# On the powerful values of polynomials over number fields

Sajad Salami,
Instituto de Matemática e Estatística
Universidade Estadual do Rio de Janeiro, Brazil

###### Abstract

In this paper, assuming a conjecture of Vojta on the bounded degree algebraic
numbers on a number field $k$, we determine explicit lower and upper bounds
for the cardinal number of the set of polynomials $f\in k[x]$ with degree
$r\geq 2$ whose irreducible factors have multiplicity strictly less than $s$
and the values $f(b_{1}),\cdots,f(b_{M})$ are $s$-powerful elements in $k^{*}$
for a certain positive integer $M$, where $b_{i}$’s belong to an arbitrary
sequence of the pairwise distinct element of $k$ that satisfy certain
conditions.

##  1 Introduction and main results

Let $k$ be a number field. We let ${\mathcal{O}}_{k}$ denote the ring of
integers of $k$. Given integer $s\geq 2$, an element $\alpha$ of
${\mathcal{O}}_{k}^{*}$ is called $s$-powerful if for each prime ideal
${\mathfrak{p}}$ of ${\mathcal{O}}_{k}$ dividing the principle ideal
$(\alpha)$ we have ${\rm ord_{\mathfrak{p}}}(\alpha)\geq s$. This definition
immediately extends to elements of $k$. Clearly, any $s$-power in $k$ is an
$s$-powerful element. Given $f\in k[x]$ of degree $r\geq s$, we say that $f$
is an $s$-powerful polynomial if each irreducible factor of $f$ has
multiplicity at least $s$. It is clear that any $s$-power in $k[x]$ is an
$s$-powerful element.

The powerful values of polynomials have been studied by several authors in the
literature, [3], [11], [15], [20], and the recent work [10] that has
considered the problem both on the number fields and the function fields. In
the case of the numebr field, in [10], the author used a conjecture of Vojta’s
on the bounded degree algebraic numbers to get his results. In this paper, we
assume the following equivalent version of the Vojta’s conjecture that is an
extension of the Roth’s theorem in Diophantine approximation to bounded degree
extensions.

We assume that ${{\bar{k}}}$ is an algebraic closure of $k$ and $h$ is the
absolute Weil height of ${\mathbb{P}}_{{\bar{k}}}^{1}$. Given
$\alpha\in{{\bar{k}}}$, we denote by $d_{k}(\alpha)$ its logarithmic
discriminant with respect to $k$. Let ${\mathcal{P}}_{k}$ to be the set of all
places of $k$, and $S\subset{\mathcal{P}}_{k}$ be a finite set containing
${\mathcal{P}}_{k}^{\infty}$, the set of infinite places of $k$. We denote by
$m_{S}(b,\alpha)$ the proximity function respect to $S$ on $k$, for any $b\in
k$ and $\alpha\in{{\bar{k}}}$ distinct from $b$. See the section 3 or [17, 16]
for more details.

###### Conjecture 1.1

Let $k$ be a number field, ${{\bar{k}}}$ its algebraic closure and
$S\subset{\mathcal{P}}_{k}$ a finite set containing
${\mathcal{P}}_{k}^{\infty}$. Let $b_{1},\cdots,b_{q}$ be pairwise distinct
elements of $k$ and $d\geq 2$ an integer. Then for any $\epsilon>0$ and
$c\in{\mathbb{R}}$, the inequality

$\sum_{i=1}^{q}m_{S}(b_{i},\alpha)\leq(2+\epsilon)h(\alpha)+d_{k}(\alpha)+c,$
(1.1)

holds for almost all $\alpha\in{{\bar{k}}}$ with $[k(\alpha):k]\leq d$ and
different from $b_{i}$’s.

We fix an arbitrary sequence ${\mathcal{B}}=\\{b_{i}\\}_{i=1}^{\infty}$ of
pairwise distinct elements in $k$. Given integers $2\leq s\leq r<n$, we let
${\bf F}_{r,s}^{{\mathcal{B}}_{n}}$ to be the set of all polynomials $f\in
k[x]$ of degree $r$ such that $f(b_{i})$ is a $s$-powerful element in $k^{*}$
for each $b_{i}\in{\mathcal{B}}_{n}$, where
${\mathcal{B}}_{n}\subset{\mathcal{B}}$ contains the terms
$b_{1},\cdots,b_{n}$. We identify two polynomials $f,g\in{\bf
F}_{r,s}^{{\mathcal{B}}_{n}}$, if $f=\alpha g$ for some $s$-powerful element
$\alpha\in k$. Denote by ${\bf G}_{r,s}^{{\mathcal{B}}_{n}}$ the subset of
this set in which all irreducible factors have multiplicity strictly smaller
than $s$.

###### Theorem 1.2

Assume Vojta’s Conjecture 1.1 . Given integers $2\leq s\leq r$, let
$M:=2r^{2}+6r+1$ if $r=s$, and $2sr^{2}+sr+1$ otherwise. Then ${\bf
G}_{r,s}^{{\mathcal{B}}_{M}}$ is a finite set. Moreover, there exist positive
constants $C_{0}$ and $C_{1}$ such that

$C_{0}\leq\\#{\bf G}_{r,s}^{{\mathcal{B}}_{M}}\leq C_{1}.$

We note that the integer $M$ given by this theorem depends only on the
integers $r$ and $s$, but it is independent of the sequence ${\mathcal{B}}$
and its subset ${\mathcal{B}}_{M}.$ In contrast, the proof of 1.2 shows that
the constants $C_{0}$ and $C_{1}$ depend on $b_{i}\in{\mathcal{B}}_{M}$, the
integers $r$ and $s$, and the basic quantities of the number field $k$. In
particular, the constant $C_{1}$ is a positive number if $k={\mathbb{Q}}$ and
goes to the infinity when the degree of $k$ tents to the infinity.

We consider the sequences ${\mathcal{C}}=\\{c_{i}\\}_{i=1}^{\infty},$
${\mathcal{D}}=\\{d_{i}\\}_{i=1}^{\infty},$ and
${\mathcal{E}}=\\{e_{i}\\}_{i=1}^{\infty},$ associated to the sequence
${\mathcal{B}}$ and given by

$c_{i}:=b_{i+1}-b_{i},\ d_{i}:=b_{i}/b_{i+1},\ e_{i}:=1/b_{i+1}-1/b_{i},\
(i=1,2,\cdots).$

As a consequence of the theorem 1.2, we have the following result.

