Clemens Fuchs

Karl-Morre Str. 46/15 8020 Graz

Quantitative Finiteness Results for Diophantine Equations

Dissertation^{‡}

Ausgeführt zum Zwecke der Erlangung

des akademischen Grades eines Doktors der Technischen Wissenschaften

Betreuer:

O.Univ.-Prof. Dr. Robert F. Tichy

Eingereicht an der

Technischen Universität Graz Technisch-Naturwissenschaftliche Fakultät

Graz, am 15.01.2002

‡This work was supported by the Austrian Science Foundation FWF, grant S8307-MAT.

Preface

The study of Diophantine equations, in general terms, is the study of solvability of equations in integers. Although researches in this field have their roots in antiquity and a history of the subject amounts more or less, to a history of mathematics itself, it is only in relatively recent times that there have emerged any general theories, and we shall begin in 1900 by referring to Hilbert’s famous list of problems.

The tenth of these asked for a universal algorithm for deciding whether or not a given Diophantine equation, that is, an equation f(x_{1}, …, x_{n}) = 0, where f denotes a polynomial with integer coefficients is solvable in integers x_{1}, …, x_{n}. Though Hilbert posed his question in terms of solvability, there are, of course, many other sorts of information that one might like to have; for instance, one might enquire as to whether a particular equation has infinitely many solutions, or one might seek some bounds on the number of solutions. In 1970, Matijasevic, proved that a general algorithm of the type sought by Hilbert does not in fact exist.

The first major advance towards a coherent theory was made by Thue in 1909 when he proved that the equation F (x, y) = m, where F denotes an irreducible binary form with integer coefficients and degree at least 3, possesses only a finite number of solutions in integers x, y.

In the midth of the 20th century two main branches have their origins. The first one started 1955 with Roth’s Theorem, for which Roth received the Fields medal. This theorem was generalized and quantified between 1965 and 1972 by W. M. Schmidt and then by many other authors; a development that had its climax in the various quantified versions of Schmidt’s Subspace Theorem. We are going to use these theorems frequently, especially they will enable us to calculate explicit upper bounds for the number of solutions of certain classes of Diophantine equations. Thereby, a remarkable feature is that these bounds will depend on very few parameters.

The second development has its roots in about the same time. In 1968, A. Baker derived lower bounds for the absolute value of linear forms of logarithms in algebraic numbers. This was a breakthrough in the field of effective methods which made it possible to compute upper bounds for the solutions themselves. For example Baker himself gave upper bounds for the solutions of the Thue equation.

In the first chapter, we will survey these main developments (this is mainly due to R. Tijdeman and can be found in [36, Chapter II]). Furthermore, we will collect some very important general methods to quantify the number of solutions of Diophantine equations. The first subsection will be about W. M. Schmidt’s celebrated Subspace Theorem. In the second subsection, we will give applications of the Subspace Theorem to S-unit equations and then in turn to the zero-multiplicity of nondegenerate linear recurring sequences. In the third subsection, we will introduce Baker’s method on linear forms of logarithms in algebraic numbers and in the fourth subsection, we state the analog of Baker’s method in algebraic function fields of one variable which is referred to as Mason’s inequality. Here we also collect applications to S-unit equations and to hyperelliptic equations in function fields.

The second chapter is devoted to the number of perfect powers in a linear recurring sequences. We will consider the Diophantine equation

G_{n} = Ex^{q},

where (G_{n})_{n=0}^{∞} is a linear recurring sequence, E is a nonzero and q a positive integer. In the introductory section, we survey some known results concerning this type of equation, in the second subsection we present our results, in the third subsection, we collect some useful lemmas and in the last subsection we present the proofs of the theorems. This chapter is identical with a joint paper with R. F. Tichy [51].

The third chapter is a generalization and continuation of the quantitative work done in the previous chapter. Now, we will consider the equation

f(G_{n}, x) = 0,

where f is polynomial in two variables that is monic in x and (G_{n})_{n=0}^{∞} is again a linear recurring sequence. We will present quantitative finiteness results for this Diophantine equation in subsection two and prove them in the following subsections.

In the fourth chapter, we turn to finiteness results concerning the equation

G_{n}(x) = G_{m}(P (x))

where (G_{n}(x))_{n=0}^{∞} is a sequence of polynomials satisfying a second order linear recurring sequence and P (x) is a fixed polynomials with deg P ≥ 1, i.e. we consider identities of polynomials. Under certain conditions, we will quantify the number of solutions (n, m) ∈ ℤ^{2} of this equation. After a introductory section, we will present the new theorems and then, we will prove them in the following subsections. This chapter is identically equal to a joint paper with R. F. Tichy and A. Pethő [49].

The fourth chapter is a continuation of the previous chapter on the equation

G_{n}(x) = G_{m}(P (x))

where now (G_{n}(x))_{n=0}^{∞} denotes a sequence of polynomials satisfying a third order linear recurring sequence. Because of Cardano’s formulae, it is possible to use the ideas from the second order case and to derive quantitative finiteness results in this case too. This chapter is equal to a manuscript which is joint work with R. F. Tichy and A. Pethő [50].

The last two chapters are about Diophantine m-tuples. Thereby, we will call a set {a_{1}, …, a_{m}} of positive integers a Diophantine m-tuple with property D(n) for n ∈ ℤ if the product of any two of them increased by n is a perfect square, i.e. a square of a positive integer. Here, we will not study the classical case but analogues problems for polynomials (which was already studied e.g. by Jones in [55], [56]). In chapter five, we prove that there does not exist a set of four polynomials with integer coefficients such that the product of two of them minus one is a square of a polynomials with integer coefficients. This chapter is equal to a joint paper with A. Dujella [33].

In the last chapter, we will consider the case where n is a linear polynomial. We will show that there are at most 26 polynomials with integer coefficients such that the product of two of them plus a linear polynomial is a square of a polynomial with integer coefficients. This chapter is equal to a manuscript which is joint work with A. Dujella and R. F. Tichy [34].

Last but not least, I want to express my gratitude to Prof. Robert F. Tichy, who is the advisor of this thesis and who is coauthor of many of the papers forming the main part of this work. Moreover, I want to thank the other coauthors of the papers involved, namely Prof. Andrej Dujella from Zagreb, Croatia and Prof. Attila Pethő from Debrecen, Hungary. I am most thankful to them for valuable suggestions and fruitful discussions. For financial support, I am indebted to the Austrian Science Foundation (FWF - “Fond zur Förderung der wissenschaftlichen Forschung”), grant S8307-MAT. Finally, I am grateful to my family, especially to my father, for permanent encouragement and support.

Graz, January 2002 Clemens Fuchs

Contents

1 General methods for quantitative finiteness results 6 1.1 The Subspace Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 S-unit equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Baker’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Mason’s inequality and applications . . . . . . . . . . . . . . . . . . . . 15

2 Perfect powers in linear recurring sequences 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Exponential–polynomial equations and linear recurrences 34 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5 Proof of Theorem 3.2 and the corollaries . . . . . . . . . . . . . . . . . 51 3.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 On the Diophantine equation G_{n}(x) = G_{m}(P (x)) 55 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3 Results for families of orthogonal polynomials . . . . . . . . . . . . . . 61 4.4 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.5 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.6 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.7 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.8 Proof of the Theorems 4.2, 4.4 and 4.6 . . . . . . . . . . . . . . . . . . 74 5 On the equation G_{n}(x) = G_{m}(P (x)) for third order linear recurring

sequences 80 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.3 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.4 Reduction to a system of equations . . . . . . . . . . . . . . . . . . . . 87 5.5 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.6 Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.7 Proof of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6 A polynomial variant of a problem of Diophantus and Euler 100 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2 Two sequences of polynomials . . . . . . . . . . . . . . . . . . . . . . . 102 6.3 Reduction to the case a = 1 . . . . . . . . . . . . . . . . . . . . . . . . 105 6.4 Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7 Diophantine m-tuples for linear polynomials 112 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.2 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.3 Elements with large degree . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.4 Elements with small degree . . . . . . . . . . . . . . . . . . . . . . . . 120

Chapter 1

General methods for quantitative finiteness results

1.1 The Subspace Theorem

In 1844, Liouville proved that transcendental numbers do exist. He derived this result from the following approximation theorem.

Theorem 1.1. (Liouville) Suppose α is a real algebraic number of degree d. Then there is a constant c(α) > 0 such that

|α − \frac{p}{q}| > c(α)q^{−d}

for every rational number p/q, q > 0 distinct from α.

Using this result, Liouville deduced that numbers like ∑_{ν=1}^{∞} 2^{−ν!} are transcendental. Furthermore, Liouville’s Theorem implies that the inequality

|α − \frac{p}{q}| < q^{−μ}  (1.1)

has only finitely many rational solutions p/q if μ > d. Thue showed in 1909 that (1.1) has only finitely many solutions if μ > d/2+1. Then, Siegel (1921) showed in his thesis that this is already true if μ > 2\sqrt{d}. A slight improvement to μ > \sqrt{2d} was made by Dyson in 1947. Finally, Roth proved in 1955 that (1.1) has only finitely many solutions if μ > 2. He received the Fields medal for this achievement. So, for d ≥ 2 this theorem, together with Lagrange’s Theorem from 1770 which says that every irrational real α admits infinitely many rational p/q such that

|α − \frac{p}{q}| < \frac{1}{q^{2}} ,

shows that the number 2 as exponent is best possible. For d = 2 Liouville’s Theorem is stronger than Roth’s one.

In a series of papers published between 1965 and 1972, W.M. Schmidt made an important step forward. One of his results is the following extension of Roth’s Theorem.

Theorem 1.2. (W. M. Schmidt) Suppose α is a real algebraic number. Let k ≥ 1 and δ > 0. Then there are only finitely many algebraic number β of degree ≤ k with

|α − β| < H(β)^{−(k+1+δ)}

where H(β) denotes the classical absolute height.

This result follows by applying the so-called Subspace Theorem which, in its simplest form, reads as follows. For x = (x_{1}, …, x_{n}) ∈ ℤ^{n} put

|x| = \sqrt{x_{1}^{2} + ⋯ + x_{n}^{2}}.

Then we have

Theorem 1.3. (Subspace Theorem, W. M. Schmidt) Suppose L_{1}(x), …, L_{n}(x) are linearly independent linear forms in x = (x_{1}, …, x_{n}) with algebraic coefficients. Given δ > 0, there are finitely many proper linear subspaces T_{1}, …, T_{w} of ℝ^{n} such that every integer point x ≠ 0 with

|L_{1}(x) ⋯ L_{n}(x)| < |x|^{−δ}

lies in one of these subspaces.

In fact, Roth’s Theorem is equivalent to n = 2 of the above stated theorem.

Up to now, apart from Liouville’s Theorem, all results stated in this section are ineffective, that is, the method of proof does not enable us to determine the finitely many exceptions. However, the method makes it possible to derive upper bounds for the number of exceptions. This was proved by Schmidt in 1989 (see [79]).

