L_{g}$. Recall that $\SS_{g}$ is the set that contains all rational number $c$ such that $\{g_{c}^{n}(0)\}$ is infinite. By Corollary 2.7 with $n^{\prime}=0$, for every $c\in(-\infty,-4L_{g}]\cup[4L_{g},\infty)$ we have $\lim_{n\to\infty}\ln|g_{c}^{n}(c)|=\infty$ and hence $\SS_{g}\supset(-\infty,-4L_{g}]\cup[4L_{g},\infty)$.

We denote by $\mathbb{U}_{g}^{0}$ the finite subset of $\SS_{g}\cap[-4L_{g},4L_{g}]$ consisting of all the rational numbers with denominator dividing $u_{d}$ and put $\mathbb{U}_{g}\coloneqq[-4L_{g},4L_{g}]\cap(\SS_{g}\backslash\mathbb{U}_{g}^{0})$. The following proposition aims at dealing the case $c\in\mathbb{U}_{g}$. It is worth noting that $\widehat{B_{0}}\geq 2$ for all $c\in\mathbb{U}_{g}$.

Proposition 2.11**.**

For an $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ and a real number $\alpha\in[-4L_{g},4L_{g}]$ such that $g(x)\neq u_{d}x^{d}$ or $\alpha\neq 0$, there exists $0<\delta<L_{g}$, $C>0$ and an integer $N\geq 0$ such that for every $c\in(\alpha-\delta,\alpha+\delta)\cap\SS_{g}$ if $\widehat{B_{n^{\prime}}}\geq 2$ for some $n^{\prime}\geq 0$, then there is a finite set $S_{c}\subset\mathbb{N}$ of bounded cardinality $N+n^{\prime}$ such that for every $n\not\in S_{c}$ we have

[TABLE]

Proof.

Let

[TABLE]

which satisfies

[TABLE]

Let $\gamma_{1},\dots,\gamma_{r}\in\mathbb{C}$ be the distinct roots of $g^{N_{0}}_{\alpha}(x)=0$ of multiplicity $m_{1},\dots,m_{r}$, respectively. Choose an $0<\epsilon<1$ small enough such that for any two distinct $i,j\in[r]$ we have $|\gamma_{i}-\gamma_{j}|>3\epsilon$.

By continuity of $g^{N}_{c}(x)$ as a function of $x$ and $c$, there exists $0<{\delta}<L_{g}$ such that for every $1\leq i\leq r$ and $\alpha^{\prime},\beta\in\mathbb{R}$ with $|\alpha^{\prime}-\alpha|<\delta$ and $|\beta|<\delta$ there are exactly $m_{i}$ roots of $g_{\alpha^{\prime}}^{N_{0}}(x)-\beta=0$ in the disk $O(\gamma_{i},\epsilon)\subset\mathbb{C}$.

Now we consider an arbitrary $c\in(\alpha-\delta,\alpha+\delta)\cap\SS_{g}$.

Let $\Gamma$ be the multiset consisting of all the roots of $g_{c}^{N_{0}}(x)=0$, i.e. two elements in $\Gamma$ could be the same. From the argument above, for every $n\geq N_{0}$ if $|g_{c}^{n}(c)|<\delta$, then there exists $1\leq i_{0}\leq r$ such that $g_{c}^{n-N_{0}}(c)\in O(\gamma_{i_{0}},\epsilon)$. We put

[TABLE]

Note that we have

[TABLE]

Now we count the distance between $g_{c}^{n-N_{0}}(c)$ and the points in $\Gamma$.