###### Corollary 1.3

Assume Vojta’s conjecture 1.1. Let $M$ denote the integer given by Theorem
1.2. If any of the sequences ${\mathcal{C}},{\mathcal{D}},$ and
${\mathcal{E}}$ is periodic with period $m\geq 1$, then there exists an
integer $M_{0}>0$, depending on $r$, $s$, the number field $k$ and elements of
${\mathcal{B}}_{M}$ such that ${\bf G}_{r,s}^{{\mathcal{B}}_{n}}=\emptyset$
for $n>M_{0}$.

Since all of the three types of the sequences are periodic with period $m=1$
for the sequence of positive integers, so 1.2 and 1.3 imply the theorem 2.1
and the corollary 2.2 in [10], if one considers the monic polynomials.

The structure of this paper is as follows. In section 2, we give the
preliminaries containing the height functions on number fields and the main
result of [5] that will use in the proof of 1.2. Section 3 is devoted to
recall the Diophantine approximation on bounded degree extensions of number
fields by focusing on the truncated version of the conjecture 1.1. The last
section 4 contains the proofs of 1.2 and 1.3.

##  2 Preliminaries

Given a number field $k$ with algebraic closure ${{\bar{k}}}$, we let
${\mathcal{P}}_{k}$ denote the set of places of $k$ that splits into two
disjoint subsets. One, ${\mathcal{P}}_{k}^{0}$ the set of the finite places,
_i.e._ , those corresponding to prime ideals ${\mathfrak{p}}$ of
${\mathcal{O}}_{k}$, and another one of the infinite places denoted by
${\mathcal{P}}_{k}^{\infty}$, _i.e._ , those corresponding to real embeddings
$\sigma:k\hookrightarrow{\mathbb{R}}$, called the real infinite places, union
with those corresponding to pair of conjugate embeddings
$\sigma,\bar{\sigma}:k\hookrightarrow{\mathbb{C}}$ that are called the complex
infinite places. For any $v\in{\mathcal{P}}_{k}$, denote by $\|\cdot\|_{v}$
its associated almost absolute value defined by

$\|\alpha\|_{v}:=\begin{cases}0&\text{if $\alpha=0$ },\\\
\\#(\frac{{\mathcal{O}}_{k}}{{\mathfrak{p}}_{v}})^{{\rm
ord_{\mathfrak{p}}}_{v}(\alpha)}&\text{if $v\in{\mathcal{P}}_{k}^{0}$
corresponds to a prime ${\mathfrak{p}}_{v}\in Spec({\mathcal{O}}_{k})$},\\\
|\sigma(\alpha)|&\text{if $v\in{\mathcal{P}}_{k}^{\infty}$ is a real infinite
place},\\\ |\sigma(\alpha)|^{2}&\text{if $v\in{\mathcal{P}}_{k}^{\infty}$ is a
complex infinite place}.\end{cases}$

Given any point $P=[\alpha_{0}:\cdots:\alpha_{n}]\in{\mathbb{P}}_{k}^{n}$, the
multiplicative  and logarithmic  heights are defined by

$H_{k}(P):=\prod_{v\in{\mathcal{P}}_{k}}\max\\{\|\alpha_{0}\|_{v},\cdots,\|\alpha_{n}\|_{v}\\},\
h_{k}(P):=\log H_{k}(P),$

where $\log^{+}x=\max\\{\log x,0\\}$ for any $x\in\mathbb{R}_{>0}$. For any
$\alpha\in k$, its multiplicative  and logarithmic  heights are defined by
$H_{k}(\alpha)=H_{k}([1:\alpha])$ and $h_{k}(\alpha)h_{k}([1:\alpha])$, where
we identified $k$ with affine space in ${\mathbb{P}}_{k}^{1}$. For any finite
extension $K|k$, $\alpha\in k$, and $P\in{\mathbb{P}}_{k}^{n}$, one has

$H_{k}(\alpha)=H_{K}(\alpha)^{1/[K:k]},\
h_{k}(\alpha)=\frac{1}{[K:k]}h_{K}(\alpha),$ $H_{k}(P)=H_{K}(P)^{1/[K:k]},\
h_{k}(P)=\frac{1}{[K:k]}h_{K}(P).$

Considering these facts, one may extend the definition of height function to
${\mathbb{P}}_{{\bar{k}}}^{n}$. In this case, they are called the absolute
multiplicative and additive Weil heights of $P\in{\mathbb{P}}_{{\bar{k}}}^{n}$
and denoted by $H(P)$ and $h(P)$, respectively. We note that the action of
Galois group of ${\mathbb{P}}_{{\bar{k}}}^{n}$ leaves the absolute heights
invariant. Moreover, for each $\alpha,\beta\in{{\bar{k}}}^{*}$ and
$n\in{\mathbb{Z}}$, one has the followings:

$h(\alpha^{n})=|n|h(\alpha),\ h(\alpha\beta)\leq h(\alpha)+h(\beta),\
h(\alpha+\beta)\leq h(\alpha)+h(\beta)+\log 2.$ (2.1)

Giving lower and upper bounds for the cardinal number of the set of bounded
degree algebraic points on a projective line is started by [13] and continued
by [14], [7], [5], and so on. Let us to recall the main result in [5] that we
will use in the proof of 1.2.

Denote by $N({\mathbb{P}}_{{\bar{k}}}^{1};r;T)$ the number of points
$\alpha\in{\mathbb{P}}_{{\bar{k}}}^{1}$ of degree at most $r$ and
$h(\alpha)\leq T$ for every constant $T>0$ and integer $r\geq 2$. Let $h_{k}$
be the class number of $k$, $\text{Reg}_{k}$ the regulator of
${\mathcal{O}}_{k}^{*}$, $w_{k}$ the number of roots of unity in $k$,
$\zeta_{k}(s)$ the Dedekind zeta-function of $k$, ${\mathfrak{d}}_{k}$ the
absolute discriminant of $k$, $m_{1}$ the number of real embedding of $k$,
$m_{2}$ the number of pairs of complex embedding of $k$, and $m=m_{1}+2m_{2}$
is the degree of $k$ over ${\mathbb{Q}}$. For more details on these
quantities, one can see [8]. Define

$a_{k,r}:=\frac{h_{k}\cdot\text{Reg}_{k}}{w_{k}\zeta_{k}(r+1)}\cdot\big{(}\frac{2^{m_{1}}(2\pi)^{m_{2}}}{{\mathfrak{d}}_{k}^{1/2}}\big{)}^{r+1}\cdot(r+1)^{m_{1}+m_{2}-1},$
(2.2)

and denote $b_{k,r}:=r\cdot a_{k,r}\cdot T^{mr(r+1)}$ and
$T_{1}=T^{mr(r+1)-r}$.

