LetA:=Z[z1, . . . , zr]⊃Zbe again a finitely generated domain. LetK denote the quotient field of the domain A, and denote by K∗ the multiplicative group of non-zero elements of K. Denote by K the algebraic closure of K and byK∗ its unit group. Let Γ be a finitely generated subgroup of K∗. Let F(X, Y) ∈ A[X, Y] be a polynomial. In 1960 Lang [44]
proved that the equation
F(x, y) = 0 in x, y ∈Γ (3.18)
(which is even more general than (3.11)) has only finitely many solutions, provided F is not divisible by any polynomial of the form
XmYn−α or Xm−αYn (3.19)
where m, n are non-negative integers, not both zero, and any α ∈ Γ. Lang’s proof of this result is ineffective. The first effective versions of this result of Lang have been proved by Bombieri and Gubler [18, p. 147, Theorem 5.4.5] for the number field case (see B´erczes, Evertse Gy˝ory and Pontreau [11] or Theorem 2.8 of the present dissertation for an explicit version) and by B´erczes [7] in its full generality, over finitely generated domains (this is Theorem 3.5 of the present dissertation). We mention that the effective results are proved under a slightly stronger condition than (3.19), namely in (3.19)α∈K is assumed instead of α∈Γ.
Denote by Γ the division group of Γ, i.e. the group defined by Γ := n
x∈K∗ | ∃ m∈N, xm ∈Γo .
Lang also conjectured ([46], [47], see also [48]) that the above equation has finitely many solutions in x, y ∈ Γ under the same condition (3.19) but with α ∈ Γ. Liardet [51], [52]
proved this conjecture of Lang. However, this famous result of Liardet is also ineffective.
An effective version of Liardet’s Theorem in the number field case is due to B´erczes, Evertse, Gy˝ory and Pontreau [11] (see Theorem 2.9 in this dissertation), however, in the general case no effective result had been proved.
In the present section we make effective the above-mentioned finiteness theorem of Liardet in the general case. Our result is not only effective, but also quantitative in the sense that an upper bound for the sizes of the solutions x, y ∈ Γ is provided. The presented result is a common generalization of the results of Bombieri and Gubler [18, p.
147, Theorem 5.4.5], B´erczes, Evertse, Gy˝ory and Pontreau [11] (see Theorem 2.9 in this
dissertation) and that of B´erczes [7] (see Theorem 3.5 in this dissertation). Further, our result is also a generalization of the result of B´erczes, Evertse and Gy˝ory [8] (see Theorem 2.4 in this dissertation) and of Evertse and Gy˝ory [32] on unit equations. The main tool of the proof is an effective specialization method introduced by Gy˝ory in the 1980’s (see [39], [40]), and improved by Evertse and Gy˝ory [32] in 2013. The main difficulty of the proof is that on one hand we have to bound also the degrees over K of the solutions from Γ, on the other hand we do not have any convenient representation for the elements of Γ.
We also mention thatthis is the first effective result for Diophantine equations considered over the division group of an arbitrary finitely generated group.
Let A := Z[z1, . . . , zr] be a finitely generated domain over Z, and let K denote its quotient field.
Let γ1, . . . , γs ∈K∗ be arbitrary non-zero elements ofK given by corresponding repre-sentation pairs (g1, h1), . . . ,(gs, hs). Define the finitely generated group
Γ :=
γ1l1. . . γsls | l1, . . . , ls∈Z , (3.20) and its division group
Γ :=
δ ∈K | ∃m ∈Z>0 : δm ∈Γ . (3.21)
Let I ⊂ Z2≥0 be a non-empty set, and let F(X, Y) = P
(i,j)∈IaijXiYj ∈ A[X, Y] be a polynomial which fulfils the following condition:
F is not divisible by any non-constant polynomial of the form
XmYn−α or Xm−αYn,where m, n∈Z≥0 and α∈K∗. (3.22) We mention that this condition is effectively decidable, as is explained in Section 8. Let N := degF denote the total degree of F, and assume that F is given by specifying a representative ˜aij of its coefficientaij for each (i, j)∈I.
Further, we assume that
( degf1, . . . ,degft,degg1, . . . ,deggs,degh1, . . . ,deghs,deg ˜aij ≤d
h(f1), . . . , h(ft), h(g1), . . . , h(gs), h(h1), . . . , h(hs), h(˜aij)≤h, (3.23) where (i, j)∈I and d, h are real numbers with d >1 and h >1.
