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

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 $|m|\ge 2$ 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 $\ge d_0$ representing $m$ is bounded by $O(|m|^{(1/d_0)+\epsilon})$} uniformly for $\vert m \vert \geq 2$. 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.


Introduction
When d, a and b are rational integers different from 0, with d ≥ 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 where the exponent β < 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]For N positive integer, we denote The set E d must also satisfy the following hypotheses (C1) : For every (a, b) = (a ′ , b ′ ) ∈ E d , at least one of ratios a/a ′ and b/b ′ is not the d-th power of a rational number, (C2): For every (a, b) = (a ′ , b ′ ) ∈ E d , at least one of ratios a/b ′ and b/a ′ is not the d-th power of a rational number.
The exponent ϑ d < 2/d is defined in [FW2,(2.1)]: for 4 ≤ d ≤ 20, When the family F is given and when the integer d is ≥ 3, the integer d † is given by the formula Our first result is the following.

Our second result is
Theorem 1.3 (General case).There exists an absolute constant η > 0 with the following property.We suppose that there exists d 0 > 0 such that, for all d ≥ d 0 , we have the inequality Then (a) For all m ∈ Z \ {−1, 0, 1} and all d ≥ 3, the set G ≥d (m) is finite.Furthermore, for all d ≥ 3 and all ǫ > 0, we have, as |m| → ∞, (b) For all d ≥ 3 and all ǫ > 0, we have, as N → ∞, (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 η = 2 −81 3 −16 , corresponding to the right-hand side of (4.1) for λ = 3.
In both Theorems 1.1 and 1.3, the proof of the bound for ♯G ≥d (m) is based on the explicit estimate (2.3).The fact that G ≥d (m) is finite for all m ∈ {−1, 0, 1} is not a consequence of the bound for ♯R ≥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.

A new definition of a regular family
Definition 2.1.We will say that an infinite set F of binary forms with discriminants different from zero and with degrees ≥ 3 is regular if there exists a positive integer A satisfying the following two conditions (i) Two forms of the family F are GL(2, Q)-isomorphic if and only if they are equal, (ii) For all ǫ > 0, there exist two positive integers N 0 = N 0 (ǫ) and d 0 = d 0 (ǫ) such that, for all N ≥ N 0 , the number of integers m in the interval [−N, N] for which there exists d ∈ Z, (x, y) ∈ Z 2 and F ∈ F d satisfying Of course the symbol F d above, denotes the subset of 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 (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 , κ)-regular family as it is introduced in [FW2].
Definition 2.2.Let A, A 1 , d 0 , d 1 be integers and let κ be a real number such that Let F be a set of binary forms with integral coefficients and with discriminants different from zero.We say that F is (A, A 1 , d 0 , d 1 , κ)-regular if it satisfies the following conditions: (i) The set F is infinite, (ii) All the forms of F have their degrees ≥ 3, (iii) For all d ≥ 3, we have the inequality (iv) Two forms of F are isomorphic if and only if they are equal, (v) For any d ≥ max{d 1 , d 0 + 1}, the following holds 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 , κ)-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 F satisfies the condition (v) of Definition 2.2.Let ǫ > 0, let N 0 sufficiently large and let d 2 > 2/ǫ.We use d 0 and d 1 as in the Definition 2.2 and we replace d 0 by max{d with X := max{|x|, |y|}.From condition (v) in Definition 2.2, we deduce From these inequalities we deduce on the one hand and on the other hand The condition (iii) of the Definition 2.2 states that family 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 (ℓ d ) d≥3 be a sequence of positive integers.Let F be the family obtained by considering the sequence of binary forms . We have the equalities 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 Choosing (ℓ d ) d≥3 to be the sequence (1, 2, 4, 1, 2, 4, 8, 1, 2, 4, 8, 16, . . . ) defined by the formula 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 F is regular in the meaning of Definition 2.2 only if for all d ≥ max{d 1 , d 0 + 1}.This follows from (2.1).For instance the condition (2.2) is not satisfied when the sequence (ℓ d ) d≥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 , κ)-regular by the assumption that the family is regular in the meaning of Definition 2.1.Namely: Theorem 2.5.Let F be a regular family of binary forms in the meaning of Definition 2.1.Then for every d ≥ 3 and every positive ε, the quantity We do not need the assumption d ≥ d 1 which occurred in [FW2,.Notice that if a family does not satisfy the condition (ii) of the Definition 2.1, then it does not satisfy the conclusion of Theorem 2.5 -the condition (ii) in Definition 2.1 is essentially optimal.
Proof of Theorem 2.5.We use the notation introduced in [FW2].The conditions (iii) and (v) of the Definition 2.2 appear in [FW2] when considering (3.5) and (3.7) to show that the cardinality of the set Firstly we remark that it suffices to bound the cardinality of the set The claimed bound immediately follows from the assumption (ii) of the Definition 2.1.
The following lemma is easy.It will be used several times Lemma 2.6.Let θ > 0. We suppose that there exists d 0 ≥ 3 such that, for m and d in Z with |m| ≥ 2 and d ≥ d 0 , the conditions with X := max{|x|, |y|}.We also suppose that the condition (1.1) is satisfied.Then (a) For every m ∈ Z \ {−1, 0, 1} and every d ≥ 3, the set G ≥d (m) is finite.
(b) For every d ≥ d 0 and every ǫ > 0, there exists N 0 such that, for N ≥ N 0 , we have the inequality The cardinality of the set G ≥d (m) is less than since, when one unknown is fixed in the equation m = ax d ′ + by d ′ , the other unknown takes two values at most.The fact that G ≥d (m) is a finite set for d ≥ 3 and |m| ≥ 2 follows from the fact that 3≤d ′ <d F d ′ is also finite.Thus the assertion (a) is a consequence of (1.1).Finally the assertion (b) follows from