###### Theorem 2.1

Notation being as above, for each $\varepsilon>0$ one has

$b_{k,r}\cdot 2^{-mr(r+1)}T^{mr(r+1)}-O_{\varepsilon}(T_{1}\cdot
T^{\varepsilon})\leq N({\mathbb{P}}_{{\bar{k}}}^{1};r;T)\leq b_{k,r}\cdot
2^{mr(r+1)}+O(T_{1}).$

In particular,

$2^{-mr(r+1)}+o(1)\leq\frac{N({\mathbb{P}}_{{\bar{k}}}^{1};r;T)}{b_{k,r}}\leq
2^{mr(r+1)}+o(1)\ \ \textit{as}\ \ T\rightarrow\infty.$

Given any polynomial $f(x)=a_{0}+a_{1}x+\cdots+a_{d}x^{d}\in k[x],$ the
absolute multiplicative  and additive heights  are defined by

$H(f):=H([a_{0}:a_{1}:\cdots:a_{d}]),\ \ h(f):=h([a_{0}:a_{1}:\cdots:a_{d}].$

Without loos of generality, we may suppose that ${{\bar{k}}}={\mathbb{C}}$ and

$f(x)=a_{0}+a_{1}x+\cdots+a_{d}x^{d}=a_{d}\prod_{j=1}^{d}(x-\alpha_{j})\in{\mathbb{C}}[x].$
(2.3)

In this case, the Mahler measure of any $f\in{\mathbb{C}}[x]$ is defined by

$M(f):=|a_{d}|\cdot\prod_{j=1}^{r}\max\\{1,|\alpha_{j}|\\},$

where $|\cdot|$ is the usual absolute value on ${\mathbb{C}}$. For
$\alpha\in{{\bar{k}}}={\mathbb{C}}$, its Mahler measure is given by
$M(\alpha)=M(f_{\alpha})$ where $f_{\alpha}\in k[x]$ denotes its minimal
polynomial.

The logarithmic discriminant  of $k$ is defined by
$d_{k}:=\log{\mathfrak{d}}_{k}/[k:{\mathbb{Q}}].$ For a tower of number fields
${\mathbb{Q}}\subseteq k\subseteq K\subset{{\bar{k}}}$ with absolute
discriminants ${\mathfrak{d}}_{k}$ and ${\mathfrak{d}}_{K}$, respectively, the
relative logarithmic discriminant of $K|k$ is

$d_{k}(K):=\frac{1}{[K:k]}\log{\mathfrak{d}}_{K/k}-\log{\mathfrak{d}}_{k},$

where ${\mathfrak{d}}_{K/k}$ is the relative discriminant  of the extension
$K|k$. The relative logarithmic discriminant of each $\alpha\in{{\bar{k}}}$ is
defined by $d_{k}(\alpha):=d_{k}(k(\alpha)).$ The following proposition gives
an upper bound for the logarithmic discriminant $d_{k}(\alpha)$ that we will
use in the proof of 1.2. For a proof, we cite to [6, 1].

###### Proposition 2.2

Let $f\in k[x]$ be of the form 2.3 with degree $d\geq 2$ and $A(d)=d\log d$ if
$k={\mathbb{Q}}$, and $A(d)=(2d-1)\log d$ otherwise.

  * (i)

$D(f)=a_{0}^{2d-2}\prod_{i>j}(\alpha_{i}-\alpha_{j})^{2},$ and $|D(f)|\leq
d^{d}\cdot M(f)^{2d-2};$

  * (ii)

If $D(f)\not=0$, then $h(D(f))\leq 2(d-1)h(f)+A(d);$

  * (iii)

If $\alpha\in{{\bar{k}}}$ is of degree $d\geq 2$, then $d_{k}(\alpha)\leq
2(d-1)h(\alpha)+A(d).$

##  3 The Value distribution on number fields

In order to get a suitable version of the conjecture 1.1, we briefly review
the basic definitions and results on the value distribution theory over number
fields. This theory is an analogue of the Nevanlinna theory in the context of
complex numbers. For more details, we cite the reader to [16] and [17].

Given a finite set $S\subset{\mathcal{P}}_{k}$ containing
${\mathcal{P}}_{k}^{\infty}$, and the distinct elements $b,\alpha\in k$, the
proximity functions with respect to $S$ are defined by

$m_{S}(\alpha):=\sum_{v\in S}\log^{+}\|\alpha\|_{v},\ \text{and}\
m_{S}(b,\alpha):=m_{S}(\frac{1}{\alpha-b}).$

Similarly, the counting functions with respect to the set $S$ are defined by

$N_{S}(\alpha):=\sum_{v\not\in S}\log^{+}\|\alpha\|_{v},\ \text{and}\
N_{S}(b,\alpha):=N_{S}(\frac{1}{\alpha-b}).$

By the properties of the logarithm function, for any $\alpha\in k$ one has

$m_{S}(\alpha)+N_{S}(\alpha)=\sum_{v\in{\mathcal{P}}_{k}}\log^{+}\|\alpha\|_{v}=h(\alpha),$
(3.1)

which is an analogue of first main theorem in classic Value distribution
theory. The proximity and counting function of any
$\alpha\in{{\bar{k}}}\backslash k$ are defined as

$m_{S}(\alpha):=\frac{1}{[K:k]}\cdot m_{T}(\alpha),\ \text{and}\
N_{S}(\alpha):=\frac{1}{[K:k]}\cdot N_{T}(\alpha),$

where $K$ is a finite extension of $k$ containing $k(\alpha)$. These
definitions are independent of the choice of the extension $K.$ For an element
$b\in k(\alpha)$ distinct from $\alpha$, one can also define

$m_{S}(b,\alpha):=\frac{1}{[k(\alpha):k]}\cdot m_{T}(b,\alpha),\ \text{and}\
N_{S}(b,\alpha):=\frac{1}{[k(\alpha):k]}\cdot N_{T}(b,\alpha).$ (3.2)

It is easy to see that $h(\alpha)=m_{S}(\alpha)+N_{S}(\alpha)$ for all
$\alpha\in{{\bar{k}}}$. and Using 3.2 and following the proof 3.1 as in [16],
one can see that the inequality

$h(\alpha)\leq
m_{S}(b,\alpha)+N_{S}(b,\alpha)+h(b)+[k(\alpha):{\mathbb{Q}}]\cdot\log 2,$
(3.3)

holds for any $\alpha\in{{\bar{k}}}$ and $b\in{{\bar{k}}}$ distinct from
$\alpha$. Applying this inequality, the conjecture 1.1 can be written as
follows.

