# Number of integers represented by families
of binary forms II: binomial forms

Étienne Fouvry
Université Paris–Saclay, CNRS,
Laboratoire de Mathématiques d’Orsay,
91405 Orsay, France
E-mail<EMAIL_ADDRESS>Michel Waldschmidt
Sorbonne Université and Université de Paris,
CNRS, IMJ–PRG,
75005 Paris, France
E-mail<EMAIL_ADDRESS>
###### Abstract

We consider some families of binary binomial forms $aX^{d}+bY^{d}$, with $a$
and $b$ integers. Under suitable assumptions, we prove that every rational
integer $m$ with $|m|\geq 2$ is only represented by a finite number of the
forms of this family (with varying $d,a,b$). Furthermore the number of such
forms of degree $\geq d_{0}$ representing $m$ is bounded by
$O(|m|^{(1/d_{0})+\epsilon})$ uniformly for $|m|\geq 2$. We also prove that
the integers in the interval $[-N,N]$ represented by one of the form of the
family with degree $d\geq d_{0}$ are almost all represented by some form of
the family with degree $d=d_{0}$.

In a previous paper we investigated the particular case where the binary
binomial forms are positive definite. We now treat the general case by using a
lower bound for linear forms of logarithms.

††2020 _Mathematics Subject Classification_ : Primary 11E76; Secondary 11D45
11D85.†† _Key words and phrases_ : Binomial binary forms, Representation of
integers by binomial binary forms, Families of Diophantine equations.

Cet article est dédié à Henryk Iwaniec,

en témoignage de notre admiration

pour ses contributions majeures

à la théorie analytique des nombres.

##  1 Introduction

When $d$, $a$ and $b$ are rational integers different from $0$, with $d\geq
3$, Theorem 1.1 of [SX] gives an asymptotic estimate for the number of
rational integers in the interval $[-N,N]$ represented by the binary form
$aX^{d}+bY^{d}$. This estimate has the shape

$C_{a,b}N^{2/d}+O(N^{\beta})\quad\hbox{ {as} $N\to\infty$},$

where the exponent $\beta<2/d$ is explicit and where the constant $C_{a,b}>0$
is also explicit (it corresponds to the constant $C_{F}=A_{F}W_{F}$ in [SX,
Corollary 1.3] associated with the binary form
$F(X,Y)=F_{a,b,d}(X,Y)=aX^{d}+bY^{d}$. For more precision see §3 below). Here
we consider the representation of integers by some element of families of such
binary binomial forms

For every integer $d\geq 3$, let $\mathcal{E}_{d}$ be a finite subset of
${\mathbb{Z}}\setminus\\{0\\}\times{\mathbb{Z}}\setminus\\{0\\}$ and let
$\mathcal{F}_{d}$ be the set of binary binomial forms $F_{a,b,d}(X,Y)$ with
$(a,b)\in\mathcal{E}_{d}$. We are interested in the representation of integers
$m\in{\mathbb{Z}}$ by some form of the family $\mathcal{F}=\bigcup_{d\geq
3}\mathcal{F}_{d}$. For $d\geq 3$ and $m$ in ${\mathbb{Z}}$, we introduce the
two sets

$\displaystyle\mathcal{G}_{\geq d}(m)=\Big{\\{}($ $\displaystyle
d^{\prime},a,b,x,y)\;\mid\;m=ax^{d^{\prime}}+by^{d^{\prime}}\;\hbox{ with }$
$\displaystyle d^{\prime}\geq
d,\;(a,b)\in\mathcal{E}_{d^{\prime}},\;(x,y)\in{\mathbb{Z}}^{2}\;\hbox{ and
}\max\\{|x|,|y|\\}\geq 2\Big{\\}}$

and

$\mathcal{R}_{\geq d}=\left\\{m\in{\mathbb{Z}}\;\mid\;\mathcal{G}_{\geq
d}(m)\not=\emptyset\right\\}.$

For $N$ positive integer, we denote

$\mathcal{R}_{\geq d}(N)=\mathcal{R}_{\geq d}\cap[-N,N].$

The set $\mathcal{E}_{d}$ must also satisfy the following hypotheses(C1) : For
every $(a,b)\neq(a^{\prime},b^{\prime})\in\mathcal{E}_{d}$, at least one of
ratios $a/a^{\prime}$ and $b/b^{\prime}$ is not the $d$–th power of a rational
number, (C2): For every $(a,b)\neq(a^{\prime},b^{\prime})\in\mathcal{E}_{d}$,
at least one of ratios $a/b^{\prime}$ and $b/a^{\prime}$ is not the $d$–th
power of a rational number.

The exponent $\vartheta_{d}<2/d$ is defined in [FW2, (2.1)]:

$\theta_{d}=\begin{cases}\frac{24\sqrt{3}+73}{60\sqrt{3}+73}=\frac{2628\sqrt{3}-1009}{5471}=0.6475\dots&\text{
{for} }d=3,\\\ \frac{2\sqrt{d}+9}{4d\sqrt{d}-6\sqrt{d}+9}&\text{ {for} }4\leq
d\leq 20,\\\ \frac{1}{d-1}&\text{ {for} }d\geq 21.\end{cases}$

When the family $\mathcal{F}$ is given and when the integer $d$ is $\geq 3$,
the integer $d^{\dagger}$ is given by the formula

$d^{\dagger}:=\begin{cases}\inf\\{d^{\prime}:d^{\prime}>d\text{ such that
}\mathcal{F}_{d^{\prime}}\neq\emptyset\\}&\hbox{if there exists $d^{\prime}>d$
such that $\mathcal{F}_{d^{\prime}}\neq\emptyset$,}\\\ \infty&\hbox{if
$\mathcal{F}_{d^{\prime}}=\emptyset$ for all $d^{\prime}>d$.}\end{cases}$

Our first result is the following.