For every $\xi\in\Gamma_{1}$, from our choice of $\epsilon$, we have

[TABLE]

For every $\xi\in\Gamma_{2}$, by Corollary 2.10 with $L\coloneqq 4L_{g}$ and $N\coloneqq N_{0}$, there is an integer $D>0$, independent of $c$, such that

[TABLE]

has at most $W(d^{N_{0}},1/10)$ rational solutions $a/b$ with $b\geq\max\{4^{20},(|u_{d}|B_{0}D)^{d^{N_{0}}}\}.$

Put

[TABLE]

Then for every $n\geq N_{1}$, by Lemma 2.5, we have

[TABLE]

If $\widehat{B_{0}}=1$, by Lemma 2.5, (2.10) and the choice of $N_{2}$, for every $n\geq N_{2}$ we have

[TABLE]

If $\widehat{B_{0}}\geq 2$, by Lemma 2.5 and (2.10) again, for every $n\geq N_{2}$ we have

[TABLE]

Therefore, there are most $W(d^{N_{0}},1/10)$ many integers $n\geq\max\{N_{1}+n^{\prime},N_{2}+n^{\prime}\}$ such that $g_{c}^{n-N_{0}}(c)$ is a rational solution to (2.24). Combined with

[TABLE]

this implies that for all but at most $\#\Gamma_{2}\cdot W(d^{N_{0}},1/10)$ many $n\geq\max\{N_{1}+n^{\prime},N_{2}+n^{\prime}\}$ we have

[TABLE]

Combining (2.22) and our assumption $0<\epsilon<1$, we have

[TABLE]

From our assumption that $g(x)\neq u_{d}x^{d}$ or $\alpha\neq 0$, we have

[TABLE]

Therefore, the previous statement implies that for all but at most

$(d-1)^{N_{0}}W(d^{N_{0}},1/10)$ many $n\geq\max\{N_{1}+n^{\prime},N_{2}+n^{\prime}\}$ we have

[TABLE]

By (2.10) and (2.21), we have

[TABLE]

By Lemma 2.5, we obtain

[TABLE]

The two inequality above implies that

[TABLE]

which completes the proof. ∎

3 Proof of Proposition 2.2

The basic idea of proving Proposition 2.2 is to show that for each $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ there exists a finite cover of $\SS_{g}$ as follows:

1. (1).

$(-\infty,-4L_{g}]\cup[4L_{g},\infty)$; 2. (2).

$(\alpha-\delta_{g,\alpha},\alpha+\delta_{g,\alpha})\cap\mathbb{U}_{g}$ for finitely many $\alpha$ in $[-4L_{g},4L_{g}]$ with $0<\delta_{g,\alpha}<L_{g}$; 3. (3).

the finite set $\mathbb{U}_{g}^{0}$,

such that $g$ has rapidly increasing numerators on every set in this cover.

Recall that for every $n\in\mathbb{N}$ we denote by $\omega(n)$ the number of its distinct prime divisors. For convenience, we put $s_{d}(n)\coloneqq\sum_{p|n}d^{\frac{n}{p}}$. Then we have the following estimation.

Lemma 3.1**.**

For every $d\geq 2$ and every $n\geq 30$, we have $s_{d}(n)\leq d^{\frac{3n}{5}}$.

Proof.

For every integer $n\geq 2$ we have $n\geq 2^{\omega(n)}$ and hence

[TABLE]

Since for every prime divisor $p$ of $n$ we have $n/p\leq n/2$. Combined with (3.1), we have

[TABLE]

On the other hand, for every $n\geq 30$ we have

[TABLE]

Together with (3.2), this finishes the proof. ∎

Lemma 3.2**.**

Given any $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ and $c\in\mathbb{Q}$, for every $n\geq 30$ we have

[TABLE]

Proof.

By Lemma 2.3, for every $n\geq 0$ we have

[TABLE]

Together with Lemma 3.1 and (2.10), this finishes the proof. ∎

Proposition 3.3**.**

Every $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ of degree $d\geq 3$ has rapidly increasing numerators on $(-\infty,-4L_{g}]\cup[4L_{g},\infty)$.

Proof.

Let $c$ be an arbitrary rational number in $(-\infty,-4L_{g}]\cup[4L_{g},\infty)$.

By Lemma 2.5, Corollary 2.7 with $n^{\prime}=0$ and $|c|\geq 4L_{g}\geq 4$, for every $n\geq 0$ we have

[TABLE]

Combined with Lemma 3.2 and $\left|c\right|\geq 4L_{g}$, this implies that for every $n\geq 30$ we have

[TABLE]

Therefore, there exists an integer $N>\max\left\{\frac{5}{2}\log_{d}3,30\right\}$, which only depends on $g$, such that for every $n\geq N$ and every rational number $c\in(-\infty,-4L_{g}]\cup[4L_{g},\infty)$ we have

[TABLE]

Thus we prove this proposition. ∎

We next prove the following.