Theorem 1.4. (Quantitative Subspace Theorem, W. M. Schmidt) Let L_{1}, …, L_{n} be linearly independent linear forms with coefficients in some algebraic number field of degree d. Consider the inequality

|L_{1}(x) ⋯ L_{n}(x)| < | det(L_{1}, …, L_{n})||x|^{−δ} where 0 < δ < 1.  (1.2)

The set of solutions of (1.2) with

x ∈ ℤ^{n}, |x| ≫ max{(n!)^{8/δ}, H(L_{1}), …, H(L_{n})}

is contained in the union of at most [(2d)^{2^{26n}δ^{−2}}] proper linear subspaces of ℝ^{n}.

A second result in this direction is due to Vojta [93]. Essentially this result says that, apart from finitely many exceptions which may depend on δ, the solutions of (1.2) are in the union of finitely many, at least in principle effectively computable, proper linear subspaces of ℝ^{n} which are independent of δ.

In 1977, Schlickewei extended Schmidt’s Subspace Theorem of 1972 to the p-adic case and to number fields. In 1990 (see [75]), he generalized Schmidt’s quantitative Subspace Theorem to the p-adic case over ℚ and later 1992 to number fields (cf. [76]). We shall see that these results posses many important applications. Vojta proved the p-adic assertion of the above mentioned result himself.

Evertse [40] derived in 1996 an improved version of Schlickewei’s and Schmidt’s quantitative Subspace Theorem. Below we state this result of Evertse and to this end we introduce suitably normalized absolute values and heights.

Let K be an algebraic number field. Denote its ring of integers by O_{K} and its collection of places by M_{K}. For v ∈ M_{K}, x ∈ K, we define the absolute value |x|_{v} by (i) |x|_{v} = |σ(x)|^{1/[K:ℚ]} if v corresponds to the embedding σ : K ↪ ℝ; (ii) |x|_{v} = |σ(x)|^{2/[K:ℚ]} = |\overline{σ}(x)|^{2/[K:ℚ]} if v corresponds to the pair of conjugate complex

embedding σ, \overline{σ} : K ↪ ℂ; (iii) |x|_{v} = (N℘)^{−ord_{℘}(x)/[K:ℚ]} if v corresponds to the prime ideal ℘ of O_{K}. Here N℘ = #(O_{K}/℘) is the norm of ℘ and ord_{℘}(x) the exponent of ℘ in the prime ideal composition of (x) with ord_{℘}(0) := ∞. In case (i) or (ii) we call v real infinite or complex infinite, respectively; in case (iii), we call v finite. These absolute values satisfy the Product formula

∏_{v∈M_{K}} |x|_{v} = 1 for x ∈ K^{∗}.  (1.3)

The height of x = (x_{1}, …, x_{n}) ∈ K^{n} with x ≠ 0 is defined as follows: for v ∈ M_{K} put

|x|_{v} = (∑_{i=1}^{n} |x_{i}|_{v}^{2[K:ℚ]})^{1/(2[K:ℚ])} if v is real infinite,
|x|_{v} = (∑_{i=1}^{n} |x_{i}|_{v}^{[K:ℚ]})^{1/[K:ℚ]}  if v is complex infinite,
|x|_{v} = max(|x_{1}|_{v}, …, |x_{n}|_{v})  if v is finite

(note that for infinite places v, | · |_{v} is a power of the Euclidean norm). Now define

H(x) = H(x_{1}, …, x_{n}) = ∏_{v} |x|_{v}.

For a linear form l(X) = a_{1}X_{1} + ⋯ + a_{n}X_{n} with algebraic coefficients we define H(l) := H(a), where a = (a_{1}, …, a_{n}) and if a ∈ K^{n} then, we put |l|_{v} = |a|_{v} for v ∈ M_{K}. Furthermore, we define the number field K(l) := K(a_{1}/a_{j}, …, a_{n}/a_{j}) for any j with a_{j} ≠ 0; this is independent of the choice of j.

We are now ready to state Evertse’s result [40]. The following notation is used:

- S is a finite set of places on K of cardinality s containing all infinite places;

- {l_{1v}, …, l_{nv}}, v ∈ S are linearly independent sets of linear forms in n variables with algebraic coefficients such that

H(l_{iv}) ≤ H, [K(l_{iv}) : K] ≤ D for v ∈ S, i = 1, …, n.

We choose for every place v ∈ M_{K} a continuation of | · |_{v} to the algebraic closure of K and denote it by | · |_{v}, too.

Theorem 1.5. (Quantitative Subspace Theorem, Evertse) Let 0 < δ < 1 and consider the inequality for x ∈ K^{n}

∏_{v∈S} ∏_{i=1}^{n} \frac{|l_{iv}(x)|_{v}}{|x|_{v}} < (∏_{v∈S} | det(l_{1v}, …, l_{nv})|_{v}) · H(x)^{−n−δ} .  (1.4)

Then the following assertions hold:

(i) There are proper linear subspaces T_{1}, …, T_{t_{1}} of K^{n}, with

t_{1} ≤ (2^{60n^{2}} · δ^{−7n})^{s} log 4D · log log 4D

such that every solution x ∈ K^{n} of (1.4) satisfying H(x) ≥ H belongs to T_{1} ∪ ⋯ ∪ T_{t_{1}}. (ii) There are proper linear subspaces S_{1}, …, S_{t_{2}} of K^{n}, with

t_{2} ≤ (150n^{4} · δ^{−1})^{ns+1}(2 + log log 2H)

such that every solution x ∈ K^{n} of (1.4) satisfying H(x) < H belongs to S_{1} ∪ ⋯ ∪ S_{t_{2}}.

Very recently (unpublished yet) Evertse and Schlickewei were able to extend the arguments in order to handle arbitrary vectors in \overline{ℚ}^{n} instead of just vectors in K^{n} for some fixed number field K. So, they derived a result which is much more general than the classical Subspace Theorem, in fact, it is an “absolute” generalization of the Subspace Theorem, dealing with vectors x in \overline{ℚ}^{n} rather than in K^{n}. The theorem is as follows and can be found in [44].

Let E be a number field. Let S be a finite subset of M_{E} of cardinality s and suppose that for each v ∈ S we have linearly independent linear forms l_{1v}, …, l_{nv} with coefficients in \overline{ℚ}. We suppose that for i = 1, …, n and for v ∈ S

[E(l_{iv}) : E] ≤ D,

and moreover that

H(l_{iv}) ≤ H, v ∈ S, i = 1, …, n.

Each normalized absolute value | · |_{v} on E has a unique extension | · |′_{v}, say, to \overline{E}_{v}, the algebraic closure of the completion E_{v}. Fix an embedding τ_{v} of \overline{ℚ} over E into \overline{E}_{v}. Then, we extend | · |_{v} from E to \overline{ℚ} by putting

|x|_{v} = |τ_{v}(x)|′_{v} for x ∈ \overline{ℚ}.

Using this notation, we obtain

Theorem 1.6. (Absolute Subspace Theorem, Evertse and Schlickewei) Let E, S and the linear forms l_{1v}, …, l_{nv} in X = (X_{1}, …, X_{n}) be as above. Let 0 < δ < 1. Then there exist proper linear subspaces T_{1}, …, T_{t_{2}} of \overline{ℚ}^{n}, all defined over E, where

t_{2} = t_{2}(n, s, D, δ) ≤ (3n)^{2ns}2^{3(n+9)^{2}}δ^{−ns−n−4} log(4D) log log(4D)

with the following property. The set of solutions x ∈ \overline{ℚ}^{n} of the inequalities

∏_{v∈S} ∏_{i=1}^{n} max_{σ∈Gal(\overline{ℚ}/E)} \frac{|l_{iv}(σ(x))|_{v}}{|σ(x)|_{v}} ≤ (∏_{v∈S} | det(l_{1v}, …, l_{nv})|_{v}) · H(x)^{−n−δ}

and

H(x) > max{n^{4n/δ}, H}  (1.5)

is contained in the union

T_{1} ∪ … ∪ T_{t_{2}}.

Observe that Evertse’s result stated before is a special case with the restriction that the solutions x lie in E^{n}. Furthermore, the bound obtained in the absolute Subspace Theorem is better than the bound in Evertse’s quantitative Subspace Theorem. However, Evertse has only to assume H(x) ≥ H instead of (1.5).

1.2 S-unit equations

In this section, we will deal with the most important application of the Subspace Theorem. Therefore, let K be an algebraically closed field of characteristic 0, n ≥ 1 an integer, α_{1}, …, α_{n} elements of K^{∗}, and Γ a finitely generated multiplicative subgroup of K^{∗}. A solution (x_{1}, …, x_{n}) of the so called weighted unit equation

α_{1}x_{1} + ⋯ + α_{n}x_{n} = 1 in x_{1}, …, x_{n} ∈ Γ  (1.6)

is called nondegenerate if

∑_{j∈J} α_{j}x_{j} ≠ 0 for each non-empty subset J of {1, …, n}  (1.7)

and degenerate otherwise. It is clear that if Γ is infinite and if (1.6) has a degenerate solution then (1.6) has infinitely many degenerate solutions. For nondegenerate solutions, we have the following result which is due to Evertse, Schlickewei and Schmidt [45].

Theorem 1.7. (Evertse, Schlickewei and Schmidt) Let K be a field of characteristic 0, let α_{1}, …, α_{n} be nonzero elements of K and let Γ be a multiplicative subgroup of (K^{∗})^{n} of rank r. Then the equation

α_{1}x_{1} + … + α_{n}x_{n} = 1

has at most

exp((6n)^{3n}(r + 1))

nondegenerate solutions (x_{1}, …, x_{n}) ∈ Γ.

This theorem is the Main Theorem on S-unit equations over fields with characteristic 0. It is a generalization of earlier results due to Evertse and Győry [41], Evertse [38], and van der Poorten and Schlickewei [72] on the finiteness of the number of nondegenerate solutions of (1.6). For a general survey on these equations and their applications, we refer to Evertse, Győry, Stewart and Tijdeman [42].

The above theorems can be applied to the following important Diophantine equation which will be needed later. Let U = (u_{m})_{m∈ℤ} be a sequence of complex numbers satisfying a recurrence relation of order q,

u_{m} = c_{1}u_{m−1} + … + c_{q}u_{m−q}

with c_{1}, …, c_{q} ∈ ℂ, c_{q} ≠ 0. As it is well-known, we have

u_{m} = ∑_{i=1}^{n} g_{i}(m)α_{i}^{m} for m ∈ ℤ,

where α_{1}, …, α_{n} are distinct, nonzero complex numbers and g_{1}, …, g_{n} ∈ ℂ[T] polynomials with

∏_{i=1}^{n}(T − α_{i})^{deg g_{i}+1} = T^{q} − c_{1}T^{q−1} − … − c_{q}.