######  Theorem 1.1 (Positive definite case).

Let $\mathcal{E}_{d}\subset{\mathbb{Z}}_{>0}\times{\mathbb{Z}}_{>0}$
satisfying (C1) and (C2) above and the equality $\mathcal{E}_{d}=\emptyset$
for odd $d$. Furthermore, we suppose that

$\frac{1}{d}\log(\sharp\mathcal{E}_{d}+1)\to 0\quad\hbox{ as }\quad
d\to\infty.$ (1.1)

Then
(a) For all $m\in{\mathbb{Z}}\setminus\\{0,1\\}$ and all $d\geq 4$, the set
$\mathcal{G}_{\geq d}(m)$ is finite. Furthermore, for all $d\geq 4$ and all
$\epsilon>0$, we have, as $|m|\to\infty$,

$\sharp\mathcal{G}_{\geq d}(m)=O\left(|m|^{(1/d)+\epsilon}\right).$

(b) For all $d\geq 4$ and all $\epsilon>0$, we have, as $N\to\infty$,

$\sharp\mathcal{R}_{\geq
d}(N)=\left(\sum_{(a,b)\in\mathcal{E}_{d}}C_{a,b}\right)N^{2/d}+O_{d,\epsilon}\Bigl{(}N^{\max\\{\vartheta_{d}+\epsilon,2/d^{\dagger}\\}}\Bigr{)}.$

(c) In the above formula, we have $C_{a,b}=A_{F_{a,b,d}}W_{F_{a,b,d}}$, where

$A_{F_{a,b,d}}=\iint_{|ax^{d}+by^{d}|\leq 1}dxdy$

and where the values of the rational positive numbers $W_{F_{a,b,d}}$ are
given in Proposition 3.3.

######  Remark 1.2.

One finds explicit values, in terms of the $\Gamma$–function, of the
fundamental area $A_{F_{a,b,d}}$ in [SX, Corollary 1.3].

The hypothesis $\sharp\mathcal{E}_{d}\leq d^{A_{1}}$ in [FW2, Theorem 1.13],
which implies the condition $(\mathrm{iii})$ in Definition 2.2 below of a
$(A,A_{1},d_{0},d_{1},\kappa)$–regular family, is replaced here by (1.1) which
cannot be omitted: for $d\geq 3$ and $N=2^{2d}+1$, each of $2^{d}$ integers of
the form $a2^{d}+1$, $a=1,2,3,\dots,2^{d}$ is represented by one of the form
$aX^{d}+Y^{d}$ with the choice $x=2$, $y=1$.

Our second result is

######  Theorem 1.3 (General case).

There exists an absolute constant $\eta>0$ with the following property. We
suppose that there exists $d_{0}>0$ such that, for all $d\geq d_{0}$, we have
the inequality

$\max_{(a,b)\in\mathcal{E}_{d}}\\{|a|,|b|\\}\leq\exp(\eta d/\log d).$

Then
(a) For all $m\in{\mathbb{Z}}\setminus\\{-1,0,1\\}$ and all $d\geq 3$, the set
$\mathcal{G}_{\geq d}(m)$ is finite. Furthermore, for all $d\geq 3$ and all
$\epsilon>0$, we have, as $|m|\to\infty$,

$\sharp\mathcal{G}_{\geq d}(m)=O\left(|m|^{(1/d)+\epsilon}\right).$

(b) For all $d\geq 3$ and all $\epsilon>0$, we have, as $N\to\infty$,

$\sharp\mathcal{R}_{\geq
d}(N)=\left(\sum_{(a,b)\in\mathcal{E}_{d}}C_{a,b}\right)N^{2/d}+O_{d,\epsilon}\Bigl{(}N^{\max\\{\vartheta_{d}+\epsilon,2/d^{\dagger}\\}}\Bigr{)}.$

(c) The properties of the constant $C_{a,b}$ are the same as in Theorem 1.1
(c).

We will prove the result with the choice

$\eta=2^{-81}3^{-16},$

corresponding to the right–hand side of (4.1) for $\lambda=3$.

In both Theorems 1.1 and 1.3, the proof of the bound for
$\sharp\mathcal{G}_{\geq d}(m)$ is based on the explicit estimate (2.3). The
fact that $\mathcal{G}_{\geq d}(m)$ is finite for all $m\not\in\\{-1,0,1\\}$
is not a consequence of the bound for $\sharp\mathcal{R}_{\geq d}(N)$ (see
Example 2.4).

Compared to [FW2], our new tool is a lower bound for linear forms in
logarithms; the finiteness of the number of representations of a given integer
$m$ depends of this estimate. As we will show in section 7, the $abc$
Conjecture would give an estimate very close to what would be deduced from
conjectures on linear forms in logarithms.

##  2 A new definition of a regular family

######  Definition 2.1.

We will say that an infinite set $\mathcal{F}$ of binary forms with
discriminants different from zero and with degrees $\geq 3$ is regular  if
there exists a positive integer $A$ satisfying the following two conditions
$(\mathrm{i})$ Two forms of the family $\mathcal{F}$ are ${\rm
GL}(2,{\mathbb{Q}})$–isomorphic if and only if they are equal,
$(\mathrm{ii})$ For all $\epsilon>0$, there exist two positive integers
$N_{0}=N_{0}(\epsilon)$ and $d_{0}=d_{0}(\epsilon)$ such that, for all $N\geq
N_{0}$, the number of integers $m$ in the interval $[-N,N]$ for which there
exists $d\in{\mathbb{Z}}$, $(x,y)\in{\mathbb{Z}}^{2}$ and
$F\in\mathcal{F}_{d}$ satisfying

$d\geq d_{0},\quad\max\\{|x|,|y|\\}\geq A\quad\hbox{and }\quad F(x,y)=m$

is bounded by $N^{\epsilon}$.

Of course the symbol $\mathcal{F}_{d}$ above, denotes the subset of
$\mathcal{F}$ of forms with degree $d$.

For the truth of Theorem 2.5 below, one cannot drop the parameter $A$, as one
sees by considering the family of cyclotomic forms [FW1] where the hypothesis
$(\mathrm{ii})$ is satisfied with $A=2$ but not with $A=1$.

Recall the definition 1.10 of a $(A,A_{1},d_{0},d_{1},\kappa)$–regular family
as it is introduced in [FW2].