Proposition 3.4**.**

Every polynomial $g(x)=u_{d}x^{d}\in\mathbb{Z}[x]$ of degree $d\geq 3$ has rapidly increasing numerators on $\left(-\frac{1}{|4u_{d}|},\frac{1}{|4u_{d}|}\right)\cap\SS_{g}$.

Since $0\notin\SS_{g}$, it is sufficient to show the following two lemmas.

Lemma 3.5**.**

Every polynomial $g(x)=u_{d}x^{d}\in\mathbb{Z}[x]$ of degree $d\geq 3$ has rapidly increasing numerators on $\left(0,\frac{1}{4|u_{d}|}\right)\cap\SS_{g}$.

Proof.

For every $c\in(0,\frac{1}{4|u_{d}|})\cap\SS_{g}$ and $n\geq 0$ we have

[TABLE]

and $\widehat{B_{0}}\geq 2.$

Combining Lemmas 2.3(1), 2.5 for $n^{\prime}=0$ with (3.4), for every $n\geq 0$ we have

[TABLE]

Together with (2.10) and lemma 3.1, this implies that for every $n\geq 30$ we have

[TABLE]

From $\widehat{B_{0}}\geq 2$, there exists an integer $N\geq 30$ such that for every $n\geq N$ we have

[TABLE]

which completes the proof. ∎

Lemma 3.6**.**

Every polynomial $g(x)=u_{d}x^{d}\in\mathbb{Z}[x]$ of degree $d\geq 3$ has rapidly increasing numerators on $\left(-\frac{1}{4|u_{d}|},0\right)\cap\SS_{g}$.

Proof.

Note that when $d$ is odd, we may replace $c$ with $-c$ and the forward orbit of [math] will be unchanged, modulo sign. By Lemma 3.5, we immediately prove this case.

Therefore, it is sufficient to study the case that $d$ is even. We first show that for every $c\in(-\frac{1}{4|u_{d}|},0)$ and every $n\geq 0$ we have

[TABLE]

For $n=0$, we have $|g_{c}^{0}(c)|=|c|$. Assume that (3.5) holds for some $n\geq 0$. Since $g_{c}(x)$ is negative and decreasing on $(-\frac{1}{4|u_{d}|},0)$, we have

[TABLE]

which proves (3.5) by induction.

Combining Lemmas 2.3(1), 2.5 with (3.5), we have

[TABLE]

Together with (2.10), Lemma 3.1 and $|c|<\frac{1}{4|u_{d}|}$, this implies that for every $n\geq 30$ we have

[TABLE]

On the other hand, for every $c\in(-\frac{1}{4|u_{d}|},0)$, we have $\widehat{B_{0}}\geq 2$. Combined with (3.6), this proves that there exists an integer $N\geq 30$ such that for every $n\geq N$ we have

[TABLE]

which completes the proof. ∎

Proposition 3.7**.**

Given any $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ of degree $d\geq 3$ and $\alpha\in[-4L_{g},4L_{g}]$ such that $g(x)\neq u_{d}x^{d}$ or $\alpha\neq 0$, there is an $0<\delta<L_{g}$ such that $g$ has rapidly increasing numerators on $c\in(\alpha-\delta,\alpha+\delta)\cap\mathbb{U}_{g}$.

Proof.