Denote by N_{U}(a) the number of integers m with

u_{m} = a.

The sequence U is called nondegenerate if no quotient α_{i}/α_{j} (1 ≤ i < j ≤ n) is equal to a root of unity. From the Theorem of Skolem-Mahler-Lech (cf. [60]) it follows that then N_{U}(a) is finite for every a ∈ ℂ. Using this, Schlickwei [77] showed that if U is nondegenerate, α_{1}, …, α_{n} are not roots of unity too, and α_{1}, …, α_{n} and the coefficients of g_{1}, …, g_{n} generate an algebraic number field K of degree d then for every a ∈ K, we have

N_{U}(a) ≤ d^{6q^{2}}2^{2^{28q!}}.

If we assume that g_{1}, …, g_{n} are all constant, we obtain the following improvement by applying Theorem 1.7 to the group generated by α_{1}, …, α_{n} which has at most rank n.

Theorem 1.8. Let U be a recurrence sequence satisfying

u_{m} = g_{1}α_{1}^{m} + … + g_{n}α_{n}^{m} for m ∈ ℤ,

where α_{1}, …, α_{n} are nonzero complex numbers such that neither α_{1}, …, α_{n} nor any of the quotients α_{i}/α_{j} (1 ≤ i < j ≤ n) are roots of unities and where g_{1}, …, g_{n} are nonzero complex numbers. Then for every a ∈ ℂ, we have

N_{U}(a) ≤ exp((n + 2) · (6n)^{3n}).

Very recently, Schmidt [82] obtained the following remarkable result concerning the zero multiplicity of an arbitrary nondegenerate complex recurring sequences (i.e., with arbitrary polynomials g_{1}, …, g_{n}).

Theorem 1.9. (W. M. Schmidt) Suppose that (G_{n})_{n∈ℤ} is a nondegenerate linear recurring sequence of complex numbers, whose characteristic polynomial has k distinct roots of multiplicity ≤ a. Then the number of solutions n ∈ ℤ of the equation

G_{n} = 0,

can be bounded from above by

c(k, a) = e^{(7k^{a})^{8k^{a}}}.

(This number of solutions is called the zero multiplicity of the recurrence.)

In recent work [22], [23], David and Philippon have proved a slight sharpening of the above results. The bound in Theorem 1.7 can be improved to

exp(exp(c_{1}n)(r + 1)).

Therefore, also the bound in Theorem 1.8 can be improved to

exp(exp(c_{1}n)).

Here c_{1} is an absolute constant.

1.3 Baker’s Method

Liouville’s Theorem was also the starting point of a development of effective methods, too. Hermite, in 1873 and Lindemann in 1882 established the transcendence of the numbers e and π, respectively. The Theorem of Lindemann-Weierstrass (1885) says that β_{1}e^{α_{1}} + ⋯ + β_{n}e^{α_{n}} ≠ 0 for any distinct algebraic numbers α_{1}, …, α_{n} and any nonzero algebraic numbers β_{1}, …, β_{n}.

A new development started in 1929 when Gelfond showed the transcendence of 2^{\sqrt{2}}. In 1934, Gelfond and Schneider, independently of each other, proved the transcendence of α^{β} for α, β algebraic, α ≠ 0, 1 and β irrational. Alternatively, this result says that for any nonzero algebraic numbers α_{1}, α_{2}, β_{1}, β_{2} with log α_{1}, log α_{2} linearly independent over the rationals, we have

β_{1} log α_{1} + β_{2} log α_{2} ≠ 0.

In 1966, Baker proved the transcendence of e^{β_{0}}α_{1}^{β_{1}} ⋯ α_{n}^{β_{n}} for algebraic numbers α_{1}, …, α_{n}, different from 0 or 1, and β_{1}, …, β_{n} with either β_{0} ≠ 0 or 1, β_{1}, …, β_{n} linearly independent over the rationals. This follows from the following result.

Theorem 1.10. (A. Baker) Let α_{1}, …, α_{n} be nonzero algebraic numbers such that their logarithms log α_{1}, …, log α_{n} are linearly independent over the field of all rational numbers. Then 1, log α_{1}, …, log α_{n} are linearly independent over the field of all algebraic numbers.

The above results have p-adic analogues and also analogues in the theory of elliptic functions.

The effective character of the above result can be expressed in form of the transcendence measures. Between 1972 and 1977, Baker [4] derived the following important estimate.

Theorem 1.11. (A. Baker) Let α_{1}, …, α_{n} be nonzero algebraic numbers with degrees at most d and (classical absolute) heights at most A_{1}, …, A_{n} (all ≥ 2), respectively. Let b_{1}, …, b_{n} be rational integers of absolute values at most B (≥ 2). Put Λ = b_{1} log α_{1} + ⋯ + b_{n} log α_{n}. Then either Λ = 0 or

log |Λ| > −(16nd)^{200n} (∏_{j=1}^{n} log A_{j}) log (∏_{j=1}^{n−1} log A_{j}) log B.

Later on, the constants have been improved and the log(∏ log) factor has been removed. The best bound known at present is due to Baker and Wüstholz [7] in the classical case and due to Waldschmidt and his colleagues, in the p-adic case.

Later, we will need the following special form of the above theorem of A. Baker [2]. Therefore, let us start by recalling the definition of the absolute logarithmic Weil height: for an algebraic number β let P_{β}(x) = x^{k} + a_{k−1}x^{k−1} + ⋯ + a_{0} ∈ ℤ[x] denote the minimal polynomial of β. Furthermore, let β_{1} = β, β_{2}, …, β_{k} denote the conjugates of β. Then we call

h(β) = \frac{1}{k} log (∏_{i=1}^{k} max\{1, |β_{i}|\})

the absolute logarithmic Weil height of β.

Theorem 1.12. (A. Baker) Let α_{1}, …, α_{k} be algebraic numbers, different from 0 or 1, K = ℚ(α_{1}, …, α_{k}) and let d be the degree of [K : ℚ]. For i = 1, …, k set

h_{i} = max \{h(α_{i}), \frac{e | log α_{i}|}{d}, \frac{1}{d}\} .

Let b_{1}, …, b_{k} ∈ ℤ, b_{k} > 0, Λ = b_{1} log α_{1} + ⋯ + b_{k} log α_{k} ≠ 0 and B = max{2, |b_{1}|, …, |b_{k−1}|}. Then we have

log |Λ| > −C(k)d^{k+2}h_{1} ⋯ h_{k} log (C(k)d^{k+2}h_{1} ⋯ h_{k−1}) log b_{k} − \frac{B}{b_{k}} ,  (1.8)

where

C(k) = 2^{26k}k^{3k}.

A proof of this theorem with the given explicit constants can be found in the monograph of Waldschmidt [95], page 309, Corollary 9.24.

Baker’s estimates from 1967 on linear forms of logarithms in algebraic numbers caused a breakthrough in computing upper bounds for the solutions of Diophantine equations themselves. Baker himself gave upper bounds for the solutions of the Thue equation [3] and the super-elliptic equation [1]

y^{m} = P (x)

where m ≥ 2 is a fixed given positive integer and P (x) ∈ ℤ[x] is a polynomial with at least three simple roots if m = 2 and at least two simple roots if m ≥ 3. Later, this conditions were weakened.

Later, Baker’s sharpening made it possible to deal with Diophantine equations which cannot be treated by the mentioned ineffective methods. E.g. Schinzel and Tijdeman showed that the hyperelliptic equation with m, x, y variable and P (x) ∈ ℤ[x] a given polynomial with at least two distinct roots implies that m is bounded. Tijdeman also showed that the Catalan equation x^{m} −y^{n} = 1 in integers m, n, x, y all > 1 implies that x^{m} is bounded by some effectively computable number [92]. The bounds obtained in equations involving a power with both base and exponent variable are, however, so large that it is not yet possible to solve such equations in practice.

The best bounds for linear forms known at present make it, however, possible to solve for example the Thue equation completely. Additional algorithms are needed to achieve this (cf. e.g. [11]).

1.4 Mason’s inequality and applications

In this section, we want to study analogues of the previous results for function fields in one variable. Therefore, let K be an algebraically closed field with characteristic 0. Corresponding to the integers there is the polynomial ring K[x] and the object of concern is the set of solutions in K[x]. The results in fact refer to a more general situation; actually, we shall deal with the solutions integral over K[x] in an arbitrary finite extension L rather than to consider just those in K[x] itself.

Let us begin by recalling the definitions of discrete valuations on the field K(x) where x is transcendental over K. For ξ ∈ K define the valuation ν_{ξ} such that for Q ∈ K(x) we have Q(x) = (x − ξ)^{ν_{ξ}(Q)}A(x)/B(x) where A, B are polynomials with A(ξ)B(ξ) ≠ 0. Furthermore, for Q = A/B with A, B ∈ K[x] we put deg Q := deg A − deg B; thus ν_{∞} := − deg is a discrete valuation on K(x). These are all discrete valuations on K(x). Now let L be a finite extension of K(x). Such an L is called a function field in one variable with constant field K. Each of the valuations ν_{ξ}, ν_{∞} can be extended in at most [L : K(x)] =: d ways to a discrete valuation on L and in this way one obtains all discrete valuations on L. A valuation on L is called finite if it extends ν_{ξ} for some ξ ∈ K and infinite if it extends ν_{∞}.

We need the following generalization of the degree from K[x] to L. Define the height of f ∈ L by

H(f) = − ∑_{ν} min{0, ν(f)}

where the sum is taken over all discrete valuations on L; thus for f ∈ K(x) the height H(f) is just the number of poles of f, counted according to multiplicity. We note that if f lies in K[x] then H(f) = d deg f. We also want to define the height of a polynomial with coefficients in L. In order to do this, let us denote for any finite set S of elements of L

ν(S) = min_{s∈S}{H(s)} and H(S) = − ∑_{ν} min{0, ν(S)}

where the sum again runs over all valuations in L. If P ∈ L[T] and S is the set of its coefficients, then the quantities ν(P) and H(P) are defined to be ν(S) and H(S) respectively.

Let O denote the ring of elements of L integral over K[x]. These elements have the property that ν(f) ≥ 0 for all finite valuations on L.

Now, we are ready to formulate Mason’s inequality which may be regarded as a form of lower bound for linear forms in logarithms of algebraic functions but we avoid the actual use of such logarithms and the consequent discussion of units in function fields.

Theorem 1.13. (Mason’s inequality, R. C. Mason) Suppose that γ_{1}, γ_{2} and γ_{3} are nonzero elements of L with γ_{1} + γ_{2} + γ_{3} = 0, and such that ν(γ_{1}) = ν(γ_{2}) = ν(γ_{3}) for each valuation ν not in the finite set V. Then either γ_{1}/γ_{2} lies in K, in which case H(γ_{1}/γ_{2}) = 0, or

H(γ_{1}/γ_{2}) ≤ |V| + 2g − 2,

where |V| denotes the number of elements of V.