######  Definition 2.2.

Let $A$, $A_{1}$, $d_{0}$, $d_{1}$ be integers and let $\kappa$ be a real
number such that

$A\geq 1,\,A_{1}\geq 1,\,d_{1}\geq d_{0}\geq 0,\,0<\kappa<A.$

Let $\mathcal{F}$ be a set of binary forms with integral coefficients and with
discriminants different from zero. We say that $\mathcal{F}$ is
$(A,A_{1},d_{0},d_{1},\kappa)$–regular if it satisfies the following
conditions:

  1. (i)

The set $\mathcal{F}$ is infinite,

  2. (ii)

All the forms of $\mathcal{F}$ have their degrees $\geq 3$,

  3. (iii)

For all $d\geq 3$, we have the inequality $\sharp\mathcal{F}_{d}\leq
d^{A_{1}}$,

  4. (iv)

Two forms of $\mathcal{F}$ are isomorphic if and only if they are equal,

  5. (v)

For any $d\geq\max\\{d_{1},d_{0}+1\\}$, the following holds

$\left.\begin{array}[]{ll}F\in\mathcal{F}_{d},\\\
(x,y)\in{\mathbb{Z}}^{2}\text{ and }F(x,y)\neq 0,\\\ \max\\{|x|,|y|\\}\geq
A,\end{array}\right\\}\Rightarrow\max\\{|x|,|y|\\}\leq\kappa\left|F(x,y)\right|^{\frac{1}{d-d_{0}}}.$

These two definitions are not independent since we have

######  Lemma 2.3.

If a family of binary forms is $(A,A_{1},d_{0},d_{1},\kappa)$–regular in the
sense of Definition 2.2 then it is also regular in the sense of Definition
2.1.

###### Proof.

Suppose that the family $\mathcal{F}$ satisfies the condition $(\mathrm{v})$
of Definition 2.2. Let $\epsilon>0$, let $N_{0}$ sufficiently large and let
$d_{2}>2/\epsilon$. We use $d_{0}$ and $d_{1}$ as in the Definition 2.2 and we
replace $d_{0}$ by $\max\\{d_{1},d_{0}+1\\}+d_{2}$ in the condition
$(\mathrm{ii})$ of the Definition 2.1. Let $N\in{\mathbb{Z}}$,
$d\in{\mathbb{Z}}$, $m\in{\mathbb{Z}}$, $(x,y)\in{\mathbb{Z}}^{2}$ and
$F\in\mathcal{F}_{d}$ such that

$N\geq N_{0},\quad d\geq\max\\{d_{1},d_{0}+1\\}+d_{2},\quad|m|\leq N,\quad
X\geq A\quad\hbox{and}\quad F(x,y)=m$

with $X:=\max\\{|x|,|y|\\}$. From condition $(\mathrm{v})$ in Definition 2.2,
we deduce

$A^{d-d_{0}}\leq X^{d-d_{0}}\leq\kappa^{d-d_{0}}|m|\leq\kappa^{d-d_{0}}N.$

From these inequalities we deduce on the one hand

$(d-d_{0})\log(A/\kappa)\leq\log N,$

which is

$d\leq d_{0}+\frac{\log N}{\log(A/\kappa)},$

and on the other hand

$X\leq\kappa N^{1/(d-d_{0})}\leq\kappa N^{1/d_{2}}.$

The condition $(\mathrm{iii})$ of the Definition 2.2 states that family
$\mathcal{F}$ contains at most $d^{A_{1}}$ forms with degree $d$. One deduces
that the number of $(d,x,y,F)$ (such that $F(x,y)=m$ with degree $F$ equal to
$d$) and also the number of $m$, are bounded by $O(N^{2/d_{2}}(\log
N)^{A_{1}+1})$. ∎

######  Example 2.4.

Let $(\ell_{d})_{d\geq 3}$ be a sequence of positive integers. Let
$\mathcal{F}$ be the family obtained by considering the sequence of binary
forms $F_{d}(X,Y)=(X-dY)^{2d}+\ell_{d}Y^{2d}$. We have the equalities

$F_{d}(d,1)=\ell_{d}\quad\hbox{and}\quad
F_{d}(d-1,1)=F_{d}(d+1,1)=\ell_{d}+1.$ (2.1)

We then check that this family is regular, in the meaning of Definition 2.1 if
and only if, when $N$ tends to infinity, we have

$\frac{1}{\log N}\log\sharp\Bigl{\\{}\\{\ell_{d}\mid d\geq
3\\}\cap[1,N]\Bigr{\\}}\to 0.$

Choosing $(\ell_{d})_{d\geq 3}$ to be the sequence
$(1,2,4,\;1,2,4,8,\;1,2,4,8,16,\;\dots)$ defined by the formula

$\ell_{d}=2^{j}\quad\hbox{ when }d=\frac{k(k+1)}{2}+j=1+2+\cdots+k+j,\quad
k\geq 2,\quad 0\leq j\leq k,$

we obtain an example of a regular family for which there exists an infinite
set of integers $m$ with infinitely many representations under the form
$m=F_{d}(x,y)$.

The family $\mathcal{F}$ is regular in the meaning of Definition 2.2 only if

$\ell_{d}\geq(d/\kappa)^{d-d_{0}}$ (2.2)

for all $d\geq\max\\{d_{1},d_{0}+1\\}$. This follows from (2.1). For instance
the condition (2.2) is not satisfied when the sequence $(\ell_{d})_{d\geq 2}$
is bounded.

We now turn our attention to the statement of [FW2, Theorem 1.11] when one
considers a family which satisfies the new notion of regularity. The
conclusion of Theorem 1.11 of our text [FW2] remains true when one replaces
the assumption that the family is $(A,A_{1},d_{0},d_{1},\kappa)$–regular by
the assumption that the family is regular in the meaning of Definition 2.1.
Namely:

######  Theorem 2.5.

Let $\mathcal{F}$ be a regular family of binary forms in the meaning of
Definition 2.1. Then for every $d\geq 3$ and every positive $\varepsilon$, the
quantity

$\displaystyle{\mathcal{R}}_{\geq
d}\left(\mathcal{F},N,A\right):=\sharp\,\bigl{\\{}m:0\leq|m|\leq N,\,\text{
there is }\,F\in\mathcal{F}\text{ with }\deg F\geq d$ $\displaystyle\text{ and
}(x,y)\in{\mathbb{Z}}^{2}\text{ with }\max\\{|x|,|y|\\}\geq A,\text{ such that
}F(x,y)=m\bigr{\\}}.$

satisfies

$\mathcal{R}_{\geq
d}(\mathcal{F},N,A)=\left(\sum_{F\in\mathcal{F}_{d}}A_{F}W_{F}\right)\cdot
N^{2/d}+O_{\mathcal{F},A,d,\varepsilon}\bigl{(}N^{\vartheta_{d}+\varepsilon}\bigr{)}+O_{\mathcal{F},A,d}\bigl{(}N^{2/d^{\dagger}}\bigr{)},$

uniformly as $N\to\infty$.

We do not need the assumption $d\geq d_{1}$ which occurred in [FW2,
Theorem1.11]. Notice that if a family does not satisfy the condition
$(\mathrm{ii})$ of the Definition 2.1, then it does not satisfy the conclusion
of Theorem 2.5 — the condition $(\mathrm{ii})$ in Definition 2.1 is
essentially optimal.

