In this section we consider the case N = 2 in the important special case, when the variety X is defined by a single linear polynomial. In this case the intersection of X with the sets Γ,Γε, C(Γ, ε) is the set of solutions in Γ,Γε, C(Γ, ε), respectively, of the Diophantine equation induced by the polynomial which defines X. Since now X is defined by a linear polynomial, thus in principle we are analyzing the solutions of a generalizedS-unit equation coming from the set Γ,Γε, C(Γ, ε), respectively.
In the literature there are various effective results onS-unit equations in two unknowns.
In this Section we present effective results in a quantitative form for the more general equation
a1x1+a2x2 = 1 in (x1, x2)∈Γ, (2.7) where a1, a2 ∈ Q
∗ and Γ is an arbitrary finitely generated subgroup of rank > 0 of the multiplicative group (Q
∗)2 = Q
∗ ×Q
∗ endowed with coordinatewise multiplication (see Theorems 2.1 and 2.2 in the present Section). Such more general results can be used to improve upon existing effective bounds on the solutions of discriminant equations and certain decomposable form equations.
In fact, in the present section we present even more general effective results for equations of the shape (2.7) with solutions (x1, x2) from a larger group, from the division group Γ = {(x1, x2) ∈ (Q
∗)2| ∃k ∈ Z>0 : (xk1, xk2) ∈ Γ}, and even with solutions (x1, x2) ‘very close’ to Γ. These results give an effective upper bound for both the height of a solution (x1, x2) and the degree of the field Q(x1, x2); see Theorems 2.3 and 2.5 and Corollary 2.4.
In the proofs of these Theorems we utilize Theorem 2.1 (on (2.7) with solutions from Γ), as well as a result of Beukers and Zagier [14], which asserts that (2.7) has at most two solutions (x1, x2)∈(Q
∗)2 with very small height.
The hard core of the proofs of our results mentioned above is a new effective lower bound for |1−αξ|v, where α is a fixed element from a given algebraic number field K, v is a place of K, and the unknown ξ is taken from a given finitely generated subgroup of K∗ (see Theorem 2.6). This result is proved using linear forms in logarithms estimates.
Our Theorem 2.6 has a consequence (cf. Theorem 2.7) which is of a similar flavour as earlier results by Bombieri [15], Bombieri and Cohen [16], [17], and Bugeaud [26] (see also Bombieri and Gubler [18, Chap. 5.4]) but it gives in many cases a better estimate.
Consequently, Theorem 2.6 leads to an explicit upper bound for the heights of the solutions of (2.7) which is in many cases sharper than what is obtainable from the work of Bombieri et al.
We consider again the equation
a1x1+a2x2 = 1 in (x1, x2)∈Γ (2.7) where a1, a2 ∈ Q
∗ and where Γ is a finitely generated subgroup of (Q
∗)2 of rank > 0.
w1, . . . ,wr∈(OS∗)2, where OS∗ denotes the group of S-units in K. Put
Then our first result reads as follows:
Theorem 2.1 (B´erczes, Evertse and Gy˝ory [8]). For every solution (x1, x2) ∈ Γ of (2.7) we have
h(x1, x2)< AH. (2.9)
We shall deduce Theorem 2.1 from Theorem 2.2 below. Let G be a finitely generated multiplicative subgroup ofQ
∗ of rankt >0, andξ1, . . . , ξra system of generators ofG/Gtors. Let K be a number field containing G, and S a finite set of places of K containing the infinite places such that G⊆ O∗S. We consider the equation
a1x1+a2x2 = 1 in x1 ∈G, x2 ∈ OS∗, (2.10) where a1, a2 ∈Q
∗. Let d:= [K :Q]. Let s be the cardinality of S,P:= maxv∈SP(v) and put
QG :=h(ξ1)· · ·h(ξr).
Theorem 2.2 (B´erczes, Evertse and Gy˝ory [8]). Under the above assumptions and notation, every solution of (2.10) satisfies
As a special case of our result, it follows an effective upper bound for heights of the solutions ofS-unit equations. As we mentioned it previously, the first such effective bound was proved by Gy˝ory [38], and it was improved by several mathematicians. The presently known best bounds are due to Bugeaud and Gy˝ory [27], Bugeaud [26], and Gy˝ory and Yu [41]. From the proof of our result it is possible to deduce an upper bound, which is comparable with these best known results.
We now consider equations such as (2.7) but with solutions (x1, x2) from a larger set.
We keep the notation introduced before Theorem 2.1.
The division group of Γ is given by Γ := n
x∈(Q
∗)2 | ∃k ∈Z>0 with xk∈Γo . For any ε >0 define the “cylinder” and “truncated cone” around Γ by
Γε :=n respectively. The set Γε was introduced by Poonen [60] and the setC(Γ, ε) by Evertse [30]
(both in a much more general context).