This theorem was generalized later on by Mason to the case of n summands [63], i.e. to the equation

γ_{1} + γ_{2} + … + γ_{n} = 0.  (1.9)

In 1986, his bound was improved by Brownawell and Masser [15] who proved

H(γ_{1}, …, γ_{n}) ≤ (n − 1)(n − 2){|V| + 2g − 2},

for every solution of (1.9) which is nondegenerate (i.e. every non-empty proper subset of {γ_{1}, …, γ_{n}} is K-linearly independent), where we have used the notation as above. This bound was independently discovered by Voloch [94] who gave a different proof.

Using Mason’s inequality, we are able to solve completely the general hyperelliptic equation over an algebraic function field L of characteristic 0. A proof of this theorem can be found in the monograph of Mason (cf. [64, Theorem 6]).

Theorem 1.14. (R. C. Mason) Let α_{1}, …, α_{n} ∈ O. All the solutions X, Y ∈ O of the hyperelliptic equation

Y^{2} = (X − α_{1})(X − α_{2}) ⋯ (X − α_{n})  (1.10)

satisfy

H(X) ≤ 26H + 8g + 4(r − 1);

here H denotes the height of the polynomial on the right hand side of (1.10), g denotes the genus of L/K and r denotes the number of infinite valuations on L.

Let us note that this bound varies only as a linear function of the height of the hyperelliptic equation, in contrast with the multiple exponential bounds for the classical case obtained by Baker [1]. This shows that the fundamental inequality due to Mason on which the proof of this theorem is based and which is the function field analog of Baker’s method of linear forms in logarithms is very sharp. Let us also mention that in the same way it is possible to construct an algorithm for the effective determination of all the solutions of a general Thue equation in the case of function fields.

Next, we will consider equation (1.6) over function fields, too. Let L be an algebraic function field in one variable with algebraically closed constant field K of characteristic 0. Thus, L is a finite extension of K(t) where t is a transcendental element of L over K. As we have seen, the field L can be endowed with a set M_{L} of additive valuations with value group ℤ for which

K = {0} ∪ {z ∈ L | ν(z) = 0 for each ν in M_{L} }

holds. Let S be a finite subset of M_{L}. An element z of L is called an S-unit if ν(z) = 0 for all ν ∈ M_{L}\S. The S-units form a multiplicative group which is denoted by U_{S}. The group U_{S} contains K^{∗} as a subgroup and U_{S}/K^{∗} is finitely generated. Now, the analogue of Theorem 1.7 for the function field case holds.

Theorem 1.15. (Evertse and Győry) Let L, K, S be as above. Let g be the genus of L/K, s the cardinality of S, and n ≥ 2 an integer. Then for every α_{1}, …, α_{n} ∈ L^{∗} the set of solutions of

α_{1}x_{1} + … + α_{n}x_{n} = 1 in x_{1}, …, x_{n} ∈ U_{S}  (1.11)
with α_{1}x_{1}, …, α_{n}x_{n} not all in K  (1.12)

is contained in the union of at most

log(g + 2) · (e(n + 1))^{(n+1)s+2}

(n − 1)-dimensional linear subspaces of L^{n}.

For deriving this upper bound the effective upper bound of Brownawell and Masser [15] for the heights of solutions of (1.11) is used. For n = 2 the theorem gives the upper bound

log(g + 2)(3e)^{3s+2}

for the number of solution of (1.11). We note that in case n = 2 Evertse [39] established an upper bound which is better and independent of g.

Theorem 1.16. (Evertse) Let L, K, S be as above. For each pair λ, μ in L^{∗} the equation

λx + μy = 1 in x, y ∈ U_{S}

has at most 2 · 7^{2s} solutions with λx/μy ∉ K. As above, s denotes the cardinality of S.

Let us mention that there exists an analog of the quantitative version of the Main Theorem on S-unit equations over fields with characteristic 0 due to Evertse, Schlickewei and Schmidt in case of the rational function field K(x), too. It is due to J. Müller and can be found in [66].

Perfect powers in linear recurring sequences

P. Corvaja and U. Zannier [20] showed for linear recurring sequences defined by G_{n} = a_{1}α_{1}^{n} + … + a_{t}α_{t}^{n} (t ≥ 2) with nonzero rational numbers a_{i} and integral characteristic roots α_{1} > … > α_{t} > 0 (α_{1}, α_{2} coprime) that the equation G_{n} = x^{q} (for q ≥ 2) has only finitely many solutions (n, x) ∈ ℕ^{2}. In this chapter we want to use a quantitative version of W. M. Schmidt’s Subspace Theorem (due to J.-H. Evertse [40]) to calculate an upper bound for the number of solutions (n, x). Combining this with an earlier result of I. Nemes and A. Pethő [67] we establish also an upper bound for the number of solutions (n, x, q).

This chapter is identically equal to a joint paper with R. F. Tichy which is to appear in Acta Arith. (cf. [51]).

2.1 Introduction

Let A_{1}, A_{2}, …, A_{k} and G_{0}, G_{1}, …, G_{k−1} be algebraic numbers over the rationals and let (G_{n}) be a k-th order linear recurring sequence given by

G_{n} = A_{1}G_{n−1} + ⋯ + A_{k}G_{n−k} for n = k, k + 1, ….  (2.1)

Let α_{1}, α_{2}, …, α_{t} be the distinct roots of the corresponding characteristic polynomial

X^{k} − A_{1}X^{k−1} − ⋯ − A_{k}.  (2.2)

Then for n ≥ 0

G_{n} = P_{1}(n)α_{1}^{n} + P_{2}(n)α_{2}^{n} + ⋯ + P_{t}(n)α_{t}^{n},

where P_{i}(n) is a polynomial with degree less than the multiplicity of α_{i}; the coefficients of P_{i}(n) are elements of the field: ℚ(G_{0}, …, G_{k−1}, A_{1}, …, A_{k}, α_{1}, …, α_{t}).

The recurring sequence is called simple, if all characteristic roots are simple. (G_{n}) is called nondegenerate, if no quotient α_{i}/α_{j} for all 1 ≤ i < j ≤ t is equal to a root of unity and degenerate otherwise. Observe that, even if (G_{n}) is degenerate, there exists a positive integer d such that, (G_{r+md}) is nondegenerate on each of the d arithmetic progressions with 0 ≤ r < d. Therefore, restricting to nondegenerate recurring sequences causes no substantial loss of generality.

In the present chapter we deal with the Diophantine equation

G_{n} = Ex^{q}, E ∈ ℤ\{0}  (2.3)

which was earlier investigated by several authors (e.g. cf. [85]).

For the Fibonacci sequence (F_{n}), Cohn [18] and Wyler [97], independently, proved that F_{n} is a square only if n = 0, 1, 2 and 13. Cohn [19] and Steiner [88] solved the equations F_{n} = 2x^{2} and F_{n} = 3x^{2}. They also proved the corresponding results for the Lucas sequence (L_{n}). London and Finkelstein [61] determined all cubes in the Fibonacci sequence; Lagarias and Weisser [59] gave another proof. Steiner [87] derived some partial results for higher powers. The proofs of these results do not depend on estimates for linear forms in logarithms. Pethő [70], [71] used the theory of linear forms in logarithms and computer calculations to determine all the cubes and fifth powers in the Fibonacci sequence.

For a nondegenerate recurring sequence (G_{n}) of order 2 induced by a (rational) integral recurrence, it has been proved, independently, by Pethő [69] and Shorey and Stewart [83] that for the solutions x ∈ ℤ, |x| > 1 and q ≥ 2 of (2.3) max(|x|, q, n) is bounded by an effectively computable constant depending only on E and the sequence (G_{n}). In fact, Pethő proved that max(|x|, q, n) is bounded by an effectively computable number depending only on the greatest prime divisor of E and on the coefficients and initial values of (G_{n}) (provided that the coefficients are coprime integers). Pethő extended this result to the equation G_{n} = bx^{q} with b ∈ S, where S is a set of integers composed solely of a finite number of primes.

Shorey and Stewart [83] proved the above finiteness result for certain recurring sequences of order > 2. Let (G_{n}) be a nondegenerate linear recurring sequence given by

G_{n} = λ_{1}α_{1}^{n} + P_{2}(n)α_{2}^{n} + ⋯ + P_{t}(n)α_{t}^{n},  (2.4)

where λ_{1} is a nonzero constant, |α_{1}| > |α_{j}| for j = 2, …, t, and G_{n} − λ_{1}α_{1}^{n} ≠ 0. Then assuming x, q > 1 the solutions q of (2.3) can be bounded by an effectively computable constant which depends on the coefficients and initial values of the recurrence. Kiss [58] proved that, in fact, q is less than a number which is effectively computable in terms of the greatest prime divisor of E and the coefficients and the initial values of the sequence (G_{n}).

Shorey and Stewart [84] considered recurring sequences (G_{n}) satisfying (2.4). Assuming that P_{2}(n) is a nonzero constant, t = 3 and |α_{1}| = |α_{2}| = |α_{3}| they showed, that (2.3) has only finitely many integral solutions E, x, q and n with the greatest prime divisor of E bounded by a given positive integer P, and |x| > 1, q > 2 and n ≥ 0.

Finkelstein [46], Williams [96] and Steiner [88] proved that 1, 2 and 5 are the only Fibonacci numbers of the form x^{2} + 1. Finkelstein [47] established a similar result for Lucas numbers. Stewart [89] and Shorey and Stewart [84] investigated the equation

G_{n} = x^{q} + c,  (2.5)

where c ∈ ℤ and (G_{n}) is a simple, nondegenerate second order recurring sequence of rational integers. Assuming |A_{2}| = 1, they showed for the integral solutions of (2.5) that the maximum of n ≥ 0, |x| > 1 and q ≥ 3 is less than an effectively computable constant depending on c, the coefficients and the initial values of the recurrence. In the case q = 2 they obtained a similar result under additional technical conditions. Furthermore, Shorey and Stewart [84] proved, that if α and β are multiplicatively independent with one root inside the unit circle, then (2.5) has only finitely many solutions in integers n, x and q with n ≥ 0, |x| > 1 and q > 2.