###### Proof of Theorem 2.5.

We use the notation

$\mathcal{Z}_{A}={\mathbb{Z}}^{2}\setminus\bigl{(}[-A,A]\times[-A,A]\bigr{)}$

introduced in [FW2]. The conditions $(\mathrm{iii})$ and $(\mathrm{v})$ of the
Definition 2.2 appear in [FW2] when considering (3.5) and (3.7) to show that
the cardinality of the set

$\left\\{(n,F,x,y):n>d^{\dagger}+d_{0},F\in\mathcal{F}_{n},(x,y)\in\mathcal{Z}_{A},|F(x,y)|\leq
B\right\\}$

is bounded by $o_{\mathcal{F}}\left(B^{2/{d^{\dagger}}}\right)$. Firstly we
remark that it suffices to bound the cardinality of the set

$\displaystyle\left\\{m:0\leq m\leq B,\right.$ $\displaystyle\hbox{ there
exists }(n,F,x,y),\;$
$\displaystyle\left.n>d^{\dagger}+d_{0},F\in\mathcal{F}_{n},(x,y)\in\mathcal{Z}_{A},|F(x,y)|=m\right\\}.$

The claimed bound immediately follows from the assumption $(\mathrm{ii})$ of
the Definition 2.1. ∎

The following lemma is easy. It will be used several times

######  Lemma 2.6.

Let $\theta>0$. We suppose that there exists $d_{0}\geq 3$ such that, for $m$
and $d$ in ${\mathbb{Z}}$ with $|m|\geq 2$ and $d\geq d_{0}$, the conditions

$d^{\prime}\geq d$, $(a,b)\in\mathcal{E}_{d^{\prime}}$, $\max\\{|x|,|y|\\}\geq
2$ and $m=ax^{d^{\prime}}+by^{d^{\prime}}$

imply the inequality

$X^{d^{\prime}}\leq|m|^{\theta}$

with $X:=\max\\{|x|,|y|\\}$. We also suppose that the condition (1.1) is
satisfied. Then
(a) For every $m\in{\mathbb{Z}}\setminus\\{-1,0,1\\}$ and every $d\geq 3$, the
set $\mathcal{G}_{\geq d}(m)$ is finite. In addition for every $d\geq d_{0}$
and every $\epsilon>0$, we have, as $|m|\to\infty$, the upper bound

$\sharp\mathcal{G}_{\geq d}(m)=O(|m|^{(\theta+\epsilon)/d}).$

(b) For every $d\geq d_{0}$ and every $\epsilon>0$, there exists $N_{0}$ such
that, for $N\geq N_{0}$, we have the inequality

$\sharp\mathcal{R}_{\geq d}(N)\leq N^{(2\theta+\epsilon)/d}.$

###### Proof.

Let $|m|\geq 2$, $d\geq d_{0}$ and $d^{\prime}\geq d$. The inequalities

$2^{d^{\prime}}\leq X^{d^{\prime}}\leq|m|^{\theta}$

imply

$d^{\prime}\leq\frac{\theta\log|m|}{\log 2}\quad\hbox{and }\quad
X\leq|m|^{\theta/d^{\prime}}.$

The cardinality of the set $\mathcal{G}_{\geq d}(m)$ is less than

$4|m|^{\theta/d}\sum_{d^{\prime}=d}^{\lfloor\frac{\theta\log|m|}{\log
2}\rfloor}\sharp\mathcal{E}_{d^{\prime}},$

since, when one unknown is fixed in the equation
$m=ax^{d^{\prime}}+by^{d^{\prime}}$, the other unknown takes two values at
most. The fact that $\mathcal{G}_{\geq d}(m)$ is a finite set for $d\geq 3$
and $|m|\geq 2$ follows from the fact that $\bigcup_{3\leq
d^{\prime}<d}\mathcal{F}_{d^{\prime}}$ is also finite. Thus the assertion (a)
is a consequence of (1.1). Finally the assertion (b) follows from

$\sharp\mathcal{R}_{\geq d}(N)\leq
4N^{2\theta/d}\sum_{d^{\prime}=d}^{\lfloor\frac{\theta\log N}{\log
2}\rfloor}\sharp\mathcal{E}_{d^{\prime}}.$ (2.3)

∎

###### Proof of Theorem 1.1.

The equality $ax^{d}+by^{d}=m$ with $a$ and $b>0$ and $d\geq 4$ even implies
the inequality $X^{d}\leq m$. Lemma 2.6 applied with $\theta=1$ proves the
part (a) of Theorem 1.1. We also check the condition $(\mathrm{ii})$ in the
Definition 2.1 of a regular family for the value $A=2$. To prove the assertion
(b) it remains to apply Theorem 2.5 since the item (i) of Definition 2.1 is
fulfilled by Corollary 3.2 below. ∎

##  3 Isomorphisms between binomial binary forms and their automorphisms

We recall the action of the group of matrices ${\rm GL}(2,{\mathbb{Q}})$ on
the set ${\rm Bin}(d,{\mathbb{Q}})$ of binary forms with degree $d$, with
rational coefficients and with non zero discriminant. If $F=F(X,Y)$ and
$\gamma=\begin{pmatrix}a_{1}&a_{2}\\\ a_{3}&a_{4}\end{pmatrix}$ respectively
belong to ${\rm Bin}(d,{\mathbb{Q}})$ and ${\rm GL}(2,{\mathbb{Q}})$, we
define

$(F\circ\gamma)(X,Y)=F(a_{1}X+a_{2}Y,a_{3}X+a_{4}Y).$

By definition, we say that the two forms $F$ and $G$ are isomorphic if and
only if there exists $\gamma\in{\rm GL}(2,{\mathbb{Q}})$ such that
$F\circ\gamma=G.$ The group of automorphisms of a form $F$ is

${\rm Aut}(F,{\mathbb{Q}})=\\{\gamma\in{\rm
GL}(2,{\mathbb{Q}}):F\circ\gamma=F\\}.$

######  Proposition 3.1.

Let $d\geq 3$ and $a$, $b$, $a^{\prime}$ and $b^{\prime}$ be integers
different from zero. Then the two binary forms $aX^{d}+bY^{d}$ and
$a^{\prime}X^{d}+b^{\prime}Y^{d}$ are isomorphic if and only if at least one
of the following two conditions hold

  1. 1.

the ratios $a/a^{\prime}$ and $b/b^{\prime}$ are both $d$–th powers of a
rational number,

  2. 2.

the ratios $a/b^{\prime}$ and $b/a^{\prime}$ are both $d$–th powers of a
rational number.