###### Conjecture 3.1

Let $k$ be a number field, ${{\bar{k}}}$ its algebraic closure and
$S\subset{\mathcal{P}}_{k}$ a finite set containing
${\mathcal{P}}_{k}^{\infty}$. Let $b_{1},\cdots,b_{n}$ be pairwise distinct
elements of $k$ and $d\geq 2$ an integer. Then for any $\epsilon>0$ and
$c\in{\mathbb{R}}$, the inequality

$(n-2-\epsilon)h(\alpha)\leq
d_{k}(\alpha)+\sum_{i=1}^{n}N_{S}(b_{i},\alpha)+c+c^{\prime},$ (3.4)

holds for almost all $\alpha\in{{\bar{k}}}$ with $[k(\alpha):k]\leq d$ and
different from $b_{i}$’s, where $c^{\prime}:=n(B+d\cdot\log 2)$ and
$B:=\max\\{h(b_{i})\\}_{i=1}^{n}.$

Let $b\in k$, $\alpha\in{{\bar{k}}}^{*}\backslash k$, and
${\mathfrak{p}}_{v}\in Spec({\mathcal{O}}_{K})$ corresponds to
$v\in{\mathcal{P}}_{K}^{0}$ where $K=k(\alpha)$. The truncated counting
function  on ${{\bar{k}}}$ is defined by

$N_{S}^{(1)}(b,\alpha):=\sum_{v\not\in
S}\min\\{1,\text{ord}_{{\mathfrak{p}}_{v}}^{+}(\alpha-b)\\}\cdot\log(\\#\frac{{\mathcal{O}}_{K}}{{\mathfrak{p}}_{v}}),$

where
$\text{ord}_{{\mathfrak{p}}_{v}}^{+}(\alpha-b):=\max\\{1,\text{ord}_{{\mathfrak{p}}_{v}}(\alpha-b)\\}.$
Here is the truncated version of the Vojta’s conjecture that is mentioned in
the introduction.

###### Conjecture 3.2

Let $k$ be a number field, ${{\bar{k}}}$ its algebraic closure and
$S\subset{\mathcal{P}}_{k}$ a finite set containing
${\mathcal{P}}_{k}^{\infty}$. Let $b_{1},\cdots,b_{n}$ be pairwise distinct
elements of $k$ and $d\geq 2$ an integer. Then for any $\epsilon>0$ and
$c\in{\mathbb{R}}$, the inequality

$(n-2-\epsilon)h(\alpha)\leq
d_{k}(\alpha)+\sum_{i=1}^{n}N_{S}^{(1)}(b_{i},\alpha)+c+c^{\prime}$ (3.5)

holds for almost all $\alpha\in{{\bar{k}}}$ with $[k(\alpha):k]\leq d$ and
different from $b_{i}$’s, where

$c^{\prime}:=n(B+d\cdot\log 2),\ \text{and }B:=\max\\{h(b_{i})\\}_{i=1}^{n}.$

The above conjecture is a special case of a general conjecture due to Vojta on
the bounded degree algebraic points on algebraic varieties, see 25.1 in [18].
Since $N_{S}^{(1)}(b,\alpha)\leq N_{S}(b,\alpha)$, so the truncated version of
the Vojta implies the non truncated one. The converse is the special case of
the theorem (3.1) in [18]. The number of elements in ${{\bar{k}}}$ of degree
at most $d$ for which the inequality of above conjecture does not hold,
depends on $b_{i}$’s, $\epsilon$, $c$, $k$, and $d$, but it is very hard to
compute it effectively in practice.

##  4 Proof of the main results

In this section, we give the proof of Theorem 1.2 and the corollary 1.3.

###  4.1 Proof of Theorem 1.2

Given integers $2\leq s\leq r$, let $M$ be the integer given by the theorem
1.2, i.e., $M=2r^{2}+6r+1$ if $r=s$ and $M=2sr^{2}+sr+1,$ otherwise. Consider
the subset ${\mathcal{B}}_{M}:=\\{b_{1},b_{2},\cdots,b_{M}\\}$ of the sequence
${\mathcal{B}},$ and $f\in{\bf G}_{r,s}^{{\mathcal{B}}_{M}}$ with
factorization $f=f_{1}^{s_{1}}\cdots f_{t}^{s_{t}}$, where $f_{j}\in k[x]$ are
irreducible polynomial of degree $d_{j}:=\deg(f_{j})$. For each
$j=1,\cdots,t$, let $\alpha_{j}\in{{\bar{k}}}$ is an arbitrary root of
$f_{j}$. We define $k_{j}:=k(\alpha_{j})$ and $g:=f_{1}\cdots f_{t}$, which is
of degree $d:=d_{1}+\cdots+d_{t}$.

Let $S\subset{\mathcal{P}}_{k}$ be a finite subset of ${\mathcal{P}}_{k}$ that
is the union of the sets ${\mathcal{P}}_{k}^{\infty}$, poles of each
$b_{i}\in{\mathcal{B}}_{M}$, and the set of zeros of $b_{i}-b_{j}$ for
$b_{i},$ $b_{j}\in{\mathcal{B}}_{M}$. Applying the truncated version 3.2 of
Vojta’s Conjecture with $b_{i}\in{\mathcal{B}}_{M}$, the set $S$, and integer
$r\geq 2$, for any $\epsilon>0$ and $c\in{\mathbb{R}}$, we conclude that

$(M-2-\epsilon)h(\alpha)\leq
d_{k}(\alpha)+\sum_{i=1}^{M}N_{S}^{(1)}(b_{i},\alpha_{j})+c+c^{\prime},$ (4.1)

holds for almost all $\alpha\in{{\bar{k}}}$ with $[k(\alpha):k]\leq r$ and
$\alpha_{j}\not=b_{i}$’s, where

$c^{\prime}:=M(B+r\cdot\log 2),\ \text{and}\ B:=\max\\{h(b_{i}):\ 1\leq i\leq
M\\}.$