Nemes and Pethő [67], [68] studied the more general equation

G_{n} = Ex^{q} + T (x),  (2.6)

where T (x) is a polynomial of degree r and of height H with integral coefficients. For fixed E ∈ ℤ and T they established bounds for the integral solutions n, q, x with |x|, q > 1. Let (G_{n}) be defined as in (2.4) and assume

|α_{1}| > |α_{2}| > |α_{j}|, for j = 3, …, t,  (2.7)

with α_{2} ≠ ±1. Nemes and Pethő showed that q < C_{1} provided that n > C_{2} and r < C_{3}q, where C_{1}, C_{2} and C_{3} are suitable positive numbers which are effectively computable in terms of E, H and the coefficients and initial values of the recurrence. Nemes and Pethő were also able to show that if q is a fixed integer larger than one and (2.6) has infinitely many integral solutions n and x, then T (x) can be characterized in terms of the Chebyshev polynomials. Kiss [58] and Shorey and Stewart [84] dealt with equation (2.6) for nondegenerate linear recurring sequences (G_{n}) of arbitrary order, under condition (2.7) and the additional assumptions that E = 1 and d is the degree of α_{1} over ℚ, α_{1} and α_{2} are multiplicatively independent and α_{2} ≠ ±1. Then they showed that there are only finitely many integers n, x and q with n ≥ 0, |x| > 1 and

q > max (\frac{d log |α_{1}|}{log(|α_{1}|/ max(1, |α_{2}|))}, d + r)

for which (2.6) holds.

Recently Corvaja and Zannier [20] considered linear recurrences defined by

G_{n} = a_{1}α_{1}^{n} + a_{2}α_{2}^{n} + ⋯ + a_{t}α_{t}^{n},

where t ≥ 2, a_{1}, a_{2}, …, a_{t} are nonzero rational numbers, α_{1} > α_{2} > ⋯ > α_{t} > 0 are integers. They used Schmidt’s Subspace Theorem [80], [81] to show that for every integer q ≥ 2 the equation

G_{n} = x^{q}  (2.8)

has only finitely many solutions (n, x) ∈ ℕ^{2} assuming that G_{n} is not identically a perfect qth power for any n in a suitable arithmetic progression.

2.2 Results

Our first main result gives a quantitative version of the above result of Corvaja and Zannier [20].

Theorem 2.1. Let (G_{n}) be a linear recurring sequence defined by

G_{n} = a_{1}α_{1}^{n} + a_{2}α_{2}^{n} + ⋯ + a_{t}α_{t}^{n},  (2.9)

where t ≥ 2, a_{1}, a_{2}, …, a_{t} are nonzero rational numbers, α_{1} > α_{2} > ⋯ > α_{t} > 0 are integers and such that for given q ≥ 2 there is no r ∈ {0, …, q−1} with G_{mq+r} a perfect qth power for all m ∈ ℕ. Then the number of solutions (n, x) ∈ ℕ^{2} of the equation

G_{n} = x^{q}

is finite and can be bounded above by an explicitly computable number depending on q, a_{1}, a_{2}, …, a_{t}, α_{1}, …, α_{t}.

Remark 2.1. Corvaja and Zannier [20] showed that (2.9) is the qth power of an integer for infinitely many n ∈ ℕ, if and only if there exist integers r ∈ {0, …, q − 1}, b ≥ 1 and

H_{n} = c_{1}β_{1}^{n} + … + c_{s}β_{s}^{n},

where c_{1}, …, c_{s} are nonzero rational numbers and β_{1} > … > β_{s} > 0 are integers as above, such that

G_{n} = b^{n−r}H_{n}^{q}.

In particular, at least one of the functions m ↦ G_{mq+r}, (r = 0, …, q −1) is a qth power in the ring of complex functions (with pointwise multiplication) of the form (2.9), or G_{n} is a perfect qth power for any n in a suitable arithmetic progression.

Remark 2.2. Observe that one can effectively determine whether G_{mq+r} is a perfect qth power or not (see again [20]). A sufficient condition is that α_{1}, α_{2} are coprime.

Remark 2.3. The example

G_{n} = 18^{n} + 2 · 6^{n} + 2^{n}

shows that the condition in Theorem 2.1 that G_{mq+r} not be a perfect qth power for every m can not be removed. Indeed, in this example the coefficients and roots of G_{n} satisfy the conditions of Theorem 2.1, but

G_{2m} = (18^{m} + 2^{m})^{2},

so G_{2m} is a perfect square for all m ∈ ℕ.

Remark 2.4. The assumption α_{1} > α_{2} > ⋯ > α_{t} > 0 guarantees that the recurring sequence (G_{n}) is nondegenerate.

Remark 2.5. We want to mention that the proof of Theorem 2.1 should also work in the case when (G_{n}) is a linear recurring sequence with algebraic characteristic roots α_{1}, …, α_{t}, which are multiplicatively independent and satisfy

|α_{1}| > |α_{i}|, ∀i = 2, …, t,

and with a_{i} ∈ ℚ(α_{1}, …, α_{t}) for all i = 1, …, t.

Our second main result extends Theorem 2.1 to the situation when also q is considered to be variable.

Theorem 2.2. Let (G_{n}) be a linear recurring sequence defined by

G_{n} = a_{1}α_{1}^{n} + a_{2}α_{2}^{n} + ⋯ + a_{t}α_{t}^{n},

where a_{1}, …, a_{t} (t ≥ 3) are nonzero rational numbers, α_{1} > … > α_{t} > 0 are integers and such that (for fixed q ≥ 2) there is no r ∈ {0, …, q − 1} with G_{mq+r} a perfect qth power for all m ∈ ℕ. Then the equation

G_{n} = x^{q}

has only finitely many integral solutions n, x > 1, q. The number of solutions can be bounded by an explicitly computable constant C depending only on the recurrence.

Remark 2.6. The assumption t ≥ 3 means no loss of generality, because for t = 2 this theorem is already well known (see [69], [83]).

Remark 2.7. Our proof of Theorem 2.2 depends on an application of the result of Nemes and Pethő [67] which was mentioned in the introduction in detail.

Remark 2.8. By Remark 2.2 and the theorem in [67] it should also be possible to obtain the following finiteness result: Let G_{n} be the nth term of a linear recurrence sequence defined by (2.9), where t ≥ 3, α_{1} …, α_{t} are multiplicatively independent algebraic numbers with

|α_{1}| > |α_{2}| > |α_{i}|, ∀i = 3, …, t,

and a_{i} ∈ ℚ(α_{1}, …, α_{t}) for all i = 1 …, t. Assuming that for fixed q ≥ 2 there is no r ∈ {0, …, q − 1} with G_{mq+r} a perfect qth power for all m ∈ ℕ, the equation

G_{n} = x^{q}

has only finitely many integral solutions n, x > 1, q.

2.3 Auxiliary results

We have collected some simple lemmas which are needed in our proofs.

Lemma 2.1. Let N_{j,k} denote the number of formal summands of (a_{1}+⋯+a_{k})^{j}, where a_{1}, …, a_{k} denote formal commuting variables. Then

N_{j,k} = \binom{k + j − 1}{j}.

This is well known from combinatorics.

Lemma 2.2. Let d be a positive integer. Then for the complex function f(z) = (1 + z)^{1/d} we have

|f(z) − ∑_{k=0}^{n} \binom{1/d}{k}z^{k}| ≤ \frac{1}{d(n + 1)(1 − |z|)} · |z|^{n+1}

for z ∈ ℂ, |z| < 1, where we have chosen the branch of (1 + z)^{1/d} which is holomorphic on ℂ\(−∞, −1] and which is equal to the positive d-th root of (1+z) for z ∈ ℝ, z > −1.

Proof. It is well-known that for z ∈ ℂ, |z| < 1 we have

f(z) = ∑_{k=0}^{∞} \binom{1/d}{k}z^{k}.

Because of

(n + 1) |\binom{1/d}{n + 1}| = \frac{\frac{1}{d} · (1 − \frac{1}{d}) · . . . · (n − \frac{1}{d})}{1 · . . . · n} < \frac{1}{d} < 1

we obtain

|f(z) − ∑_{k=0}^{n} \binom{1/d}{k}z^{k}| ≤ ∑_{k=n+1}^{∞} |\binom{1/d}{k}| · |z|^{k} ≤

≤ \frac{1}{d(n + 1)} ∑_{k=n+1}^{∞} |z|^{k} =
= \frac{1}{d(n + 1)(1 − |z|)} · |z|^{n+1},

and therefore the proof is complete. □

Lemma 2.3. Let a, b ≥ 0 and let x ∈ ℝ be the largest solution of x = a + b log x. If b > e^{2} then

x < 2 (a + b log b) .

This lemma is due to A. Pethő and B.M.M. de Weger [86].

2.4 Proof of Theorem 2.1

According to Theorem 1.9 the number of solutions of (2.8) of the form (n, 0), n ∈ ℕ can be estimated by

c_{1}(t) ≤ e^{(7t)^{8t}}.

Therefore we can restrict ourselves to solutions of the form (n, x) ∈ ℕ^{2} with x ≠ 0. These solutions are denoted by (n, x_{n}) ∈ ℕ^{2} with n ∈ Σ, where Σ is a set of positive integers.

Let us now consider the expansion of the function f(z) = (1 + z)^{1/q} around the origin

(1 + z)^{1/q} = ∑_{j=0}^{∞} \binom{1/q}{j}z^{j}, with |z| ≤ 1, z ≠ −1.

We approximate G_{n}^{1/q} by defining

H_{m} := (a_{1}α_{1}^{r})^{1/q} · α_{1}^{m} [1 + ∑_{j=1}^{R} \binom{1/q}{j} · (∑_{i=2}^{t} \frac{a_{i}α_{i}^{mq+r}}{a_{1}α_{1}^{mq+r}} )^{j}] ,

where R ≥ 1 is an integer to be chosen later and where we have set n = mq + r with n ∈ ℕ, r ∈ {0, …, q − 1}. We write

H_{m} = ∑_{i=1}^{h} d_{i} (\frac{e_{i}}{b} )^{m} ,

where d_{i} ∈ ℚ ((a_{1}α_{1}^{r})^{1/q})^{∗}, e_{i}, b are integers, b > 0, and the e_{i}/b are nonzero distinct rational numbers. Clearly, H_{m} is nondegenerate (the roots are all positive) and we have

[ℚ ((a_{1}α_{1}^{r})^{1/q}) : ℚ] ≤ q.

By Lemma 2.1 we obtain

h ≤ \binom{R + t − 1}{R}.

On the other hand, we have

|∑_{i=2}^{t} \frac{a_{i}α_{i}^{mq+r}}{a_{1}α_{1}^{mq+r}} | ≤ (t − 1)c (\frac{α_{2}}{α_{1}} )^{mq+r} ≤ \frac{1}{2} < 1

where

c := max \{|\frac{a_{i}}{a_{1}} | | i = 2, …, t\} ,

if

m ≥ \frac{log 2(t − 1)c}{q log \frac{α_{1}}{α_{2}}}.  (2.10)

Therefore, by Lemma 2.2 for m we get for m large enough

|H_{m} − x_{mq+r}| = |G_{mq+r}^{1/q} − H_{m}| ≤
≤ |a_{1}^{1/q}| · α_{1}^{r/q} · α_{1}^{m} · \frac{1}{q(R + 1) (1 − |∑_{i=2}^{t} \frac{a_{i}α_{i}^{mq+r}}{a_{1}α_{1}^{mq+r}} |)} |∑_{i=2}^{t} \frac{a_{i}α_{i}^{mq+r}}{a_{1}α_{1}^{mq+r}} |^{R+1} ≤

≤ |a_{1}^{1/q}| · α_{1} · α_{1}^{m} · \frac{2}{q(R + 1)} [(t − 1)c (\frac{α_{2}}{α_{1}} )^{r} (\frac{α_{2}}{α_{1}} )^{mq}]^{R+1} ≤
≤ |a_{1}^{1/q}| · α_{1}^{r/q} · [(t − 1)c]^{R+1} · α_{1}^{m} · (\frac{α_{2}}{α_{1}} )^{mq(R+1)} .