Isomorphisms between binomial binary forms and their automorphisms
We recall the action of the group of matrices GL(2, Q) on the set Bin(d, Q) of binary forms with degree d, with rational coefficients and with non zero discriminant.If F = F (X, Y ) and γ = a 1 a 2 a 3 a 4 respectively belong to Bin(d, Q) and GL(2, Q), we define By definition, we say that the two forms F and G are isomorphic if and only if there exists 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 ′ and b ′ 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 γ = a 1 a 2 a 2 a 4 with a 1 a 2 a 3 a 4 = 0, the equality cannot hold.Indeed, if it holds, a computation leads to the equalities .
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 γ = 0.
The following corollary is straightforward 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 Aut(F, Q), more precisely on lattices defined by some subgroups of Aut(F, 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].
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.
(b) For every d ≥ 3, there exists N 0 > 0 such that, for every N ≥ N 0 , we have 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) ∈ 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 λ > 2 is thus natural.
One cannot drop the condition a = 0. Indeed, if a = 0, every m ≤ N is represented by some form (take y = 1, b = m and d sufficiently large).Similarly, one cannot drop the condition b = 0.
One cannot forget the hypothesis max{|x|, |y|} ≥ 2: take x = 1 and y = −1, then every integer m in the interval 1 ≤ m ≤ 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|} ≤ exp(µd/ log d) by max{|a|, |b|} ≤ 2 d , as it can be seen by the example 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 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.

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 H(p/q) = max{|p|, q} and its logarithmic height is h(p/q) = log 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 Then, if a b 1 1 a b 2 2 = 1, we have the inequality This lower bound follows from Corollary 9.22 in [W, p. 308] by taking and the constant C(m) defined in [W, p. 252].
Corollary 5.2.Let d, a, b, x and y be rational integers.Let We suppose d ≥ 2, A ≥ 2, X ≥ 2 and ax d + by d = 0. Then we have the lower bound The conclusion is obviously false when one of the parameters d, X, A equals 1.
Corollary 5.2 implies the lower bound that we write as Thus we can use Lemma 2.6 with θ = λ ′ /2.
We now consider the pairs (a, b) ∈ d ′ ≥d E d ′ such that A := max{|a|, |b|} satisfies A ≥ 2. Since we supposed that A ≤ exp(µd ′ / log d ′ ), we have and by supposing that b 1 log a 1 + • • • + b n log a n = 0, we have the lower bound • Actually, we will only use a weak form of this conjecture : we will suppose the existence of a number ǫ > 0 for which Conjecture 7.1 holds.
Theorem 7.2.Let ǫ > 0 such that Conjecture 7.1 is verified for this value of ǫ.Let λ > 2. Let d 0 be a sufficiently large integer and let X 0 ≥ 2. We suppose with c(ǫ) > 0 depending only on ǫ.Let λ ′ in the interval 2 < λ ′ < λ and let d 0 sufficiently large to ensure the inequality allows to use Lemma 2.6.