There are a finite number of elements in ${{\bar{k}}}$ of degree at most $r$
for which 4.1 does not hold. Let us to denote by $N_{r}^{{\mathcal{B}}_{M}}$
the set of such elements in ${{\bar{k}}}$ and denote its cardinal number by
$n_{r}^{{\mathcal{B}}_{M}}$. Since we are going to estimate $\\#{\bf
G}_{r,s}^{{\mathcal{B}}_{M}}$, so for a while we ignore the polynomials
$f\in{\bf G}_{r,s}^{{\mathcal{B}}_{M}}$ that have some roots in the set
$N_{r}^{{\mathcal{B}}_{M}}$. We recall them in the moment of estimating the
$\\#{\bf G}_{r,s}^{{\mathcal{B}}_{M}}$. Thus,

$(M-2-\epsilon)h(\alpha_{j})\leq
d_{k}(\alpha_{j})+\sum_{i=1}^{M}N_{S}^{(1)}(b_{i},\alpha_{j})+c+c^{\prime},$
(4.2)

where $\alpha_{i}$ is a root of $f_{j}$ for each $j=1,\cdots,t.$ Applying the
part (iii) of theorem 2.2 to each of $\alpha_{j}$’s and using $d_{j}\leq d$,
leads to

$d_{k}(\alpha_{j})\leq 2(d_{j}-1)h(\alpha_{j})+A(d_{j})\leq
2(d-1)h(\alpha_{j})+A(d),$

where $A(d)=d\log d$ if $k={\mathbb{Q}}$ and $(2d-1)\log d$ otherwise. Putting
this into 4.2 and using the fact that $A(d)\leq A(r)\leq 2r\log r$ leads to

$(M-2d-\epsilon)h(\alpha_{j})\leq\sum_{i=1}^{M}N_{S}^{(1)}(b_{i},\alpha_{j})+c+c_{1},$
(4.3)

where $c_{1}:=M(B+r\cdot\log 2)+2r\log r.$ Then, multiplying the both side
with $d_{j}$ and summing-up, one can obtain that

$\sum_{j=1}^{t}(M-2d-\epsilon)h_{k_{j}}(\alpha_{j})\leq\sum_{j=1}^{t}\sum_{i=1}^{M}d_{j}N_{S}^{(1)}(b_{i},\alpha_{j})+r(c+c_{1}).$
(4.4)

We are going to give an upper bound for the term involving the truncated
function in 4.4. But to do this, we need the following lemma.

###### Lemma 4.1

Let $D(g)$ be the discriminant the polynomial $g=f_{1}\cdots f_{t}$ of degree
$d\geq 2$ and let $A(d)$ be as above. Then

$h(D(g))\leq 2(d-1)\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})+4d(d-1)+A(d).$

Proof. We assume that $\alpha_{ji}$ are the roots of $f_{j}$ for $1\leq i\leq
d_{j}$. Since the absolute heights are invariant by the action of the Galois
group of ${\mathbb{P}}_{{{\bar{k}}}}^{1}$, so

$\displaystyle\sum_{j=1}^{t}h(f_{j})$
$\displaystyle\leq\sum_{j=1}^{t}(\sum_{i=1}^{d_{j}}h(\alpha_{ji})+(d_{j}-1)\log(2))$
$\displaystyle\leq\sum_{j=1}^{t}d_{j}h(\alpha_{j})+\sum_{j=1}^{t}(d_{j}-1)\log(2)$
$\displaystyle\leq\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})+d-t\log(2).$

Hence, using the properties of heights functions, we have

$\displaystyle h(g)=h(f_{1}\cdots f_{t})$
$\displaystyle\leq\sum_{j=1}^{t}[h(f_{j})+(d_{j}+1)\log(2)]$
$\displaystyle=\sum_{i=1}^{t}h(f_{j})+d+t\log(2)\leq\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})+2d$

By the part (ii) of 2.2, we obtain the desired inequality,

$h(D(g))\leq 2(d-1)h(g)+A(d)\leq
2(d-1)\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})+4d(d-1)+A(d).$

Let $D$ be the reduced divisor on $Spec({\mathcal{O}}_{k})$ whose support
consists of the union of the sets $S$, the zeros of $D(g)$ and the poles of
the $\alpha_{j}$’s.

###### Lemma 4.2

With notation as above, we have:

$\sum_{j=1}^{t}\sum_{i=1}^{M}d_{j}N_{S}^{(1)}(b_{i},\alpha_{j})\leq\big{[}\frac{Ms^{+}}{s}+d(2d-1)\big{]}\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})+rc_{2},$

where $s^{+}:=\max\\{s_{1},\cdots,s_{t}\\}$, $d=d_{1}+\cdots+d_{t}$,
$a(S):=\sum_{{\mathfrak{p}}\in S}\log\\#({\mathcal{O}}_{k}/{\mathfrak{p}})$,
and

$c_{2}:=\frac{M(B+\log 2)}{s}+a(S)+A(r)+4r(r-1).$

Proof. By changing the order of sums in the left hand side of 4.4 and
following the last part of the proof of Lemma 4.9 in [10], we have

$\displaystyle\sum_{i=1}^{M}\sum_{j=1}^{t}d_{j}N_{S}^{(1)}(b_{i},\alpha_{j})$
$\displaystyle\leq\frac{1}{s}\sum_{i=1}^{M}\sum_{j=1}^{t}s_{j}d_{j}h(b_{i}-\alpha_{j})+d\deg(D)$
$\displaystyle\leq\frac{1}{s}\sum_{i=1}^{M}\big{(}\sum_{j=1}^{t}s_{j}d_{j}[h(b_{i})+h(\alpha_{j})+\log
2]\big{)}+d\deg(D).$

Since $t\leq r=\sum_{j=1}^{t}s_{j}d_{j}$, and $s_{j}\leq s^{+}$, so we have

$\displaystyle\sum_{i=1}^{M}\sum_{j=1}^{t}d_{j}N_{S}^{(1)}(b_{i},\alpha_{j})$
$\displaystyle\leq\frac{1}{s}\sum_{i=1}^{M}[\sum_{j=1}^{t}s_{j}d_{j}h(\alpha_{j})+$
$\displaystyle\hskip 56.9055pt\sum_{j=1}^{t}s_{j}d_{j}h(b_{i})+t\log
2]+d\deg(D)$
$\displaystyle\leq\frac{1}{s}\sum_{i=1}^{M}[\sum_{j=1}^{t}s_{j}h_{k_{j}}(\alpha_{j})+r(h(b_{i})+\log
2)]+d\deg(D)$
$\displaystyle\leq\frac{M}{s}\sum_{j=1}^{t}s_{j}h_{k_{j}}(\alpha_{j})+\frac{Mr(B+\log
2)}{s}+d\deg(D)$
$\displaystyle\leq\frac{Ms^{+}}{s}\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})+\frac{Mr(B+\log
2)}{s}+d\deg(D).$