Note that for every $c\in\mathbb{U}_{g}$ we have $\widehat{B_{0}}\geq 2$. By Proposition 2.11 with $n^{\prime}=0$, there is a $3$-tuple $0<\delta<L_{g}$, $C>0$ and $N_{1}>0$ such that for every $c\in(\alpha-\delta,\alpha+\delta)\cap\mathbb{U}_{g}$ there is a finite set $S_{c}\subset\mathbb{N}$ of bounded cardinality $N_{1}$ such that for every $n\not\in S_{c}$ we have

[TABLE]

and therefore

[TABLE]

On the other hand, by Lemma 2.5, we have

[TABLE]

Combined with Lemma 3.2, (3.7) and $|c|\leq 5L_{g}$, this implies that there exists an integer $N_{2}\geq 30$ such that for every rational number $c\in(\alpha-\delta,\alpha+\delta)$ and $n\in\{N_{2},N_{2}+1,\dots,\}\backslash S_{c}$ we have

[TABLE]

Taking $N\coloneqq N_{1}+N_{2}$, we prove this proposition. ∎

Now we turn our attention to the finite set $\mathbb{U}_{g}^{0}$.

Proposition 3.8**.**

Every $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ of degree $d\geq 3$ has rapidly increasing numerators on the finite set $\mathbb{U}_{g}^{0}$.

Proof.

It is sufficient to show that for each individual rational number in $\mathbb{U}_{g}^{0}$ there are finite many $n\in\mathbb{N}$ satisfying (2.1).

Let $c$ be an arbitrary rational number in $\mathbb{U}_{g}^{0}$. We first show that there must exist an integer $n^{\prime}$ such that either $|g_{c}^{n^{\prime}}(c)|>4L_{g}$ or $\widehat{B_{n^{\prime}}}\geq 2$. Suppose that for every $n\geq 0$ we have $\widehat{B_{n}}=1$, i.e., $B_{n}|u_{d}$. Since $c\in\SS_{g}$ and there are only finitely many integers in $[-4L_{g},4L_{g}]$ with denominator dividing $u_{d}$, we know that there must exist an $n^{\prime}\geq 0$ such that $|g_{c}^{n^{\prime}}(c)|>4L_{g}$.

(1) When $|c|\leq 4L_{g}$, $\widehat{B_{0}}=1$ and there exists an integer $n^{\prime}\geq 1$ such that $|g_{c}^{n^{\prime}}(c)|>4L_{g}$. Combining these conditions with Corollary 2.7 and Lemma 3.2, for every $n\geq\max\{30,n^{\prime}\}$ we have

[TABLE]

Clearly, there exists an integer $N>\max\{30,n^{\prime}\}$ such that for every $n\geq N$ we have

[TABLE]

(2) When $|c|\leq 4L_{g}$, $\widehat{B_{0}}=1$ and there is an integer $n^{\prime}\geq 1$ such that $\widehat{B_{n^{\prime}}}\geq 2$. Similar to Proposition 3.7, we combine Lemma 2.5 with Proposition 2.11, and obtain a finite set $S_{c}\subset\mathbb{N}$, an integer $N$, $\delta_{c}>0$ and $C_{c}>0$ such that for every $n\geq n^{\prime}$ if $n\not\in S_{c}$, then

[TABLE]

and therefore

[TABLE]

On the other hand, by Lemma 2.5, we have

[TABLE]

Combined with Lemma 3.2, (3.8) and $|c|\leq 4L_{g}$, this implies that there exists an integer $N_{1}>\max\{n^{\prime},30\}$ such that for every $n\in\{N_{1},N_{1}+1,\dots\}\backslash S_{c}$ we have

[TABLE]

Put $N\coloneqq\#S_{c}+N_{1}$. Then we complete the proof. ∎

Proof of Proposition 2.2.

By Proposition 3.7, for every $g(x)\neq u_{d}x^{d}$ or $\alpha\neq 0$ and every real number $\alpha\in[-4L_{g},4L_{g}]$, there is an $0<\delta_{g,\alpha}<L_{g}$ such that $g$ has rapidly increasing numerators on $c\in(\alpha-\delta_{g,\alpha},\alpha+\delta_{g,\alpha})\cap\mathbb{U}_{g}$.

For $g(x)=u_{d}x^{d}$ and $\alpha=0$, if we put $\delta_{g,\alpha}\coloneqq\frac{1}{4|u_{d}|}$, then we proved in Proposition 3.4 that $g$ has rapidly increasing numerators on $c\in(-\delta_{g,\alpha},\delta_{g,\alpha})\cap\mathbb{U}_{g}$.