Theorem 3.6 (B´erczes [6]). LetA be a finitely generated domain as above,Γ the above-defined division group and F(X, Y) ∈ A[X, Y] a polynomial which fulfils the condition (3.22). Define the set
C :={(x, y)∈(Γ)2|F(x, y) = 0}. (3.24)
(i) Then there exists a positive integer m with m ≤exp
N6(2d)exp{C3(r+s)}(h+ 1)4s (3.25)
with C3 an effectively computable absolute constant such that
xm ∈Γ and ym∈Γ, for every (x, y)∈ C.
(ii) More precisely, there exists an effectively computable absolute constant C4, such that for all (x, y)∈ C there are integers t1,x, . . . , ts,x, t1,y, . . . , ts,y with
ti,x, ti,y ≤exp exp
N12(2d)exp{C4(r+s)}(h+ 1)8s
(3.26) for i= 1, . . . , s, such that
xm =γ1t1,x. . . γsts,x, ym =γ1t1,y. . . γsts,y. (3.27)
Part II
Proofs
Chapter 4
Proof of the results from Section 2.2
4.1 Proof of Theorems 2.6 and 2.7
We need several auxiliary results.
Keeping the notation of Section 2.2, letK be an algebraic number field of degreedand assume that it is embedded in C. Let
Λ =αb11· · ·αbnn−1, (4.1) where α1, . . . , αn are n (≥2) non-zero elements of K, and b1, . . . , bn are rational integers, not all zero. Put
B∗ = max{|b1|, . . . ,|bn|}.
LetA1, . . . , An be reals with
Ai ≥max{dh(αi), π} (i= 1, . . . , n). (4.2) Theorem A. (Matveev [56]) Let n≥2. Suppose that Λ6= 0, bn=±1, and letB be a real number with
B ≥max
B∗,2emax nπ
√2, A1, . . . , An−1
An
. (4.3)
Then we have
log|Λ|>−c1(n, d)A1· · ·Anlog(B/(√
2An)), (4.4)
where
c1(n, d) = minn
1.451(30√
2)n+4(n+ 1)5.5, π26.5n+27o
d2log(ed).
Proof. This is a consequence of Corollary 2.3 of Matveev [56]; see Proposition 4 in Gy˝ory
and Yu [41]. 2
Let B and Bn be real numbers satisfying
B ≥max{|b1|, . . . ,|bn|}, B ≥Bn ≥ |bn|. (4.5) Denote byp a prime ideal of the ring of integers OK and letep and fp be the ramification index and the residue class degree ofp, respectively. ThusN(p) =pfp, wherepis the prime number below p.
Theorem B. (Yu [78]) Let n≥2. Assume that ordpbn ≤ordpbi for i= 1, . . . , n, and set h0i = max{h(αi),1/(16e2d2)}, i= 1, . . . , n.
If Λ 6= 0, then for any real δ with 0< δ ≤1/2 we have ordpΛ≤c2(n, d)enp N(p)
(logN(p))2·
·max
h01· · ·h0nlog(M δ−1), δB Bnc3(n, d)
,
(4.6)
where
c2(n, d) =(16ed)2(n+1)n3/2log(2nd) log(2d), c3(n, d) =(2d)2n+1log(2d) log3(3d),
and
M =Bnc4(n, d)N(p)n+1h01. . . h0n−1, with
c4(n, d) = 2e(n+1)(6n+5)
d3nlog(2d).
Proof. This is the second consequence of the Main Theorem in Yu [78]. 2 The following theorem is a consequence of Theorems A and B.
Theorem C. Let n≥2 and v ∈MK. Suppose that in (4.1) we have Λ6= 0, bn=±1 and that α1, . . . , αn−1 are not roots of unity. Let
Qα :=h(α1)· · ·h(αn−1), H := max(h(αn),1).
If
B ≥max(|b1|, . . . ,|bn−1|,2e(3d)2nQαH), (4.7) then
log|Λ|v >−c5(n, d) P(v)
logP(v)QαHlog∗
BP(v) H
, (4.8)
where P(v) is defined in (2.4) and
c5(n, d) =λ(16ed)3n+2(log∗d)2, with λ= 1 or 12 according as n ≥3 or n = 2.
To deduce Theorem C from Theorems A and B, we need the following.
Lemma 4.1. (Voutier [76]) Suppose that α is a non-zero algebraic number of degree d which is not a root of unity. Then
dh(α)≥
log 2 if d = 1, 2/(log 3d)3 if d ≥2.