To give an upper bound on the $\deg(D)$ in terms of $h(\alpha_{j})$’s, we
assume that $S^{\prime}$ and $S_{j}$ are the subsets of
${\mathcal{P}}_{k}^{0}$ such that $D(g)$ vanished at ${\mathfrak{p}}$,
$\alpha_{j}$ has a pole above ${\mathfrak{p}}$, respectively. We let
$S^{\prime\prime}$ to be the union of $S_{j}$ for $j=1,\cdots,t$. Then,

$\displaystyle\deg(D)$ $\displaystyle=\sum_{{\mathfrak{p}}\in
S^{\prime\prime}}\deg({\mathfrak{p}})+\sum_{{\mathfrak{p}}\in
S^{\prime}}\deg({\mathfrak{p}})+\sum_{{\mathfrak{p}}\in
S}\deg({\mathfrak{p}})$ $\displaystyle=\sum_{j=1}^{t}\sum_{{\mathfrak{p}}\in
S_{j}}\deg({\mathfrak{p}})+\\#S^{\prime}+a(S)$
$\displaystyle=\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})+h(D(g))+a(S).$

Using 4.1, and $A(d)\leq A(r)$ for $d\leq r$, we get that

$\displaystyle\deg(D)$
$\displaystyle\leq\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})+2(d-1)\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})+a(S)+A(r)+4r(r-1)$
$\displaystyle\leq(2d-1)\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})+a(S)+A(r)+4r(r-1).$

Multiplying the last inequality by $d$, gives that

$d\deg(D)\leq
d(2d-1)\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})+r[a(S)+A(r)+4r(r-1)].$

Putting all of the above inequalities together leads to the desired one.

By the lemma 4.2, one can rewrite 4.4 as follows,

$\sum_{j=1}^{t}[M(1-\frac{s^{+}}{s})-2d^{2}-d-\epsilon]h_{k_{j}}(\alpha_{j})\leq
r(c+c_{1}+c_{2}).$ (4.5)

###### Lemma 4.3

For integers $2\leq s\leq r$, let $M=2r^{2}+6r+1$ if $r=s$ and $2sr^{2}+sr+1$
otherwise. Then, for each $1\leq d\leq r$, we have

$M(1-\frac{s^{+}}{s})-2d^{2}-d\geq\frac{1}{r}.$ (4.6)

Proof. If $f\in{\bf G}_{r,s}^{{\mathcal{B}}_{M}}$ with irreducible
factorization $f=f_{1}^{s_{1}}\cdots f_{t}^{s_{t}}$, then

$s^{+}:=\max\\{s_{1},\cdots,s_{t}\\}\leq s-1,$

and hence $r-s^{+}\geq d-1$. Indeed, if $j_{0}$ is an index such that
$s_{j_{0}}=s^{+}$, then

$r=\sum_{j=1}^{t}s_{j}d_{j}\geq s^{+}d_{j_{0}}+\sum_{j\not=j_{0}}^{t}d_{j}\geq
s^{+}+d_{j_{0}}-1+\sum_{j\not=j_{0}}^{t}d_{j}=s^{+}+d-1.$

Using $r-s^{+}\geq d-1$, we have

$1-\frac{s^{+}}{s}=\frac{r-s^{+}}{s}\geq\begin{cases}\frac{d-1}{r}&\mbox{if
}s=r\\\ \frac{1}{s}&\mbox{otherwise}.\end{cases}$ (4.7)

In the case $s=r$, since $M=2r^{2}+6r+1$ and $1-s^{+}/s\geq d-1\geq 1$, so we
have

$\displaystyle M(1-\frac{s^{+}}{s})-2d^{2}-d$ $\displaystyle\geq
M(\frac{d-1}{r})-2d^{2}-d$
$\displaystyle\geq\frac{d-1}{r}(M-\frac{2rd^{2}+rd}{d-1})$
$\displaystyle\geq\frac{d-1}{r}(M-2rd-3r-\frac{3r}{d-1}).$

Since $3r/(d-1)\leq 3r$ and $(d-1)/r\geq 1/r$ for $d-1\geq 1$, and $d\leq r$,
so we get that

$\displaystyle M(1-\frac{s^{+}}{s})-2d^{2}-d$
$\displaystyle\geq\frac{d-1}{r}(M-2rd-6r)$
$\displaystyle\geq\frac{1}{r}(M-2r^{2}-6r)\geq\frac{1}{r}.$

In the case $s<r$, we have $M=2sr^{2}+sr+1$ and $1-s^{+}/s\geq 1/s$. Hence,
$d\leq r$ and $M-2sr^{2}-sr\geq 1$, gives that

$\displaystyle M(1-\frac{s^{+}}{s})-2d^{2}-d$ $\displaystyle\geq M/s-2d^{2}-d$
$\displaystyle\geq\frac{1}{s}(M-2sd^{2}-sd)$
$\displaystyle\geq\frac{1}{r}(M-2sr^{2}-sr)\geq\frac{1}{r}.$

Using 4.6, in either cases, one can rewrite 4.5 as

$(\frac{1}{r}-\epsilon)\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})\leq
r(c+c_{1}+c_{2}).$ (4.8)

Since the constants $\epsilon$ and $c$ are arbitrary, so we assume that

$\epsilon:=1/(r+1),\ c:=\frac{1}{mr^{3}(r+1)^{2}}-c_{1}-c_{2},\
c_{3}:=\frac{1}{mr(r+1)},$ (4.9)

where $m$ is the degree of the number field $k$. Then, 4.8 implies that

$h(\alpha_{j})=d_{j}h(\alpha_{j})=h_{k_{j}}(\alpha_{j})\leq\sum_{j=1}^{t}h_{k_{j}}(\alpha_{j})\leq
r^{2}(r+1)[c+c_{1}+c_{2}]\leq c_{3}.$

We note that the consonant $c_{3}$ depends only on $r$ and $m$, but it is
independent of an special $f\in{\bf G}_{r,s}^{{\mathcal{B}}_{M}}$. Denote by
$N({\mathbb{P}}_{{\bar{k}}}^{1};r;c_{3})$ the number of algebraic numbers
$\alpha\in{{\bar{k}}}$ of degree at most $r$ and height at most $c_{3}$. By
the famous Northcott’s theorem [9], $N({\mathbb{P}}_{{\bar{k}}}^{1};r;c_{3})$
is a positive number. Letting $c_{4}:=c_{3}^{mr(r+1)-r}$ and applying the
theorem 2.1 with constants $\varepsilon:=1$, $T:=c_{3}$ and $T_{1}:=c_{4}$,
gives us two constants $c_{5},c_{6}>0$ ( depending on $r$ and $K$ but not on
$b_{i}\in{\mathcal{B}}_{M}$) such that