Conjecture abc
Let R(m) be the radical of a positive integer m: R(m) = p prime, p|m p.
The well known Conjecture abc (see for example [W, §1.2]) asserts that for all ǫ > 0, there exists a constant κ(ǫ) such that if a, b, c are coprime positive integers such that a + b = c, then the following inequality holds c ≤ κ(ǫ)R(abc) 1+ǫ .
Like in Section 7.1, we will only assume the existence of a number ǫ > 0 for which the property holds.Let ∆ be the greatest common divisor of ax d and |b|y d and let P be the set of prime divisors of ∆.For p ∈ P , we write Thus we have δ p = min{α p + dξ p , β p + dη p }.
We also define Thanks to (7.3) we conclude the proof of Lemma 7.3.
Theorem 7.4.Let ǫ > 0 such that Conjecture abc is verified for this value of ǫ.Let λ > 2 + 2ǫ, let d 0 be a sufficiently large integer and let X 0 ≥ 2.
We suppose 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 ≥ 3, let E d be a finite subset of Z \ {0} × Z \ {0} and let F d be the set of binary binomial forms F a,b,d (X, Y ) with (a, b) ∈ E d .We are interested in the representation of integers m ∈ Z by some form of the family F = d≥3 F d .For d ≥ 3 and m in Z, we introduce the two sets G ≥d (m) = (d ′ , a, b, x, y) | m = ax d ′ + by d ′ with d ′ ≥ d, (a, b) ∈ E d ′ , (x, y) ∈ Z 2 and max{|x|, |y|} ≥ 2 and R ≥d = {m ∈ Z | G ≥d (m) = ∅} .

Theorem 1. 1 (
Positive definite case).Let E d ⊂ Z >0 × Z >0 satisfying (C1) and (C2) above and the equality E d = ∅ for odd d.Furthermore, we suppose that 1 d log(♯E d + 1) → 0 as d → ∞. (1.1) Then (a) For all m ∈ Z \ {0, 1} and all d ≥ 4, the set G ≥d (m) is finite.Furthermore, for all d ≥ 4 and all ǫ > 0, we have, as |m| → ∞, ♯G ≥d (m) = O |m| (1/d)+ǫ .(b) For all d ≥ 4 and all ǫ > 0, we have, as N → ∞, d + O d,ǫ N max{ϑ d +ǫ,2/d † } .(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 = |ax d +by d |≤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 Γ-function, of the fundamental area A F a,b,d in [SX, Corollary 1.3].The hypothesis ♯E d ≤ d A 1 in [FW2, Theorem 1.13], which implies the condition (iii) in Definition 2.2 below of a (A, A 1 , d 0 , d 1 , κ)-regular family, is replaced here by (1.1) which cannot be omitted: for d ≥ 3 and N = 2 2d +1, each of 2 d integers of the form a2 d + 1, a = 1, 2, 3, . . ., 2 d is represented by one of the form aX d + Y d with the choice x = 2, y = 1.

. 3 )
Proof of Theorem 1.1.The equality ax d + by d = m with a and b > 0 and d ≥ 4 even implies the inequality X d ≤ m.Lemma 2.6 applied with θ = 1 proves the part (a) of Theorem 1.1.We also check the condition (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.

Proposition 3. 1 .
Let d ≥ 3 and a, b, a ′ and b ′ be integers different from zero.Then the two binary forms aX d + bY d and a ′ X d + b ′ Y d are isomorphic if and only if at least one of the following two conditions hold 1. the ratios a/a ′ and b/b ′ are both d-th powers of a rational number, 2. the ratios a/b ′ and b/a ′ are both d-th powers of a rational number.

Corollary 3. 2 .
Let F be a family of binomial forms (F a,b,d ) with d ≥ 3, (a, b) ∈ E d , where E d satisfies the conditions (C1) and (C2) of §1.Then F satisfies the item (i) of Definition 2.1 and the item (iv) of Definition 2.2.
Proposition 3.3.Let F a,b,d (X, Y ) = aX d + bY d be a binary binomial form with ab = 0 and with d ≥ 3. Then we have • If a/b is not a d-th power of a rational number, then