(4.9) Proof. For d ≥ 2 this is due to Voutier [76]. He showed also that for d ≥ 2 this lower
bound may be replaced by (1/4)(log logd/logd)3. 2
Proof of Theorem C. First assume that v is infinite. We apply Theorem A with Ai = max{dh(αi), π} for i= 1, . . . , n. Then using (4.9), it is easy to see that
A1· · ·An≤(2.52d)2nQαH.
Further, we have √
2An> H/P(v) and 2emax
nπ
√2, A1, . . . , An−1
An≤2e(3d)2nQαH.
Now (4.7) implies (4.3), and (4.8) follows from the inequality (4.4) of Theorem A.
Next assume thatv is finite. Keeping the notation of Theorem B and using again (4.9), we infer that
h0i =h(αi) for i= 1, . . . , n−1 h0n=h(αn).
Hence h0n =H if h(αn)≥ 1 and H = 1 otherwise. Choosing δ =h01· · ·h0n/B and Bn = 1 in Theorem B, (4.7) implies thatδ ≤ 12. Using the fact that|Λ|v =N(p)−ordpΛ, after some
computation (4.8) follows from (4.6) of Theorem B. 2
Theorem 2.6 will be proved by combining Theorem C with the following result from the geometry of numbers. Lett be a positive integer. A convex distance function on Rt is a functionf :=Rt →R≥0 such that
f(x+y)≤f(x) +f(y) for x,y∈Rt, f(λx) = |λ|f(x) for x∈Rt, λ∈R, f(x) = 0 ⇐⇒ x=0.
Lemma 4.2. Let f be a convex distance function on Rt. Let {a1, . . . ,at} be any basis of Zt for which the product f(a1)· · ·f(at) is minimal. Let x ∈ Zt and suppose that x = b1a1+· · ·+btat with b1, . . . , bt ∈Z. Then
max(|b1|f(a1), . . . ,|bt|f(at))≤c6(t)f(x), (4.10) where c6(t) = t2t.
Remark. Schlickewei [66] proved that there exists a basis {a1, . . . ,at} of Zt satisfying (4.10) with 4t instead of c6(t), but it is not clear whether for this basis, the product f(a1)· · ·f(at) is minimal. In our proof of Theorem 2.6, the minimality off(a1)· · ·f(at) is crucial, while an improvement of c6(t) would have only little influence on the final result.
Proof. LetC ={x∈Rt :f(x)≤1}. This is a compact, convex body which is symmetric around0. Let λ1, . . . , λt denote the successive minima of C with respect to the latticeZt. Since λ1 ≤ · · · ≤ λt, it follows from a result of Mahler (see e.g. Cassels [28], pp. 135-136, Lemma 8) that there exists a basis y1, . . . ,yt of Zt such that f(yi) ≤ max(1, i/2)λi. Together with Minkowski’s theorem on successive minima, this gives
f(a1)· · ·f(at)≤f(y1)· · ·f(yt)≤2t!·Vol(C)−1, (4.11) where Vol(C) denotes the volume of C.
By Jordan’s theorem or John’s Lemma (see e.g. Schmidt [68], pp. 87–89) there is a t-dimensional ellipsoid E in Rt such that E ⊆ C ⊆ (√
t)E. Further, there is a t×t real non-singular matrix A such that E = {x ∈ Rt : kAxk ≤ 1}, where k · k denotes the Euclidean norm. Thus
√1
tkAxk ≤f(x)≤ kAxk for x∈Rt. (4.12) Consequently,
V(t)|det(A)|−1 ≤Vol(C)≤tt/2V(t)|det(A)|−1, (4.13) where V(t) denotes the volume of the t-dimensional unit ball.
Now let x=b1a1 +· · ·+btat with b1, . . . , bt∈ Z. Then Ax=b1(Aa1) +· · ·+bt(Aat).
Let B be the matrix with columns Aa1, . . . , Aat. Since |det(a1, . . . ,at)| = 1, we have
|det(B)| =|det(A)|. By this fact, Cramer’s rule and Hadamard’s inequality, we have for i= 1, . . . , t,
|bi|=|det(Aa1, . . . , Aai−1, Ax, Aai+1, . . . Aat)|
|det(B)|
≤ kAa1k · · · kAai−1k · kAxk · kAai+1k · · · kAatk
|det(A)|.
Together with (4.12), (4.11) and (4.13), this implies
|bi|f(ai)≤t(t−1)/2 f(a1)· · ·f(at)/|det(A)|
f(x)
≤t(t−1)/2 ·2t!V(t)−1f(x) for i= 1, . . . , t.
