In this chapter we turn our attention to varieties of arbitrary dimension N. Let Γ be a finitely generated subgroup of (Q
∗)N, where N ≥ 2. Further, let Γ, Γε and C(Γ, ε) be defined as in (2.1). Choose a basis{w1, . . . ,wr} of Γ modulo Γtors and put
h0 := max 1, h(w1), . . . , h(wr) .
Denote by K the smallest number field such that Γ⊂(K∗)N, and putd:= [K :Q]. Let S be the minimal finite set of places of K containing all the infinite places of K and having the property that Γ⊂(O∗S)N. Denote by s the cardinality of S and letP be the quantity defined in (2.5).
Let
X :={x∈(Q
∗)N : fi(x) = 0, i= 1, . . . , m}
be a subvariety of (Q
∗)N, wheref1, . . . , fm are non-constant polynomials inQ[X1, . . . , XN] each consisting of 2 or 3 monomials. Put
δ:= max(degf1, . . . ,degfm), H := max(1, h(f1), . . . , h(fm)).
Further, let L be the smallest number field containing K and the coefficients of the poly-nomialsfi (i= 1, . . . , m).
The stabilizer of X is given by Stab(X) =
n
x∈(Q
∗)N | xX ⊆ Xo ,
where xX = {xy : y ∈ X }. Stab(X) is clearly an algebraic subgroup of (Q
∗)N, and it can be computed effectively in terms of the polynomials f1, . . . , fm defining X.
Put
C∗ := e11d3rr+3
(δh0)rs· P
logP·log∗max(dsP, δh0), (2.30)
and (
C2 :=C∗N(2δ)N−1,
C3 :=C∗·2mh0(r4r+1·d(log 3d)3·mδh0)r. (2.31) Theorem 2.11 (B´erczes, Evertse, Gy˝ory and Pontreau [11]). Let X be a variety satisfying the conditions listed above, and put H:= Stab(X).
(i) Suppose that H is finite. Then for every x∈ X ∩Γ we have h(x)≤C2H.
(ii) Suppose that His not finite. ThenX ∩Γ is contained in some finite union of translates x1H ∪ · · · ∪xTH,
with
xiH ⊂ X, xi ∈Γ, h(xi)≤C3H for i= 1, . . . , T. (2.32) Our results for X ∩Γε and X ∩C(Γ, ε) are as follows.
Theorem 2.12 (B´erczes, Evertse, Gy˝ory and Pontreau [11]). Put ε:= 0.03
4δ . (2.33)
(i) Assume that H:= Stab(X) is finite. Then for every x∈ X ∩Γε we have
h(x)< rh0δC2+C2H, [L(x) :L]≤2m+NδN. (2.34) (ii) Assume that H is not finite. Then X ∩Γε is contained in a finite union of translates
x1H ∪ · · · ∪xTH,
where for i = 1, . . . , T, we have xi ∈ X ∩Γε, xiH ⊂ X, and where h(xi) and [L(xi) : L]
are bounded above by effectively computable numbers depending only on Γ, f1, . . . , fm. Remark. It is possible in principle to give explicit expressions for the effectively com-putable numbers in part (ii) of Theorem 2.12, but these are rather complicated.
Theorem 2.13 (B´erczes, Evertse, Gy˝ory and Pontreau [11]). Let
ε := 0.03
4δ(C2δrh0+ 2C2H). (2.35)
Assume that Stab(X) is finite. Then for every x∈ X ∩C(Γ, ε) we have h(x)≤2rh0δC2+ 2C2H, [L(x) :L]≤2m+NδN.
Remark. IfH:= Stab(X) is not finite, then in general X ∩C(Γ, ε) need not be contained in a finite union of translates x1H ∪ · · · ∪xTH. Indeed, suppose that dimX >dimH, and that H ∩Γ contains points of infinite order. Pick any x0 ∈ X. Choose a point u∈ H ∩Γ of infinite order. Thus h(u)>0. Then for any sufficiently large integer n,
h(x0)≤ε(1 +nh(u)−h(x0))≤ε(1 +h(x0un)).
Hence x:=x0un∈x0H ∩C(Γ, ε). That is, every translatex0H with x0 ∈ X contains ele-ments fromC(Γ, ε). If X ∩C(Γ, ε) were contained in a finite union of translates ∪ti=1xiH, then so were X, which is impossible.