###### Proof.

The proof is an extension of the proof of [FW2, Lemma 1.14] which worked under
the restrictions that $d$ is an even integer and $a$, $b$, $a^{\prime}$ and
$b^{\prime}$ are all positive. We quickly give the necessary modifications to
obtain Proposition 3.1. Indeed the beginning of the proof of [FW2, Lemma 1.14]
does not require these restrictions. They are only used at the very last item
of the proof where we appeal to them to prove that if
$\gamma=\begin{pmatrix}a_{1}&a_{2}\\\ a_{2}&a_{4}\end{pmatrix}$ with
$a_{1}a_{2}a_{3}a_{4}\neq 0$, the equality

$F_{a,b,d}\circ\gamma=F_{a^{\prime},b^{\prime},d}$

cannot hold. Indeed, if it holds, a computation leads to the equalities

$\frac{a_{1}}{a_{3}}=-\frac{b}{a}\left(\frac{a_{4}}{a_{2}}\right)^{d-1},\
\left(\frac{a_{1}}{a_{3}}\right)^{2}=-\frac{b}{a}\left(\frac{a_{4}}{a_{2}}\right)^{d-2}.$

Dividing the second equality by the first one we obtain the equality
$a_{1}/a_{3}=a_{2}/a_{4}$. This is impossible since $\det\gamma\neq 0$. ∎

The following corollary is straightforward

######  Corollary 3.2.

Let $\mathcal{F}$ be a family of binomial forms $(F_{a,b,d})$ with $d\geq 3$,
$(a,b)\in\mathcal{E}_{d}$, where $\mathcal{E}_{d}$ satisfies the conditions
(C1) and (C2) of §1. Then $\mathcal{F}$ satisfies the item (i) of Definition
2.1 and the item (iv) of Definition 2.2.

We now recall the values of the constants $W_{F_{a,b,d}}$. More generally, for
any binary form $F$, the constant $W_{F}$ is a rational number only depending
on the group ${\rm Aut}(F,{\mathbb{Q}})$, more precisely on lattices defined
by some subgroups of ${\rm Aut}(F,{\mathbb{Q}})$. The constant $W_{F}$ has a
rather intricate definition but in the case of the binomial forms, the
corresponding group of automorphisms is rather simple (see [SX, Lemma 3.3]).
The following proposition is the first part of [SX, Corollary 1.3].

######  Proposition 3.3.

Let $F_{a,b,d}(X,Y)=aX^{d}+bY^{d}$ be a binary binomial form with $ab\not=0$
and with $d\geq 3$. Then we have

  * •

If $a/b$ is not a $d$–th power of a rational number, then

$W_{F_{a,b,d}}=\begin{cases}1&\text{if $d$ is odd},\\\ 1/4&\text{if $d$ is
even}.\end{cases}$

  * •

If $a/b$ is a $d$–th power of a rational number say $a/b=(A/B)^{d}$ then

$W_{F_{a,b,d}}=\begin{cases}1-1/(2|AB|)&\text{if $d$ is odd},\\\
(1-1/(2|AB|))/4&\text{if $d$ is even}.\end{cases}$

##  4 On the integers represented by binary binomial forms with large degree.

The following result gives an asymptotic upper bound for the number of
integers represented by binary forms with high degree and for the number of
representations of such integers.

######  Theorem 4.1.

Let $d_{0}\geq 3$ be an integer. Let $\lambda$ and $\mu$ be two real numbers
such that $\lambda>2$ and

$0<\mu<2^{-81}3^{-15}\frac{\lambda-2}{\lambda}\cdotp$ (4.1)

We suppose that

$\mathcal{E}_{d}\subset\left\\{(a,b)\in{\mathbb{Z}}\times{\mathbb{Z}}\,\;\mid\;ab\not=0,\;\max\\{|a|,|b|\\}\leq\exp(\mu
d/\log d)\right\\}$

for all $d\geq d_{0}$. Then
(a) For every $m\in{\mathbb{Z}}\setminus\\{-1,0,1\\}$ and every $d\geq 3$, the
set $\mathcal{G}_{\geq d}(m)$ is finite. Furthermore, for every $\epsilon>0$
and as $|m|\to\infty$, we have

$\sharp\mathcal{G}_{\geq d}(m)=O\left(|m|^{\epsilon+\lambda/(2d)}\right).$

(b) For every $d\geq 3$, there exists $N_{0}>0$ such that, for every $N\geq
N_{0}$, we have

$\sharp\mathcal{R}_{\geq d}(N)\leq N^{\lambda/d}.$

The set of $m$ that we are considering in the assertion (b) contains the set
of $m$ for which the hypotheses are satisfied with $(a,b)\in\mathcal{E}_{d}$.
By the result of [SX] (see [BDW] for the particular case of binary binomial
forms), each of these forms with degree $d$ contributes to this number of $m$
by $N^{2/d}$, up to some positive constant. The hypothesis $\lambda>2$ is thus
natural.

One cannot drop the condition $a\neq 0$. Indeed, if $a=0$, every $m\leq N$ is
represented by some form (take $y=1$, $b=m$ and $d$ sufficiently large).
Similarly, one cannot drop the condition $b\neq 0$.

One cannot forget the hypothesis $\max\\{|x|,|y|\\}\geq 2$: take $x=1$ and
$y=-1$, then every integer $m$ in the interval $1\leq m\leq N$ satisfies the
equality $m=a-b$ with $d$, $a$, $b$ satisfying the conditions of Theorem 4.1.

One cannot replace the condition $\max\\{|a|,|b|\\}\leq\exp(\mu d/\log d)$ by
$\max\\{|a|,|b|\\}\leq 2^{d}$, as it can be seen by the example $x=2$,
$y=a=1$, $b=m-2^{d}$, $m=1,\dots,2^{d}-1$. In §7 we will see to what extent
one can hope to weaken this hypothesis by assuming either Conjecture 1 of [L,
p. 212] or Conjecture $abc$. In this connection, in [FW2, Theorem 1.13], there
is no hypothesis concerning $\max\\{|a|,b|\\}$ when $(a,b)$ is in the set
$\mathcal{E}_{d}$: the only condition deals with the number of elements which
must be less than $d^{A_{1}}$ for [FW2, Theorem 1.13], and must satisfy
condition (1.1) for Theorem 1.1. The example of the family
$X^{d}+(d-2^{d})Y^{d}$ shows that such a result cannot be extended to the case
where the binary forms has real zeroes.