By inserting V(t) = πt/2/(t/2)! ift is even andV(t) =π(t−1)/2 1
2 · 32· · ·2t
if t is odd, we
get the bound in (4.10). 2
Lemma 4.3. LetGbe a finitely generated multiplicative subgroup ofK∗ of rank t >0. Let δ1, . . . , δt∈G be multiplicatively independent such that h(δ1)≤ · · · ≤h(δt). Then G/Gtors has a basis {γ1, . . . , γt} such that
h(γi)≤max(1, i/2)h(δi) for i= 1, . . . , t. (4.14) Proof. Let {ρ1, . . . , ρt}be a basis for G/Gtors. Then we can write
δi =ζiρb1i1· · ·ρbtit, i= 1, . . . , t, where ζi ∈Gtors, and
b1 = (b11, . . . , b1t), . . . ,bt= (bt1, . . . , btt) are linearly independent vectors in Zt.
Let S ⊂ MK be minimal such that S contains all infinite places and G ⊆ OS∗. We define
f(x) := 1 2
X
v∈S
|x1log|ρ1|v +· · ·+xtlog|ρt|v|,
wherex= (x1, . . . , xt)∈Rt. This is a convex distance function. Further, by (2.6) we have f(bi) =h(δi) for i= 1, . . . , t. (4.15) Using again Mahler’s result mentioned above, we infer that there is a basisai = (ai1, . . . , ait) (i= 1, . . . , t) of Zt for which
f(ai)≤max(1, i/2)f(bi) for i= 1, . . . , t. (4.16) Putting γi = ρa1i1· · ·ρatit for i = 1, . . . , t, we infer that {γ1, . . . , γt} is a basis for G/Gtors ,
which in view of (4.15), (4.16) satisfies (4.14). 2
We first prove Theorem 2.6 and then Theorem 2.7.
Proof of Theorem 2.6. SinceGhas rankt >0, there aret multiplicatively independent elements among the generators ξ1, . . . , ξr, say ξ1, . . . , ξt. Then by Lemma 4.1
h(ξ1)· · ·h(ξt)≤c7(d)r−tQG, (4.17) where c7(d) = d2(log 3d)3 if d ≥ 2 and c7(d) = (log 2)−1 if d = 1. Let δ1, . . . , δt be multiplicatively independent elements of G such that h(δ1)· · ·h(δt) is minimal. Then
h(δ1)· · ·h(δt)≤h(ξ1)· · ·h(ξt). (4.18) Further, by Lemma 4.3, G/Gtors has a basis {γ1, . . . , γt} such that
h(γ1)· · ·h(γt)≤c8(t)h(δ1)· · ·h(δt), (4.19) withc8(t) :=t!/2t−1. We may assume that{γ1, . . . , γt}is such a basis ofG/Gtors for which h(γ1)· · ·h(γt) is minimal.
For ξ∈G, we can write
ξ=ζγ1b1· · ·γtbt, (4.20) where ζ ∈ Gtors and b = (b1, . . . , bt) ∈ Zt. As in the proof of Lemma 4.3, consider the following convex distance function on Rt:
f(x) := 1 2
X
v∈S
|x1log|γ1|v +· · ·+xtlog|γt|v|,
where x = (x1, . . . , xt) ∈ Rt and S is the same as in the proof of Lemma 4.3. Then f(b) = h(ξ). Consider the standard basis a1 = (1,0, . . . ,0), a2 = (0,1,0, . . . ,0), . . ., at= (0, . . . ,0,1) in Zt. Then
f(ai) = h(ξi) for i= 1, . . . , t, and f(a1)· · ·f(at) is minimal among the bases ofZt.
We can now apply Lemma 4.2 to this basis a1, . . . ,at, and infer that
|bi|h(γi) =|bi|f(ai)≤c6(t)f(b) =c6(t)h(ξ), i= 1, . . . , t.
Together with Lemma 4.1 this gives
max(|b1|, . . . ,|bt|)≤c6(t)c7(d)h(ξ). (4.21) We apply now Theorem C with v ∈S and with
Λ = 1−αξ = 1−α0γ1b1· · ·γtbt,
where α0 =ζα. Let Qγ :=h(γ1)· · ·h(γt). First assume that
Together with (4.17), (4.18), (4.19) and (4.24), Theorem C gives (2.24) after some compu-tation.
Consider now the case when at least one of (4.22) and (4.23) does not hold. We cover this remaining case by assuming that
h(ξ)< 1
2c2(r, d, t)QGH
with the c2(r, d, t) occurring in Theorem 2.6. By the product formula and Liouville’s inequality we get Proof of Theorem 2.7. Together with the estimate (2.24) of Theorem 2.6, (2.25) gives
(2.26), and then (2.27) easily follows. 2