Thus we derive

|H_{m} − x_{mq+r}| ≤ c_{2}(R) · l_{1}^{m}  (2.11)

where we have set

l_{1} := α_{1} · (\frac{α_{2}}{α_{1}} )^{q(R+1)}

and

c_{2}(R) := |a_{1}^{1/q}| · α_{1} · [(t − 1)c]^{R+1} .

Now choose

R > max \{1, \frac{1}{q} \frac{log α_{1}}{log \frac{α_{1}}{α_{2}}} − 1\} .  (2.12)

Then 0 < l_{1} < 1. Put

l := \frac{l_{1} + 1}{2}.

Then for m large enough, to be more precise, for

m > \frac{log c_{2}(R)}{log \frac{l}{l_{1}}}  (2.13)

we have

c_{2}(R) · l_{1}^{m} = c_{2}(R) (\frac{l_{1}}{l} )^{m} · l^{m} < l^{m}.

Consequently we obtain

|H_{m} − x_{mq+r}| < l^{m} with 0 < l < 1,  (2.14)

provided that R satisfies (2.12) and m satisfies (2.10) and (2.13).

Now let S be the set of places of ℚ consisting of ∞ and all primes dividing some of the e_{i} or b. Extend each place in S to K := ℚ ((a_{1}α_{1}^{r})^{1/q}) in some way, the infinite place being extended such that it coincides with the complex absolute value in the given embedding of K in ℂ. Define the linear forms L_{i,v} for v ∈ S and i = 1, …, h as follows: L_{0,∞} := L := X_{0} − ∑_{i=1}^{h} d_{i}X_{i}, L_{i,∞} := X_{i} for i = 1, …, h, while for v ∈ S, v ≠ ∞, put L_{i,v} := X_{i} for i = 1, …, h. Then {L_{0,v}, …, L_{h,v}}, v ∈ S are linearly independent sets of linear forms in h + 1 variables with coefficients in K. Furthermore we have

H(L_{i,v}) = \{ \begin{matrix}H(1, −d_{1}, …, −d_{h}) =: H̃ & for i = 0, v = ∞ \\ H(0, …, 0, 1, 0, …, 0) = 1 & else. \end{matrix}

We set H := max{1, H̃} then it follows H(L_{i,v}) ≤ H, for v ∈ S, i = 0, …, h. Then ℚ(L_{i,v}) = ℚ, for v ≠ ∞, [ℚ(L_{0,∞}) : ℚ] ≤ q and therefore

[ℚ(L_{i,v}) : ℚ] ≤ q ∀v ∈ S, i = 0, …, h.

For n ∈ Σ define the vector x_{m} = (b^{m}x_{mq+r}, e_{1}^{m}, …, e_{h}^{m}) ∈ ℤ^{h+1}.

From (2.14) we obtain |L_{0,∞}(x_{m})| ≤ (bl)^{m}. Recall that S includes all primes dividing b and that the x_{mq+r} are integers. Thus by the product formula (1.3),

∏_{v∈S\{∞}} |L_{0,v}(x_{m})|_{v} = ∏_{v∈S\{∞}} |b^{m}x_{mq+r}|_{v} ≤ ∏_{v∈S\{∞}} |b^{m}|_{v} = b^{−m}.

Moreover, since S includes also the primes dividing the numbers e_{i} the product formula (1.3) gives

∏_{v∈S} ∏_{i=1}^{h} |L_{i,v}(x_{m})|_{v} = ∏_{v∈S} ∏_{i=1}^{h} |e_{i}^{m}|_{v} = 1.

Thus we obtain

∏_{v∈S} ∏_{i=0}^{h} \frac{|L_{i,v}(x_{m})|_{v}}{|x_{m}|_{v}} ≤ (∏_{v∈S} |x_{m}|_{v})^{−h−1} · l^{m}.

Since the coordinates of the vectors x_{m} are integers we have |x_{m}|_{v} ≤ 1 for v ∈ M_{ℚ}\{∞}. Further, we have

|x_{m}|_{∞} ≤ A^{m}

for some real A independent of m. Indeed, we have

|x_{mq+r}| ≤ |x_{mq+r} − H_{m}| + |H_{m}| ≤ l^{m} + h · c̃ · a^{m} ≤ 1 + h · c̃ · a^{m} ≤ ã^{m},

with

c̃ := max{|d_{i}| | i = 1, …, h},
a := max \{|\frac{e_{i}}{b} | | i = 1, …, h\} ,

and ã := (1 + h · c̃)(1 + a). Hence

|x_{m}|_{∞} = (|b^{m}x_{mq+r}|^{2} + ∑_{i=1}^{h} |e_{i}|^{2})^{1/2} ≤
≤ ((bã)^{2m} + h(ba)^{2m})^{1/2} ≤ A^{m}

with A := (h + 1)ãb. It follows that

H(x_{m}) = ∏_{v∈M_{ℚ}} |x_{m}|_{v} ≤ ∏_{v∈S} |x_{m}|_{v} ≤ |x_{m}|_{∞} ≤ A^{m}.

Lastly we have

det(L_{0,v}, …, L_{h,v}) = | \begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ ∗ & 1 & 0 & ⋯ & 0 \\ ∗ & 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ ∗ & 0 & 0 & ⋯ & 1 \end{matrix} | = 1,

which yields

| det(L_{0,v}, …, L_{h,v})|_{v} = 1 ∀v ∈ S.

Combining our estimates we get

∏_{v∈S} ∏_{i=0}^{h} \frac{|L_{i,v}(x_{m})|_{v}}{|x_{m}|_{v}} < (∏_{v∈S} | det(L_{0,v}, …, L_{h,v})|_{v}) · H(x_{m})^{−h−1−δ}

for all m with (2.13), provided that δ < log(1/l)/ log A. By Theorem 1.5 there exist finitely many nonzero rational linear forms Λ_{1}(X_{0}, …, X_{h}), …, Λ_{g}(X_{0}, …, X_{h}) with

g ≤ (2^{60(h+1)^{2}} · δ^{−7(h+1)})^{s} log 4q · log log 4q+
+(150(h + 1)^{4} · δ^{−1})^{(h+1)s+1}(2 + log log 2H),

such that each vector x_{m} is a zero of some Λ_{j}.

Suppose first Λ_{j} does not depend on X_{0}. Then, if Λ_{j}(x_{m}) = 0, we have a nontrivial relation

∑_{i=1}^{h} u_{i} (\frac{e_{i}}{b} )^{m} = 0, u_{i} ∈ ℚ, i = 1, …, h.

By Theorem 1.9 this can hold for at most a finite number of m. More precisely, the number of solutions m can be estimated by

c_{1}(h) = e^{(7h)^{8h}},

since (H_{m}) is nondegenerate.

Suppose that Λ_{j} depends on X_{0} and that Λ_{j}(x_{m}) = 0. Then we have

∑_{i=1}^{h} v_{i} (\frac{e_{i}}{b} )^{m} = x_{mq+r}, v_{i} ∈ ℚ, i = 1, …, h.  (2.15)

Set

U_{m} := ∑_{i=1}^{h} v_{i} (\frac{e_{i}}{b} )^{m} ,

then U_{m} is a nondegenerate, simple recurring sequence and we obtain

U_{m}^{q} = x_{mq+r}^{q} = G_{mq+r}.

Hence

V_{m} := (∑_{i=1}^{h} v_{i} (\frac{e_{i}}{b} )^{m})^{q} − ∑_{i=1}^{t} a_{i}α_{i}^{r}(α_{i}^{q})^{m},

has the form

V_{m} = ∑_{i=1}^{p} b_{i}β_{i}^{m}

with b_{i} ∈ ℚ, β_{i} ∈ ℚ^{+}, i = 1, …, p. Therefore V_{m} is a nondegenerate, simple recurring sequence, and we conclude by Lemma 2.1

p ≤ t + \binom{h + q − 1}{q} ≤ t + \binom{\binom{R+t−1}{t} + q − 1}{q}.

Observe that by our assumptions V_{m} = 0 does not hold for every m hence an i with b_{i} ≠ 0 exists. Again by Theorem 1.9 we can bound the number of solutions of (2.15) by

c_{1}(p) = e^{(7p)^{8p}}.

Therefore the number of solutions of (2.8) can be estimated by

C̃(q) :=

= e^{(7t)^{8t}} + \frac{log 2(t − 1)c}{log \frac{α_{1}}{α_{2}}} + q · [\{(2^{60(h̃+1)^{2}} · δ^{−7(h̃+1)})^{s} ·

· log 4q · log log 4q + (150(h̃ + 1)^{4} · δ^{−1})^{(h̃+1)s+1} · (2 + log log 2H)} ·

· \{e^{(7h̃)^{8h̃}} + e^{(7p̃)^{8p̃}}\} + \frac{log c_{2}(R)}{log \frac{l}{l_{1}}}],

where

h̃ = \binom{R + t − 1}{R}, p̃ = \binom{h̃ + q − 1}{q} + t

H = max{1, H(1, −d_{1}, …, −d_{h})}

s = |S|,

c_{2}(R) = |a_{1}^{1/q}| · α_{1} · [(t − 1)c]^{R+1} ,

δ < log(1/l)/ log A,

and l is as in (2.14). □

2.5 Proof of Theorem 2.2

Assume that n, x > 1, q ≥ 2 is a solution of (2.8) and write

x^{q} = G_{n} = a_{1}α_{1}^{n} + B(n).

We distinguish two cases:

Case 1: B(n) = 0. Here we get

|B(n)| = |a_{2}α_{2}^{n} (1 + ∑_{i=3}^{t} \frac{a_{i}}{a_{2}} (\frac{α_{i}}{α_{2}} )^{n})| ≥ |a_{2}|α_{2}^{n} |1 − |∑_{i=3}^{t} \frac{a_{i}}{a_{2}} (\frac{α_{i}}{α_{2}} )^{n}|| > 0,

since

|∑_{i=3}^{t} \frac{a_{i}}{a_{2}} (\frac{α_{i}}{α_{2}} )^{n}| ≤

≤ max \{|\frac{a_{i}}{a_{2}} | | i = 3, …, t\} · (t − 2) · (\frac{α_{3}}{α_{2}} )^{n} ≤
≤ tc|a_{2}|^{−1} (\frac{α_{3}}{α_{2}} )^{n} < 1,

where c = max{|a_{i}| | i = 1, …, t}, whenever

n > \frac{log(tc|a_{2}|^{−1})}{log \frac{α_{2}}{α_{3}}} =: n_{1}.