##  5 A diophantine result

The central tool in the proof of Theorem 4.1 is a lower bound coming from the
theory of linear forms in logarithms, more precisely [W, Corollary 9.22]. The
usual height of the rational number $p/q$, written under its irreducible form,
is defined by $\mathrm{H}(p/q)=\max\\{|p|,q\\}$ and its logarithmic height is

$\mathrm{h}(p/q)=\log\mathrm{H}(p/q)=\log\max\\{|p|,q\\}.$

######  Proposition 5.1.

Let $a_{1},a_{2}$ be rational numbers, $b_{1},b_{2}$ be positive integers,
$A_{1},A_{2},B$ be real positive numbers. We suppose for $j=1,2$ that

$B\geq\max\\{\mathrm{e},\;b_{1},\;b_{2}\\},\quad\log
A_{j}\geq\max\\{\mathrm{h}(a_{j}),\;1\\}.$

Then, if $a_{1}^{b_{1}}a_{2}^{b_{2}}\not=1$, we have the inequality

$|a_{1}^{b_{1}}a_{2}^{b_{2}}-1|\geq\exp\left\\{-C(\log B)(\log A_{1})(\log
A_{2})\right\\}$

with $C=2^{79}3^{15}$.

This lower bound follows from Corollary 9.22 in [W, p. 308] by taking

$D=1,\quad m=2,\quad\alpha_{1}=a_{1},\quad\alpha_{2}=a_{2}$

and the constant $C(m)$ defined in [W, p. 252].

######  Corollary 5.2.

Let $d$, $a$, $b$, $x$ and $y$ be rational integers. Let

$\mathcal{A}:=\max\\{|a|,|b|\\},\quad X:=\max\\{|x|,|y|\\}.$

We suppose $d\geq 2$, $\mathcal{A}\geq 2$, $X\geq 2$ and
$ax^{d}+by^{d}\not=0$. Then we have the lower bound

$|ax^{d}+by^{d}|\geq\max\\{|ax^{d}|,|by^{d}|\\}\exp\\{-4C(\log d)(\log
X)(\log\mathcal{A})\\}.$

The conclusion is obviously false when one of the parameters $d$, $X$,
$\mathcal{A}$ equals $1$.

###### Proof.

By symmetry one can suppose that $|ax^{d}|\leq|by^{d}|$. We use Proposition
5.1 with

$b_{1}=d,\quad b_{2}=1,\quad a_{1}=\frac{x}{y},\quad a_{2}=-\frac{a}{b},$
$B=\begin{cases}d&\hbox{if $d\geq 3$}\\\ \mathrm{e}&\hbox{if
$d=2$,}\end{cases}\quad A_{1}=\begin{cases}X&\hbox{if $X\geq 3$}\\\
\mathrm{e}&\hbox{if $X=2$,}\end{cases}\quad
A_{2}=\begin{cases}\mathcal{A}&\hbox{if $\mathcal{A}\geq 3$}\\\
\mathrm{e}&\hbox{if $\mathcal{A}=2$.}\end{cases}$

We conclude using the inequality $1/(\log 2)^{3}<4$. ∎

Corollary 5.2 implies the lower bound

$|ax^{d}+by^{d}|\geq X^{d}\exp\\{-4C(\log d)(\log X)(\log\mathcal{A})\\},$

that we write as

$|ax^{d}+by^{d}|\geq X^{d-4C(\log d)(\log\mathcal{A})}.$ (5.1)

##  6 Proof of Theorems 4.1 and 1.3

###### Proof of Theorem 4.1.

Let $\lambda^{\prime}$ in the interval $2<\lambda^{\prime}<\lambda$ such that

$\mu=\frac{\lambda^{\prime}-2}{4C\lambda^{\prime}},$

where $C$ is defined in Proposition 5.1. Let
$m=ax^{d^{\prime}}+by^{d^{\prime}}$ with $|m|\geq 2$, $d^{\prime}\geq d$,
$(a,b)\in\mathcal{E}_{d^{\prime}}$ and $X\geq 2$. When $ax^{d^{\prime}}$ and
$by^{d^{\prime}}$ have the same sign, we have $|m|\geq X^{d^{\prime}}$ and we
use Lemma 2.6 with $\theta=1$. When $ax^{d^{\prime}}$ and $by^{d^{\prime}}$
have opposite signs, in order to use Lemma 2.6, we can suppose that $m\geq 2$,
$a,x,y>0$ and $b<0$.

We are firstly interested in the pairs $(a,b)\in\bigcup_{d^{\prime}\geq
d}\mathcal{E}_{d^{\prime}}$ satisfying $\max\\{|a|,|b|\\}=1$. By our
hypotheses, we have $a=1$ and $b=-1$. There is no restriction to suppose that
$d\geq\lambda^{\prime}/(\lambda^{\prime}-2)$, since when $m\neq 0$ is given,
the equation $x^{d}-y^{d}=m$ has at most $(d-1)\cdot\sharp\\{k\mid
k|m\\}=O_{d}(|m|^{\epsilon})$ solutions. For $d^{\prime}\geq d$, we write

$m=x^{d^{\prime}}-y^{d^{\prime}}=(x-y)(x^{d^{\prime}-1}+x^{d^{\prime}-2}y+\cdots+xy^{d^{\prime}-2}+y^{d^{\prime}-1})>X^{d^{\prime}-1}\geq
X^{2d^{\prime}/\lambda^{\prime}}.$

Thus we can use Lemma 2.6 with $\theta=\lambda^{\prime}/2$.

We now consider the pairs $(a,b)\in\bigcup_{d^{\prime}\geq
d}\mathcal{E}_{d^{\prime}}$ such that $\mathcal{A}:=\max\\{|a|,|b|\\}$
satisfies $\mathcal{A}\geq 2$. Since we supposed that $\mathcal{A}\leq\exp(\mu
d^{\prime}/\log d^{\prime})$, we have

$\frac{d^{\prime}}{(\log
d^{\prime})(\log\mathcal{A})}\geq\frac{1}{\mu}=\frac{4C\lambda^{\prime}}{\lambda^{\prime}-2}\cdotp$

Let $X:=\max\\{|x|,|y|\\}$. We deduce from (5.1) the inequality

$X^{d^{\prime}-4C(\log d^{\prime})(\log\mathcal{A})}\leq m$

with

$d^{\prime}-4C(\log d^{\prime})(\log\mathcal{A})\geq
d^{\prime}(1-4C\mu)=\frac{2d^{\prime}}{\lambda^{\prime}},$

which allows to use Lemma 2.6 with the choice $\theta=\lambda^{\prime}/2$.