Now for every $x^{2}$-divisible $g(x)\in\mathbb{Z}[x]$ we obtain a cover of $\SS_{g}$ as

[TABLE]

Note that

[TABLE]

is an open cover of the closed interval $[-4L_{g},4L_{g}]$, which has a finite cover. We use the center $\alpha$ to represent the interval $(\alpha-\delta_{g,\alpha},\alpha+\delta_{g,\alpha})$ in this finite cover, and put $T$ to be the index set of $\alpha$.

Therefore, we obtain a finite cover of $\SS_{g}$ as follows:

[TABLE]

By Propositions 3.3, 3.4, 3.7 and 3.8, we know that $g$ has rapidly increasing numerators on each set in this cover. It implies that $g$ also has rapidly increasing numerators on $\SS_{g}$, which completes the proof. ∎

4 Theorem 1.4 implies Theorem 1.2

At the end of this section, we will prove that Theorem 1.4 implies the following proposition which leads to Theorem 1.2.

Proposition 4.1**.**

For every $x^{2}$-divisible polynomial $g(x)\in\mathbb{Q}[x]$ there is a constant $\mathbf{M}_{g}>0$, depending only on $g$, such that

[TABLE]

for every $c\in\SS_{g}$.

For every $n\in\mathbb{Z}\backslash\{0\}$, we denote by $\omega(n)$ the number of distinct prime factors of $n$.

Lemma 4.2**.**

Let $g(x)\in\mathbb{Q}[x]$ be $x^{2}$-divisible and $a\in\mathbb{Z}\backslash\{0\}$. Then with $h(x)\coloneqq\frac{1}{a}g(ax)$ we have

[TABLE]

Proof.

Consider that

[TABLE]

For every $p\nmid u$ we have $p$ is a primitive prime divisor of $g^{n}_{c}(0)$ if and only if it is a primitive prime divisor of $h^{n}_{c/a}(0)$. Therefore, the difference between $\#\mathcal{Z}(g_{c},0)$ and $\#\mathcal{Z}(h_{c/a},0)$ can not exceed the number of prime factors of $a$, which completes the proof. ∎

Lemma 4.3**.**

For any $x^{2}$-divisible $g(x)\in\mathbb{Q}[x]$ and $t\in\mathbb{Q}\backslash\{0\}$, Proposition 4.1 holds for $g$ if and only if it holds for $\frac{1}{t}g(tx)$.

Proof.

Let $t=\frac{a}{b}$ be an arbitrary rational number. Put

[TABLE]

Note that $h_{1}(x)=\frac{1}{b}h(bx)$.

By Lemma 4.2 for every $c\in\mathbb{Q}$ we have

[TABLE]

which implies

[TABLE]

Since $\omega(a)$ and $\omega(b)$ are both independent of $c$, we complete the proof. ∎

Proposition 4.4**.**

Theorem 1.4 implies Theorem 1.2.

Proof.

Step I: We first prove that Theorem 1.4 implies Proposition 4.1.

Given any $x^{2}$-divisible polynomial $g(x)\in\mathbb{Q}[x]$ of degree $d\geq 2$, if $d=2$, then we can find $t\in\mathbb{Q}$ such that $\frac{1}{t}g(tx)=x^{2}$. Combining Lemma 4.3 with [6, Theorem 1.1], we prove Proposition 4.1 for this case.

If $d\geq 3$, there is $t\in\mathbb{Z}$ such that $\frac{1}{t}g(tx)\in\mathbb{Z}[x]$. Combined with Lemma 4.3, this proves Proposition 4.1 for this case.

Step II: Now we prove that Proposition 4.1 implies Theorem 1.2.

For every polynomial $g(x)\in\mathbb{Q}[x]$ with a critical point $u\in\mathbb{Q}$, if we put $f(x)\coloneqq g(x+u)-u$, then we know that [math] is a critical point of $f$ and for every $c\in\mathbb{Q}$ we have $f_{c}^{n}(0)=g_{c}^{n}(u)-u$ and hence

[TABLE]

This completes the proof. ∎