Therefore n ≤ n_{1} must hold and we obtain from a_{1}α_{1}^{n} = x^{q} and x ≥ 2

q = \frac{log(|a_{1}|α_{1}^{n})}{log x} ≤ \frac{log(cα_{1}^{n_{1}})}{log 2}.

Case 2: B(n) ≠ 0. In this case we first set

δ := \frac{1}{2} (1 − \frac{log α_{2}}{log α_{1}} ) .

Then we get

|B(n)| ≤ tcα_{2}^{n} < \frac{1}{2}α_{1}^{n(1−δ)},

if

n > \frac{2 log(2ct)}{log \frac{α_{1}}{α_{2}}} =: n_{2}.

Further

\frac{x^{q}}{a_{1}α_{1}^{n}} = 1 + \frac{B(n)}{a_{1}α_{1}^{n}},  (2.16)

so

1 − (|a_{1}|α_{1}^{δn})^{−1} ≤ |a_{1}|^{−1}α_{1}^{−n}x^{q} ≤ 1 + (|a_{1}|α_{1}^{δn})^{−1},  (2.17)

where we have used that (|a_{1}|α_{1}^{δn})^{−1} < \frac{1}{2}, if

n > \frac{log(2|a_{1}|^{−1})}{δ log α_{1}} =: n_{3}.

Taking logarithms and using the inequalities | log(1 + x)| ≤ x and | log(1 − x)| ≤ 2x for 0 ≤ x < \frac{1}{2}, we derive by (2.17)

−2|a_{1}|^{−1}α_{1}^{−δn} ≤ − log |a_{1}| − n log α_{1} + q log x ≤ 2|a_{1}|^{−1}α_{1}^{−δn}.

Thus

| − log |a_{1}| − n log α_{1} + q log x| ≤ 2|a_{1}|^{−1}α_{1}^{−δn}.  (2.18)

Put Λ = − log |a_{1}| − n log α_{1} + q log x. From (2.16) and the fact that B(n) ≠ 0, we get Λ ≠ 0. Thus we can employ Theorem 1.12 and obtain for n ≥ 2

|Λ| > exp \{−C(3)h_{1}h_{2} log (C(3)h_{1}h_{2}) e log x log q − \frac{n}{q}\} ,  (2.19)

where C(3) = 2^{78}3^{9} and

h_{1} = max{h(|a_{1}|^{−1}), e | log |a_{1}||, 1},
h_{2} = max{h(α_{1}), e log α_{1}, 1} = e log α_{1}.

Set

c_{3} := C(3)h_{1}h_{2} log (C(3)h_{1}h_{2}) e.

A comparison of (2.18) and (2.19) reveals that

−c_{3} log q log x − \frac{n}{q} < log(2|a_{1}|^{−1}) − nδ log α_{1}.  (2.20)

However, for n > max{n_{1}, n_{2}},

\frac{1}{2}|a_{1}|α_{1}^{n} ≤ |a_{1}|α_{1}^{n} − |B(n)| ≤ x^{q} ≤ |a_{1}|α_{1}^{n} + |B(n)| ≤ ctα_{1}^{n}.

Thus, for

n > max \{\frac{log ct}{log α_{1}} , \frac{2 log(2|a_{1}|^{−1})}{log α_{1}}\} =: n_{4}

CHAPTER 2. PERFECT POWERS IN RECURRENCES 33 we obtain

\frac{log α_{1}}{2} n < q log x < 2 log α_{1}n.

Writing this as

\frac{n}{q} < \frac{2 log x}{log α_{1}}, \frac{log x}{2 log α_{1}} q < n,

(2.20) can be reformulated as

q < \frac{2 log(\frac{1}{2}|a_{1}|^{−1})}{δ log 2} + \frac{4}{δ log α_{1}} + \frac{2c_{3}}{δ} log q.

Thus by Lemma 2.3

q < 2 (\frac{2 log(\frac{1}{2}|a_{1}|^{−1})}{δ log 2} + \frac{4}{δ log α_{1}} + \frac{2c_{3}}{δ} log (\frac{2c_{3}}{δ} )) =: C_{1},

if n > max{2, n_{2}, n_{3}, n_{4}} =: n_{5}. Otherwise, we have

q ≤ \frac{log (ctα_{1}^{n_{5}})}{log 2}.

Altogether we derive

q ≤ max \{\frac{log (ctαc̄ ; 1)}{log 2}, C_{1}(q)\} =: C,

where c̄:= max{2, n_{1}, n_{2}, n_{3}, n_{4}}. For the number of solutions n, x > 1, q ≥ 2 of (2.8) we finally obtain the upper bound

∑_{q=2}^{C} C̃(q),

and therefore the proof is finished. □

Exponential–polynomial equations and linear recurrences

Let K be an algebraic number field and let (G_{n}) be a linear recurring sequence defined by G_{n} = λ_{1}α_{1}^{n} + P_{2}(n)α_{2}^{n} + ⋯ + P_{t}(n)α_{t}^{n}, where λ_{1}, α_{1}, …, α_{t} are nonzero elements of K and where P_{i}(x) ∈ K[x] for i = 2, …, t. Furthermore let f(z, x) ∈ K[z, x] monic in x. In this chapter we want to study the exponential–polynomial Diophantine equation f(G_{n}, x) = 0. We want to use a quantitative version of W. M. Schmidt’s Subspace Theorem (due to J.-H. Evertse [40]) to calculate an upper bound for the number of solutions (n, x) under some additional assumptions.

This chapter is similar to a preprint of my paper [48].

3.1 Introduction

Let A_{1}, A_{2}, …, A_{k} and G_{0}, G_{1}, …, G_{k−1} be algebraic numbers over the rationals and let (G_{n}) be a k-th order linear recurring sequence given by

G_{n} = A_{1}G_{n−1} + ⋯ + A_{k}G_{n−k} for n = k, k + 1, ….  (3.1)

Let α_{1}, α_{2}, …, α_{t} be the distinct roots of the corresponding characteristic polynomial

X^{k} − A_{1}X^{k−1} − ⋯ − A_{k}.  (3.2)

Then for n ≥ 0

G_{n} = P_{1}(n)α_{1}^{n} + P_{2}(n)α_{2}^{n} + ⋯ + P_{t}(n)α_{t}^{n},  (3.3)

where P_{i}(n) is a polynomial with degree less than the multiplicity of α_{i}; the coefficients of P_{i}(n) are elements of the field: ℚ(G_{0}, …, G_{k−1}, A_{1}, …, A_{k}, α_{1}, …, α_{t}). We shall be interested in linear recurring sequences (G_{n}), where G_{n} defined as in (3.3) for which P_{1}(n) is a nonzero constant, λ_{1} say. Thus

G_{n} = λ_{1}α_{1}^{n} + P_{2}(n)α_{2}^{n} + ⋯ + P_{t}(n)α_{t}^{n}.  (3.4)

The recurring sequence is called simple, if all characteristic roots are simple. (G_{n}) is called nondegenerate, if no quotient α_{i}/α_{j} for all 1 ≤ i < j ≤ t is equal to a root of unity and degenerate otherwise. Observe that, even if (G_{n}) is degenerate, there exists a positive integer d such that, (G_{r+md}) is nondegenerate on each of the d arithmetic progressions with 0 ≤ r < d. Therefore, restricting to nondegenerate recurring sequences causes no substantial loss of generality.

Let f ∈ ℚ[z, x] be a polynomial, which is monic in x. In the present chapter we deal with the Diophantine equation

f(G_{n}, x) = 0,  (3.5)

which was earlier investigated by several authors in the special case f(z, x) = Ex^{q} − z, E ∈ ℤ\{0}, which yields the Diophantine equation

G_{n} = Ex^{q}, E ∈ ℤ\{0}.  (3.6)

A survey about this equation can be found in [51] (see also Chapter 2). We cite here only those papers, which are of interest for us. We will repeat some results also we have stated them already in the introduction of the previous chapter.

Let us repeat the following finiteness result due to Shorey and Stewart [83]. Let (G_{n}) be an integral nondegenerate linear recurring sequence given by

G_{n} = λ_{1}α_{1}^{n} + P_{2}(n)α_{2}^{n} + ⋯ + P_{t}(n)α_{t}^{n},  (3.7)

where λ_{1} is a nonzero constant, |α_{1}| > |α_{j}| for j = 2, …, t, and G_{n} − λ_{1}α_{1}^{n} ≠ 0. Then assuming x, q > 1 the solutions q of (3.6) can be bounded by an effectively computable constant which depends on E and the coefficients and initial values of the recurrence. Kiss [58] proved that, in fact, q is less than a number which is effectively computable in terms of the greatest prime divisor of E and the coefficients and the initial values of the sequence (G_{n}).

Nemes and Pethő [67], [68] studied the more general equation

G_{n} = Ex^{q} + T (x),  (3.8)

where T (x) is a polynomial of degree r and of height H with integral coefficients. For fixed E ∈ ℤ and T they established bounds for the integral solutions n, q, x with |x|, q > 1. Let (G_{n}) be defined as in (3.7) and assume

|α_{1}| > |α_{2}| > |α_{j}|, for j = 3, …, t,  (3.9)

with α_{2} ≠ ±1. Nemes and Pethő showed that q < C_{1} provided that n > C_{2} and r < C_{3}q, where C_{1}, C_{2} and C_{3} are suitable positive numbers which are effectively computable in terms of E, H and the coefficients and initial values of the recurrence. Kiss [58] and Shorey and Stewart [84] dealt with equation (3.8) for nondegenerate linear recurring sequences (G_{n}) of arbitrary order, under condition (3.9) and the additional assumptions that d is the degree of α_{1} over ℚ, α_{1} and α_{2} are multiplicatively independent and α_{2} ≠ ±1. Then they showed that there are only finitely many integers n, x and q with n ≥ 0, |x| > 1 and

q > max (\frac{d log |α_{1}|}{log(|α_{1}|/ max(1, |α_{2}|))}, d + r)

for which

G_{n} = x^{q} + T (x)

holds.

Recently Corvaja and Zannier [20] considered linear recurrences defined by

G_{n} = a_{1}α_{1}^{n} + a_{2}α_{2}^{n} + ⋯ + a_{t}α_{t}^{n},

where t ≥ 2, a_{1}, a_{2}, …, a_{t} are nonzero rational numbers, α_{1} > α_{2} > ⋯ > α_{t} > 0 are integers. They used Schmidt’s Subspace Theorem [80], [81] to show that for every integer q ≥ 2 the equation

G_{n} = x^{q}  (3.10)

has only finitely many solutions (n, x) ∈ ℕ^{2} assuming that G_{n} is not identically a perfect qth power for any n in a suitable arithmetic progression. Tichy and the author [51] gave a quantitative version of the above result of Corvaja and Zannier (cf. Theorem 2.1).