$b_{k,r}\cdot 2^{-mr(r+1)}+c_{5}\cdot c_{4}\cdot c_{3}\leq
N({\mathbb{P}}_{{\bar{k}}}^{1};r;c_{3})\leq b_{k,r}\cdot
2^{mr(r+1)}+c_{6}\cdot c_{4},$

where $b_{k,r}=r\cdot a_{k,r}\cdot c_{3}^{mr(r+1)}$ and $a_{k,r}$ is given by
2.2. Let ${\mathcal{A}}_{r}$ be the union of all $\alpha\in{{\bar{k}}}$ of
degree at most $r$ and height at most $c_{3}$ with the set
$N_{r}^{{\mathcal{B}}_{M}}$ of algebraic numbers not satisfying the inequality
4.1. Then

$b_{k,r}\cdot 2^{-mr(r+1)}+c_{5}\cdot c_{4}\cdot
c_{3}\leq\\#{\mathcal{A}}_{r}\leq b_{k,r}\cdot 2^{mr(r+1)}+c_{6}\cdot
c_{4}+n_{r}^{{\mathcal{B}}_{M}}.$

Since for each $f\in{\bf G}_{r,s}^{{\mathcal{B}}_{M}}$ has at most $r$ roots
in ${{\bar{k}}}$, so we conclude that

$b_{k,r}\cdot 2^{-mr(r+1)}+c_{5}\cdot c_{4}\cdot c_{3}\leq\\#{\bf
G}_{r,s}^{{\mathcal{B}}_{M}}\leq r\cdot(b_{k,r}\cdot 2^{mr(r+1)}+c_{6}\cdot
c_{4}+n_{r}^{{\mathcal{B}}_{M}}).$

Therefore, we obtain the desired lower and upper bounds for $\\#{\bf
G}_{r,s}^{{\mathcal{B}}_{M}}$, i.e., $C_{0}\leq\\#{\bf
G}_{r,s}^{{\mathcal{B}}_{M}}\leq C_{1}$, where

$C_{0}:=b_{k,r}\cdot 2^{-mr(r+1)}+c_{5}\cdot c_{4}\cdot c_{3},$

and

$C_{1}:=r\cdot(b_{k,r}\cdot 2^{mr(r+1)}+c_{6}\cdot
c_{4}+n_{r}^{{\mathcal{B}}_{M}}).$

This complete the proof of Theorem 1.2.

###  4.2 Proof of Corollary 1.3.

We start with the following result that is used in the proof of the corollary
1.3.

###### Lemma 4.4

If any of the sequences ${\mathcal{C}},{\mathcal{D}}$ and ${\mathcal{E}}$ is
periodic with period $m\geq 1$, then for each $\ell=qm+p$ with $q\geq 0$ and
$1\leq p\leq m$, we have:

  * (i)

$c_{\ell}=c_{p}$ implies that $b_{\ell}=b_{p}+q(b_{m+1}-b_{1}),$ in
particular,
$b_{qm}=b_{m}+(q-1)(b_{m+1}-b_{1});$

  * (ii)

$d_{\ell}=d_{p}$ implies that $b_{\ell+1}=b_{p+1}(b_{m+1}/b_{1})^{q},$ in
particular, $b_{qm}=b_{m}(b_{m+1}/b_{1})^{q-1};$

  * (iii)

$e_{\ell}=e_{p}$ implies that $1/b_{\ell}=1/b_{p}+q(1/b_{m+1}-1/b_{1}),$ in
particular,
$1/b_{qm}=1/b_{m}+(q-1)(1/b_{m+1}-1/b_{1}).$

Proof. We prove just the part (i) of this lemma by induction on $q$, and leave
the other cases to the reader. We assume that $\ell=m+p$ and $1\leq p\leq m$.
Then, we have

$\displaystyle b_{m+p}-b_{m+1}$
$\displaystyle=c_{m+1}+c_{m+2}+\cdots+c_{m+p-1}$
$\displaystyle=c_{1}+c_{2}+\cdots+c_{p-1}=b_{p}-b_{1}.$

Hence, the assertion is true in the case $q=1$, i.e.,
$b_{m+p}-b_{p}=b_{m+1}-b_{1}$. We assume that it is true for $\ell=qm+p$ with
$1\leq p\leq m$. Since

$\displaystyle b_{(q+1)m+p}-b_{qm+p}$
$\displaystyle=c_{qm+p}+\cdots+c_{(q+1)m}+c_{(q+1)m+1}+\cdots+c_{(q+1)m+p-1}$
$\displaystyle=c_{p}+c_{p+1}+\cdots+c_{m}+c_{1}+\cdots+c_{p-1}=b_{m+1}-b_{1},$

so $b_{(q+1)m+p}=b_{qm+p}+b_{m+1}-b_{1}$. By the hypothesis of the induction,
we have $b_{qm+p}=b_{p}+q(b_{m+1}-b_{1})$ and hence
$b_{(q+1)m+p}-b_{p}=(q+1)(b_{m+1}-b_{1})$ as desired. Clearly, the general
case implies the particular one.

Given integers $2\leq s\leq r$, let $M$ be the integer given by the theorem
1.2. Let $N_{0}$ denote the cardinal number of ${\bf
G}_{r,s}^{{\mathcal{B}}_{M}}$. We assume that ${\mathcal{C}}$ or
${\mathcal{D}}$ and or ${\mathcal{F}}$ is periodic with period $m\geq 1$. To
prove the corollary, we show that ${\bf G}_{r,s}^{{\mathcal{B}}_{n}}$ is empty
set for each $n\geq M_{0}$, where

$M_{0}:=\begin{cases}m\cdot(N_{0}+M+2)&\text{if $m\leq M$;}\\\
m\cdot(N_{0}+2)&\text{if $m>M$.}\end{cases}$

By contrary, we assume that ${\bf G}_{r,s}^{{\mathcal{B}}_{n}}\not=\emptyset$
and $f\in{\bf G}_{r,s}^{{\mathcal{B}}_{n}}$ for some $n\geq M_{0}.$ We prove
the existence of $N_{0}+1$ pairwise distinct polynomials
$f_{1},\cdots,f_{N_{0}},f_{N_{0}+1}$ in ${\bf G}_{r,s}^{{\mathcal{B}}_{M}}$.
This gives a contradiction and complete the proof of the corollary. Let us to
assume that the sequence ${\mathcal{C}}$ is periodic with period $m\geq 1$.
For each $1\leq j\leq N_{0}+1$, the polynomials $f_{j}(x)$ defined by