We emphasize that points from Γ, Γε or C(Γ, ε) do not have their coordinates in a prescribed number field. So for effective results on Diophantine equations with solutions from Γ, Γε or C(Γ, ε), we need an effective upper bound not only for the height of each solution, but also for the degree of the field which it generates. We fix a1, a2 ∈ Q
∗ and define
K :=Q(Γ), K0 :=Q(a1, a2,Γ).
The quantitiesd, s,P, H andQΓ will have the same meaning as in Theorem 2.1 andAwill be the constant defined in (2.8). Further, we put
h0 := max{h(ξ1), . . . , h(ξr), h(η1), . . . , h(ηr)},
where wi = (ξi, ηi) for i= 1, . . . , r is the chosen system of generators for Γ/Γtors. Consider now the equation
a1x1 +a2x2 = 1 in (x1, x2)∈Γε. (2.15)
Theorem 2.3 (B´erczes, Evertse and Gy˝ory [8]). Suppose that (x1, x2) is a solution of (2.15) and that
ε <0.0225. (2.16)
Then we have
h(x1, x2)≤Ah(a1, a2) + 3rh0A (2.17) and
[K0(x1, x2) :K0]≤2. (2.18) The following consequence is immediate:
Corollary 2.4 (B´erczes, Evertse and Gy˝ory [8]). With the above notation and as-sumptions, let (x1, x2) be a solution of
a1x1+a2x2 = 1 in (x1, x2)∈Γ. (2.19) Then h(x1, x2)≤Ah(a1, a2) + 3rh0A and [K0(x1, x2) :K0]≤2.
Finally we consider the equation
a1x1+a2x2 = 1 in (x1, x2)∈C(Γ, ε). (2.20) Theorem 2.5 (B´erczes, Evertse and Gy˝ory [8]). Suppose that (x1, x2) is a solution of (2.20) and that
ε < 0.09
8Ah(a1, a2) + 20rh0A. (2.21) Then we have
h(x1, x2)≤3Ah(a1, a2) + 5rh0A (2.22) and
[K0(x1, x2) :K0]≤2. (2.23) The paper [12] of Beukers and Schlickewei gives explicit upper bounds for the number of solutions of (2.19), while from the result of Evertse, Schlickewei and Schmidt [34] one can deduce explicit upper bounds for multivariate generalizations of (2.19), (2.15), (2.20). The sets Γε and C(Γ, ε) have been defined in the much more general context of semi-abelian varieties (see [60], [63]). In [61], [62], R´emond proved quantitative analogues of the work of Evertse, Schlickewei and Schmidt [34] for subvarieties of abelian varieties and subvarieties of tori. We mention that the results of [12], [34], [60], [61], [62] and [63] are all ineffective, i.e. it does not even provide a theoretical algorithm to find the solutions. In contrast, our
above presented results are effective, i.e. they provide a theoretical algorithm to find all solutions of the equations in question, even though this algorithm is not practical.
The proof of the Theorems presented in this section is based on our two Diophantine approximation theorems presented below.
Let againK be an algebraic number field of degreed,MK the set of places on K, andG a finitely generated multiplicative subgroup of K∗ of rank t > 0. Further, let {ξ1, . . . , ξr} be a system of (not necessarily multiplicatively independent) generators of G such that ξ1, . . . , ξr are not roots of unity. Put again
QG :=h(ξ1)· · ·h(ξr).
Further, for any v ∈MK letP(v) be as in (2.4).
Theorem 2.2 and then subsequently Theorem 2.1 will be deduced from the following theorem.
It should be observed that c2(d, t) does not contain a tt factor.
The following theorem is in fact an immediate consequence of Theorem 2.6.
Theorem 2.7 (B´erczes, Evertse and Gy˝ory [8]). Let α ∈ K∗ with max(h(α),1) ≤
with the constant c2(r, d, t) specified in Theorem 2.6.
In particular, if r=t and{ξ1, . . . , ξt} is a basis ofG/Gtors, then (2.26) and (2.27) hold with c2(d, t) instead of c2(r, d, t).
We note that when applying Theorem 2.7 to equation (2.10), inequality (2.26) yields better bounds in Theorem 2.2 than (2.27).
The main tool in the proofs of Theorems 2.6 and 2.7 is the theory of logarithmic forms, i.e. Baker’s method, more precisely Theorem C in Section 4.1. Bombieri [15] and Bombieri and Cohen [16], [17] have developed another effective method in Diophantine approxima-tion, based on an extended version of the Thue-Siegel principle, the Dyson lemma and some geometry of numbers. Bugeaud [26], following their approach and combining it with esti-mates for linear forms in two and three logarithms, obtained sharper results than Bombieri and Cohen. Our above result is comparable with that of Bugeaud, and in most of the parameters (except in some cases the parameters H and QG) it is sharper than the result of Bugeaud.