To conclude, we add the three values $m=0$ and $m=\pm 1$. ∎

###### Proof of Theorem 1.3.

It mimics the proof of Theorem 1.1 in Section 2: combining Corollary 3.2 and
Theorem 4.1 (b), one deduces that the family $\mathcal{F}$ satisfies the
definition 2.1 of a regular family. ∎

##  7 Conjectures

Let $X_{0}$ be an integer $\geq 2$. we introduce the following subset of
$\mathcal{R}_{\geq d}$:

$\displaystyle\mathcal{R}_{\geq d,X_{0}}=\Bigl{\\{}m\in{\mathbb{Z}}\;\mid$
$\displaystyle\hbox{ there exists }(d^{\prime},a,b,x,y)\;\hbox{ such that
}m=ax^{d^{\prime}}+by^{d^{\prime}}\;$ with $\displaystyle d^{\prime}\geq
d,\;(a,b)\in\mathcal{E}_{d^{\prime}},\;(x,y)\in{\mathbb{Z}}^{2}\;\hbox{ and
}\max\\{|x|,|y|\\}\geq X_{0}\Big{\\}},$

such that $\mathcal{R}_{\geq d}=\mathcal{R}_{\geq d,2}$. For $N$ a positive
integer we also denote

$\mathcal{R}_{\geq d,X_{0}}(N)=\mathcal{R}_{\geq d,X_{0}}\cap[-N,N].$

After the statement of Theorem 4.1 in Section 4, we gave the example of the
equation $2^{d}-b=m$ to show that one cannot replace the condition
$\max\\{|a|,|b|\\}\leq\exp(\mu d/\log d)$ by $\max\\{|a|,|b|\\}\leq 2^{d}$.
Introducing a parameter $X_{0}$ and assuming $\max\\{|x|,|y|\\}\geq X_{0}$,
one might expect a more general result to hold. It is interesting to notice
that a conjecture on lower bounds for linear forms in logarithms and the $abc$
Conjecture would produce very similar results.

###  7.1 Conjecture 1 of [L]

We state Conjecture 1 in [L, Introduction to Chapters X and XI, p. 212] as
follows.

######  Conjecture 7.1.

Let $\epsilon>0$. There exists a constant $C(\epsilon)>0$ only depending on
$\epsilon$ such that, if $a_{1},\dots,a_{n}$ are rational positive numbers and
if $b_{1},\dots,b_{n}$ are integers, by defining

$B_{j}=\max\\{|b_{j}|,1\\},\quad
A_{j}=\max\\{{\mathrm{e}}^{\mathrm{h}(a_{j})},1\\},\quad B=\max_{1\leq j\leq
n}B_{j}$

and by supposing that $b_{1}\log a_{1}+\cdots+b_{n}\log a_{n}\not=0$, we have
the lower bound

$|b_{1}\log a_{1}+\cdots+b_{n}\log a_{n}|>\frac{C(\epsilon)^{n}B}{(B_{1}\cdots
B_{n}A_{1}^{2}\cdots A_{n}^{2})^{1+\epsilon}}\cdotp$

Actually, we will only use a weak form of this conjecture : we will suppose
the existence of a number $\epsilon>0$ for which Conjecture 7.1 holds.

######  Theorem 7.2.

Let $\epsilon>0$ such that Conjecture 7.1 is verified for this value of
$\epsilon$. Let $\lambda>2$. Let $d_{0}$ be a sufficiently large integer and
let $X_{0}\geq 2$. We suppose

$\mathcal{E}_{d}\subset\left\\{(a,b)\in{\mathbb{Z}}\times{\mathbb{Z}}\,\;\mid\;ab\not=0,\;\max\\{|a|,|b|\\}\leq
X_{0}^{d/d_{0}}\right\\}$

for all $d\geq d_{0}$. Then for every $d\geq d_{0}$, we have

$\sharp\mathcal{R}_{\geq d,X_{0}}(N)\leq N^{\lambda/d}.$

###### Proof.

Conjecture 7.1 with $n=2$, $B=B_{1}=B_{2}=d$, $A_{1}=X$, $A_{2}=\mathcal{A}$
would allow to replace the conclusion of Corollary 5.2 by

$|ax^{d}+by^{d}|\geq X^{d}\exp\\{-c(\epsilon)-(1+\epsilon)\log
d-(2+\epsilon)\log X-(2+\epsilon)\log\mathcal{A}\\}$

with $c(\epsilon)>0$ depending only on $\epsilon$. Let $\lambda^{\prime}$ in
the interval $2<\lambda^{\prime}<\lambda$ and let $d_{0}$ sufficiently large
to ensure the inequality

$1-\frac{2(2+\epsilon)}{d_{0}}-\frac{c(\epsilon)+(1+\epsilon)\log
d_{0}}{d_{0}\log 2}>\frac{2}{\lambda^{\prime}}\cdotp$

For $d^{\prime}\geq d$ and $m=ax^{d^{\prime}}+by^{d^{\prime}}$, the resulting
upper bound

$X^{d^{\prime}}\leq|m|^{\lambda^{\prime}/2}$

allows to use Lemma 2.6. ∎

###  7.2 Conjecture $abc$

Let $R(m)$ be the radical of a positive integer $m$:

$R(m)=\prod_{p\;\mathrm{prime},\;p\mid m}p.$

The well known Conjecture $abc$ (see for example [W, §1.2]) asserts that for
all $\epsilon>0$, there exists a constant $\kappa(\epsilon)$ such that if $a$,
$b$, $c$ are coprime positive integers such that $a+b=c$, then the following
inequality holds

$c\leq\kappa(\epsilon)R(abc)^{1+\epsilon}.$

Like in Section 7.1, we will only assume the existence of a number
$\epsilon>0$ for which the property holds.

######  Lemma 7.3.

Let $\epsilon>0$ such that Conjecture $abc$ holds. Then under the hypotheses
of Corollary 5.2, we have the inequality

$X^{d-2-2\epsilon}\leq\kappa(\epsilon)\mathcal{A}^{1+2\epsilon}|m|^{1+\epsilon}.$

###### Proof.

Let $m=ax^{d}+by^{d}$. Without loss of generality, one can suppose
$|ax^{d}|\geq|by^{d}|$. If $|m|\geq|ax^{d}|$, the conclusion is obvious. Now
suppose $|ax^{d}|>|m|$. After a potential change of signs, we can also suppose
that $a>0$, $x>0$, $y>0$ and $b<0$.