Chapter 3
Results over arbitrary finitely generated domains
Let A:=Z[z1, . . . , zr]⊃Z be a finitely generated integral domain over Z. In this chapter under finitely generated domain we mean an integral domainZ[z1, . . . , zr]⊃Z, which may contain both algebraic and transcendental elements over Q. Finiteness results for several kinds of Diophantine equations over A date back to the middle of the last century. In his book [45] and paper [44] S. Lang generalized several earlier results on Diophantine equations over the integers to results over A, including results concerning unit equations, Thue-equations and integral points on curves. However, all his results were ineffective. The first effective finiteness results for Diophantine equations over finitely generated domains were published in the 1980’s, when Gy˝ory [39], [40] developed his new effective specialization method. This enabled him to prove effective results over finitely generated domains of a special type containing also transcendental elements over Q. He proved such results for unit equations, norm form equations, index form equations, discriminant form equations [39] and for polynomials and integral elements of given discriminant [40]. Later Brindza proved such results for superelliptic equations [22] and the generalized Catalan equation [23], Brindza and Pint´er obtained such results for equal values of binary forms [24], and V´egs˝o [75] for the Schinzel-Tijdeman equation.
In 2013 Evertse and Gy˝ory [32] combined the method of Gy˝ory with newer results of Aschenbrenner [1] such that they were able to prove effective results for unit equations ax+by= 1 in x, y ∈A∗ over arbitrary finitely generated domains A of characteristic 0.
In this chapter on one hand I present effective finiteness results for Thue equations, hyper- and superelliptic equations, and the Schinzel-Tijdeman equation over arbitrary finitely generated domains. These are joint results with Jan-Hendrik Evertse and K´alm´an
Gy˝ory published in [10]. On the other hand I present effective finiteness results for gen-eralized unit points and division points on curves over finitely generated domains. These results have been published in my papers [7] and [6].
3.1 Finitely generated domains
Before presenting our results we introduce some concepts and notation.
Let r > 0 and let A := Z[z1, . . . , zr] be a domain of characteristic 0 which is finitely generated over Z. Clearly, A can be expressed as a factor ring
A∼=Z[X1, . . . , Xr]/I, (3.1) where I is the ideal of R := Z[X1, . . . , Xr] which consists of all polynomials f ∈ R with the property f(z1, . . . , zr) = 0. The ideal I is finitely generated, so we may write
I = (f1, . . . , ft) with f1, . . . , ft∈Z[X1, . . . , Xr]. (3.2) In fact in this way the polynomialsf1, . . . , ftfix a representation for the domain A. Recall that A is a domain of characteristic 0 if and only if I is a prime ideal, and I ∩Z = ∅.
Given a set of generators f1, . . . , ft for I this property can be checked effectively (see [1]
and [42]).
Let K denote the quotient field of A. We say that the polynomial f ∈ R represents α ∈ A if we have f(z1, . . . , zr) = α. Further we say that the pair (f, g) ∈ R2 represents β ∈K if g 6∈ I (i.e. g(z1, . . . , zr)6= 0) and f(zg(z1,...,zr)
1,...,zr) =β. We will also use the terminology that f is a representative for α, or (f, g) is a pair of representatives for β. Clearly, any elementα ∈Ahas infinitely many representatives, and anyβ ∈Khas infinitely many pairs of representatives. However, since one can effectively decide whether a given polynomial of R belongs to a given ideal of R or not (see [1]), one can also effectively decide if two polynomials represent the same element ofA, or if two pairs of polynomials of R represent the same element of K. Indeed, two polynomials f, f0 ∈ R represent the same element α∈A if and only if f−f0 ∈ I, and two pairs of polynomials (f, g),(f0, g0)∈R2 represent the same element β ∈K if and only if f g0 −f0g ∈ I.
We shall measure elements of A by their representatives. For a non-zero polynomial f ∈ R let us denote by degf the total degree of f and by h(f) the absolute logarithmic height of f, i.e. the logarithm of the maximum of the absolute values of its coefficients.
Further we define the size of f by
s(f) := max(1,degf, h(f)).
For the constant 0 polynomial we defines(0) := 1.
Throughout the dissertation we shall use the notation O(·) to denote a quantity which isctimes the expression between the parentheses, wherecis an effectively computable posi-tive absolute constant which may be different at each occurrence of theO-symbol. Further, throughout the dissertation we write log∗a:= max(1,loga) for a >0, and log∗0 := 1.