$f_{j}(x):=f(x+j(b_{m+1}-b_{1})),$

are distinct elements of ${\bf G}_{r,s}^{{\mathcal{B}}_{M}}$. To see this, it
is enough to check that $f_{j}(b_{i})$ is an $s$-powerful element in $k^{*}$,
for $1\leq i\leq M$ and $1\leq j\leq N_{0}+1$. If $m>M$, then by definition of
$f_{j}$’s and part (i) of above lemma, we have

$f_{j}(b_{i})=f(b_{i}+j(b_{m+1}-b_{1}))=f(b_{jm+i}).$

Since $jm+i\leq(N_{0}+1)m+M\leq(N_{0}+2)m\leq n$ and $f\in{\bf
G}_{r,s}^{{\mathcal{B}}_{n}}$, so $f(b_{jm+i})$ and hence $f_{j}(b_{i})$ is an
$s$-powerful element in $k^{*}$. In the case $m\leq M$, for each $1\leq i\leq
M$ we write $i=i_{1}m+i_{2}$ for some $0\leq i_{1}\leq M$ and $0\leq i_{2}\leq
m$. Then, by part (i) of above lemma, we have

$\displaystyle f_{j}(b_{i})$ $\displaystyle=f(b_{i}+j(b_{m+1}-b_{1}))$
$\displaystyle=f(b_{i_{2}}+i_{1}(b_{m+1}-b_{1})+j(b_{m+1}-b_{1}))$
$\displaystyle=f(b_{i_{2}}+(i_{1}+j)(b_{m+1}-b_{1}))=f(b_{(i_{1}+j)m+i_{2}}).$

Since $(i_{1}+j)m+i_{2}\leq(N_{0}+M+1)m+m=(N_{0}+M+2)m\leq n$ and $f\in{\bf
G}_{r,s}^{{\mathcal{B}}_{n}}$, so $f(b_{(i_{1}+j)m+i_{2}})$ and hence
$f_{j}(b_{i})$ is an $s$-powerful element in $k^{*}$.

In the case that the sequence ${\mathcal{D}}$ or ${\mathcal{E}}$ is periodic
with period $m\geq 1$, one can get the result by a similar arguments. Indeed,
for each $f\in{\bf G}_{r,s}^{{\mathcal{B}}_{n}}$ it is enough to consider

$f_{j}(x):=f((b_{m+1}/b_{1})^{(j-1)}\cdot x),$
$f_{j}(x):=(1/x+j(1/b_{m+1}-1/b_{1}))^{r}f\left(\frac{1}{1/x+j(1/b_{m+1}-1/b_{1})}\right),$

respectively, where $1\leq j\leq M$ and $1\leq j\leq N_{0}+1$, We note that
given any polynomial $f\in k[x]$ of degree $\geq 2$ and any $c\in k^{*}$, the
function $g(x)=(1/x+c)^{r}f(\frac{1}{1/x+c})$ is a polynomial of degree $r$.

## References

  * [1] Bèrczes, A., Evertse, J. H. and Gyory, L.: Effective results for hyper- and supperelliptic equations over number fields, Publ. Math. Debresen, 82, No. 3-4, (2013), 354-358.
  * [2] Bombieri, E. and Gubler, W.: Heights in Diophantine Geometry, Cambridge University press, New york (2006).
  * [3] Davenport, H., Lewis, D. J., and Schinzel, A.: Polynomials of certain special types, Acta Arith., 9, (1964), 107-116.
  * [4] Hindry, M. and Silverman, J. H.: Diophantine Geometry: An introduction, Graduate Texts in Mathematics, Vol. 201, Springer-Verlag, New york (2001).
  * [5] Ih, Su-Ion: Algebraic points on the projective line, J. Korean Math. Soc., 45, No. 6, (2008), 1635-1646.
  * [6] Mahler, K. : An inequality for the discriminant of a polynomial, Michigan Math. J., 11, No. 3, (1964), 257-262.
  * [7] Masser, D. and Vaaler, J. D. : Counting algebraic numbers with large height. II, Trans. Amer. Math. Soc., 359, No. 1, (2007), 427-445.
  * [8] Neukrich, J.: Algebraic Number theory: translated from the Geman by Norbert Schappacher, Springer-Verlag, Berlin- Heidelberg, 1999.
  * [9] Northcott, D. G.: An inequality in the theory of arithmetic on algebraic varieties, Proc. Camb. Phil. Soc., 45, (1949), 502-509.
  * [10] Pasten, H.: Powerful values of polynomials and a conjecture of Vojta, J. Number Theory, 133, (2014), 2964-2998.
  * [11] Ribenboim, P.: Polynomials whose values are powers, Collection of articles dedicated to Helmut Hasse on his seventy- fifth birthday, J. Reine Angew. Math. II, 268/269, (1974), 34-40.
  * [12] Salami, S.: On some different related problems in Diophantine geometry, Ph.D thesis, Universidade Federal do Rio de Janeiro, Brazil, (2017).
  * [13] Schmidt, W. M.: Northcott’s theorem on heights. I, Monatsh. Math., 115, No. 1-2, (1993), 169–181.
  * [14] Schmidt, W. M.: Northcott’s theorem on heights. II. The quadratic case, Acta Arith., 70, No. 4, (1995), 343–375.
  * [15] Shinzel, A. and Tijdeman, R.: On the equation $y^{m}=P(x)$, Acta Arith., 31, (1976), 199-204.
  * [16] Vojta, P.: Diophantine Approximation and Nevanlinna Theory, in Arithmetic Geometry, Lecture Notes in Mathematics, 2009, (2009), 111-224.
  * [17] Vojta, P.: Diophantine Approximation and Value distribution theory, Lecture Notes in Mathematics, 1239, Springer-Verlag, (1987).
  * [18] Vojta, P.: A more general ABC conjecture, Int. Math. Res. Notices, 21, Springer-Verlag, (1998), 1103-1116.
  * [19] Waldschmidt, M.: Diophantine approximation on linear algebraic group, Springer-Verlag, New York- Berlin, (2000).
  * [20] Walsh, P. G.: On a conjecture of Schinzel and Tijdeman, Zakopane-Koscielisko, de Gruyter, 1, (1997), 577-582.