Let $\Delta$ be the greatest common divisor of $ax^{d}$ and $|b|y^{d}$ and let
$P$ be the set of prime divisors of $\Delta$. For $p\in P$, we write

$\alpha_{p}=v_{p}(a),\;\beta_{p}=v_{p}(b),\;\xi_{p}=v_{p}(x),\;\eta_{p}=v_{p}(y),\;\delta_{p}=v_{p}(\Delta).$

Thus we have

$\delta_{p}=\min\\{\alpha_{p}+d\xi_{p},\;\beta_{p}+d\eta_{p}\\}.$

We also define

$a=\tilde{a}\prod_{p\in P}p^{\alpha_{p}},\;|b|=\tilde{b}\prod_{p\in
P}p^{\beta_{p}},\;x=\tilde{x}\prod_{p\in
P}p^{\xi_{p}},\;y=\tilde{y}\prod_{p\in P}p^{\eta_{p}},$

so that, for $p\in P$, we have the equalities
$v_{p}(\tilde{a})=v_{p}(\tilde{b})=v_{p}(\tilde{x})=v_{p}(\tilde{y})=0$.

Let $\tilde{m}=\Delta^{-1}m$; we have

$\tilde{m}=\Delta^{-1}ax^{d}-\Delta^{-1}|b|y^{d}\ $

with

$\Delta^{-1}ax^{d}=\tilde{a}\tilde{x}^{d}\prod_{p\in
P}p^{\alpha_{p}+d\xi_{p}-\delta_{p}}\quad\hbox{and
}\quad\quad\Delta^{-1}|b|y^{d}=\tilde{b}\tilde{y}^{d}\prod_{p\in
P}p^{\beta_{p}+d\eta_{p}-\delta_{p}}.$

The radical of $\Delta$ is $\tilde{\Delta}:=\prod_{p\in P}p$. The integers
$\Delta^{-1}ax^{d}$ and $\Delta^{-1}|b|y^{d}$ are coprime integers, the
radical of their product is less than
$\tilde{\Delta}\tilde{a}\tilde{b}\tilde{x}\tilde{y}$. We use Conjecture $abc$
for

$c=\Delta^{-1}ax^{d}=\tilde{m}+\Delta^{-1}|b|y^{d}.$

It gives the inequality

$\Delta^{-1}ax^{d}\leq\kappa(\epsilon)(\tilde{\Delta}\tilde{a}\tilde{b}\tilde{x}\tilde{y}\tilde{m})^{1+\epsilon},$

which is

$ax^{d}\leq\kappa(\epsilon)(a\,|b|\,xym)^{1+\epsilon}\Delta\prod_{p\in
P}p^{(1+\epsilon)(1-\alpha_{p}-\beta_{p}-\xi_{p}-\eta_{p}-\delta_{p})}.$

Since $\delta_{p}\geq 1$ we have $\alpha_{p}+\eta_{p}\geq 1$,
$\beta_{p}+\xi_{p}\geq 1$ and we obtain

$ax^{d}\leq\kappa(\epsilon)(a\,|b|\,xym)^{1+\epsilon},$

that we write as

$x^{d-1-\epsilon}\leq\kappa(\epsilon)a^{\epsilon}(|b|\,ym)^{1+\epsilon}.$
(7.1)

We now use the bound $|b|y^{d}\leq ax^{d}$ by writing the inequality

$y\leq(a/|b|)^{1/d}x.$ (7.2)

Then we have

$y^{1+\epsilon}\leq(a/|b|)^{(1+\epsilon)/d}x^{1+\epsilon}$

and (7.1) gives

$\displaystyle x^{d-2-2\epsilon}$
$\displaystyle\leq\kappa(\epsilon)a^{\epsilon}(a/|b|)^{(1+\epsilon)/d}(|b|\,m)^{1+\epsilon}$
$\displaystyle=\kappa(\epsilon)a^{(1/d)+\epsilon+(\epsilon/d)}|b|^{1-(1/d)+\epsilon-(\epsilon/d)}m^{1+\epsilon}.$
(7.3)

We again use (7.1) and (7.2) to obtain

$\displaystyle y^{d-2-2\epsilon}$
$\displaystyle\leq\kappa(\epsilon)(a/|b|)^{1-(2/d)-2(\epsilon/d)}a^{(1/d)+\epsilon+(\epsilon/d)}|b|^{(1-(1/d)+\epsilon-(\epsilon/d))}m^{1+\epsilon}$
$\displaystyle=\kappa(\epsilon)a^{(1-(1/d)+\epsilon-(\epsilon/d))}|b|^{(1/d)+\epsilon+(\epsilon/d)}m^{1+\epsilon}.$

Thanks to (7.2) we conclude the proof of Lemma 7.3. ∎

######  Theorem 7.4.

Let $\epsilon>0$ such that Conjecture $abc$ is verified for this value of
$\epsilon$. Let $\lambda>2+2\epsilon$, let $d_{0}$ be a sufficiently large
integer and let $X_{0}\geq 2$. We suppose

$\mathcal{E}_{d}\subset\left\\{(a,b)\in{\mathbb{Z}}\times{\mathbb{Z}}\,\;\mid\;ab\not=0,\;\max\\{|a|,|b|\\}\leq
X_{0}^{d/d_{0}}\right\\}$

for all $d\geq d_{0}$. Then for every $d\geq d_{0}$, we have

$\sharp\mathcal{R}_{\geq d,X_{0}}(N)\leq N^{\lambda/d}.$

###### Proof.

Let $\lambda^{\prime}$ in the interval
$2<\lambda^{\prime}<\lambda/(1+\epsilon)$ and let $d_{0}$ be a sufficiently
large integer such that the following inequality holds

$1-\frac{3+4\epsilon}{d_{0}}-\frac{\log\kappa(\epsilon)}{d_{0}\log
2}>\frac{2}{\lambda^{\prime}}\cdotp$

Let $d^{\prime}\geq d$ and $m=ax^{d^{\prime}}+by^{d^{\prime}}$. From Lemma 7.3
we deduce the bound $X^{d^{\prime}}\leq|m|^{\theta}$ with
$\theta=\lambda^{\prime}(1+\epsilon)/2$, which allows to apply Lemma 2.6. ∎

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