Tichy and the author [51] also showed by combining their result with the previously mentioned result of Nemes and Pethő [67] that the following is true (cf. Theorem 2.2): Let (G_{n}) be a linear recurring sequence defined as above, such that (for fixed q ≥ 2) there is no r ∈ {0, …, q − 1} with G_{mq+r} a perfect qth power for all m ∈ ℕ. Then the equation

G_{n} = x^{q}

has only finitely many integral solutions n, x > 1, q. The number of solutions can be bounded by an explicitly computable constant C depending only on the recurrence.

Very recently, Pethő [71] used the above result of Corvaja and Zannier to show that there are only finitely many perfect powers in a third order linear recurring sequence G_{n}, if we assume that the characteristic polynomial of G_{n} is irreducible and has a dominating root.

3.2 Results

Our main result is the generalization of the above quantitative result to the Diophantine equation f(G_{n}, x) = 0, where (G_{n}) is defined by (3.4). This will generalize and quantify a very resent result due to Corvaja and Zannier [21].

Theorem 3.1. Let K be an algebraic number field and let (G_{n}) be a linear recurring sequence defined by

G_{n} = λ_{1}α_{1}^{n} + P_{2}(n)α_{2}^{n} + ⋯ + P_{t}(n)α_{t}^{n},

where t ≥ 2, λ_{1} is a nonzero elements of K, P_{i}(x) ∈ K[x] for all i = 2, …, t and where α_{1}, …, α_{t} are multiplicatively independent numbers with 1 ≠ |α_{1}| > |α_{j}| for all j = 2, …, t. Let f(z, x) ∈ K[z, x] be monic in x and suppose that there do not exist nonzero algebraic (over K) numbers β_{j} and polynomials d_{j}(n) ∈ \overline{K}[n] for j = 1, …, k such that

f(G_{n}, ∑_{j=1}^{k} d_{j}(n)β_{j}^{n}) = 0  (3.11)

for all n in an arithmetic progression. Then the number of solutions (n, x) ∈ ℕ × K of the equation

f(G_{n}, x) = 0

is finite and can be bounded by an explicitly computable number C depending on f and on the coefficients and the initial values of the recurrence.

Remark 3.1. Corvaja and Zannier showed in [21], under the restriction that the recurrence is simple, that the assumption of f(G_{n}, x) = 0 having infinitely many solutions implies that there exist d_{j}, β_{j} ∈ \overline{K}^{×}, j = 1, …, k, and an arithmetic progression P such that

f(G_{n}, ∑_{j=1}^{k} d_{j}β_{j}^{n}) = 0, for n ∈ P.

Remark 3.2. The assumption that the roots of the recurrence α_{1}, …, α_{t} are multiplicatively independent is also necessary for our proof. It would be possible to relax this condition. In fact, we only need that certain linear recurrences, where the roots lie in the multiplicative group generated by powers of the α_{i}, are nondegenerate.

In fact, the proof of Theorem 3.1 does also work in the case when the roots are different positive (rational) integers.

Theorem 3.2. Let (G_{n}) be a linear recurring sequence defined by

G_{n} = λ_{1}α_{1}^{n} + P_{2}(n)α_{2}^{n} + ⋯ + P_{t}(n)α_{t}^{n},

where t ≥ 2, λ_{1} is a nonzero rational numbers, P_{i}(x) ∈ ℚ[x] for all i = 2, …, t and where α_{1} > α_{2} > … > α_{t} > 0 are integers. Let f(z, x) ∈ ℚ[z, x] be monic in x and suppose that there do not exist nonzero algebraic (over ℚ) numbers β_{j} and polynomials d_{j}(n) ∈ \overline{ℚ}[n] for j = 1, …, k such that

f(G_{n}, ∑_{j=1}^{k} d_{j}(n)β_{j}^{n}) = 0

for all n in an arithmetic progression. Then the number of solutions (n, x) ∈ ℕ × ℚ of the equation

f(G_{n}, x) = 0

is finite and can be bounded by an explicitly computable number C depending on f and on the coefficients and the initial values of the recurrence.

Remark 3.3. Observe that one can effectively determine whether there do exist nonzero numbers β_{j} ∈ \overline{K} and polynomials d_{j}(n) ∈ \overline{K}[n] for j = 1, …, k such that (3.11) holds for all n in an arithmetic progression or not (see [21]). This will also follow from the proof of Theorem 3.1. The following example shows that this condition is necessary: let

G_{n} = 18^{n} + 2 · 6^{n} + 2^{n},

and f(z, x) = x^{2} − z. The coefficients and roots have the desired properties, but

G_{2k} = (18^{k} + 2^{k})^{2},

so f(G_{2k}, 18^{k} + 2^{k}) = 0 for all k ∈ ℕ. Another much simpler example is obtained by taking f(z, x) = x − z.

Remark 3.4. We want to note that the above assumption (3.11) also means (for k = 0) that f(z, 0) = 0 does not hold identically or equivalently x is not a divisor of f(z, x). For example f(z, x) = x − z · x, which yields solutions (n, 0) ∈ ℕ^{2} for all n ∈ ℕ, is excluded.

Remark 3.5. Let us mention that the condition on the dominant root α_{1} is crucial. The proof of the theorem heavily depends on that assumption.

Remark 3.6. For simplicity we have introduced the condition that f(z, x) is monic in x. There is no problem at all, if we assume that the leading coefficient of f with respect to x does not depend on z. Moreover, it is a well-known trick how to get rid of the this assumption (with a corresponding modification of the theorem); namely we may replace f(z, x) with the polynomial a(z)^{d−1}f(z, x/a(z)), where a(z), d is the leading coefficient, the degree respectively of f with respect to x.

Next we want to state some conclusion concerning special cases of the above result.

Corollary 3.1. Let (G_{n}) be a linear recurring sequence defined by

G_{n} = λ_{1}α_{1}^{n} + P_{2}(n)α_{2}^{n} + ⋯ + P_{t}(n)α_{t}^{n},

where t ≥ 2, λ_{1} is a nonzero rational number, P_{i}(x) ∈ ℚ[x] for all i = 2, …, t and where α_{1}, …, α_{t} are multiplicatively independent rational numbers with 1 ≠ |α_{1}| > |α_{j}| for all j = 2, …, t. Let P (x) ∈ ℚ[x] be monic and suppose that there do not exist nonzero algebraic (over ℚ) numbers β_{j} and polynomials d_{j}(n) ∈ \overline{ℚ}[n] for j = 1, …, k such that

G_{n} = P ( ∑_{j=1}^{k} d_{j}(n)β_{j}^{n})  (3.12)

for all n in an arithmetic progression. Then the number of solutions (n, x) ∈ ℕ × ℚ of the equation

G_{n} = P (x)

is finite and can be bounded by an explicitly computable number C depending on P and on the coefficients and the initial values of the recurrence.

Remark 3.7. The above corollary is also true, if we assume P (x) ∈ ℚ(x), say P (x) = f(x)/g(x), where f(x) is a monic polynomial.

Remark 3.8. Also the classical case P (x) = x^{q}, concerning the number of perfect powers in the linear recurring sequence (G_{n}) is included. So we get a generalisation of the results stated in [51]. Observe that from [20] it follows that condition (3.12) is equivalent to the assumption that G_{n} is not equal to a perfect qth power for all n in an arithmetic progression (cf. also [98]).

Last we want to discuss some families of Diophantine equations related to the above types of equations.

Corollary 3.2. Let (G_{n}) be an integral linear recurring sequence with (3.4), where t ≥ 2, λ_{1} is a nonzero element of K = ℚ(α_{1}, …, α_{t}), P_{i}(x) ∈ K[x] for all i = 2, …, t and where α_{1}, …, α_{t} are multiplicatively independent algebraic integers with 1 ≠ |α_{1}| > |α_{j}| for all j = 2, …, t. Furthermore, suppose that there do not exist nonzero numbers β_{j} ∈ \overline{K} and polynomials d_{j}(n) ∈ \overline{K}[n] for j = 1, …, k such that

G_{n} = ( ∑_{j=1}^{k} d_{j}(n)β_{j}^{n})^{q}  (3.13)

for all n in an arithmetic progression and for given q ≥ 2. Then the number of solutions (n, x, q) ∈ ℕ^{3} with n, x, q > 1 of the equation

G_{n} = x^{q}

is finite and can be bounded by an explicitly computable number C depending only on the coefficients and the initial values of the recurrence.

Remark 3.9. Observe that condition (3.13) can be verified effectively, because under the other assumptions one can calculate an upper bound for q first. Then condition (3.13) must only be verified for q smaller than this bound.

Corollary 3.3. Let (G_{n}) be an integral linear recurring sequence with (3.4), where t ≥ 2, λ_{1} is a nonzero rational element of K = ℚ(α_{1}, …, α_{t}), P_{i}(x) ∈ K[x] for all i = 2, …, t and where α_{1}, …, α_{t} are multiplicatively independent algebraic integers with 1 ≠ |α_{1}| > |α_{2}| > |α_{j}| for all j = 3, …, t. Let α_{2} ≠ ±1 and let T (x) be a polynomial with integer coefficients and degree r; we take r = 0 if T (x) is the zero polynomial. Furthermore, suppose that there do not exist nonzero numbers β_{j} ∈ \overline{K} and polynomials d_{j}(n) ∈ \overline{K}[n] for j = 1, …, k such that

G_{n} = ( ∑_{j=1}^{k} d_{j}(n)β_{j}^{n})^{q} + T ( ∑_{j=1}^{k} d_{j}(n)β_{j}^{n})  (3.14)

for all n in an arithmetic progression and for given q ≥ 2. Then there are only finitely many integers n, x and q with n ≥ 0, q ≥ 1 and |x| > 1 for which

G_{n} = x^{q} + T (x)

holds.

3.3 Auxiliary results

We need some results from the theory of algebraic functions fields, which can be found in the monographs of Eichler [37] and Iwasawa [54], namely the theory of Puiseux expansions.

Let K be an algebraic number field, which is generated over the field of rational numbers ℚ. We assume that f(x, y) is an absolutely irreducible polynomial in x and y, with coefficients in the algebraic number field K, that is f is irreducible over the algebraic closure \overline{K} of K. We denote by F the field obtained by adjoining a root of f(x, y) to \overline{K}(x), the field of rational functions in x with coefficients in the algebraic closure of K. Then F is an algebraic function field over the algebraically closed field \overline{K} of characteristic 0.

Theorem 3.3. (Puiseux’s Theorem) Let F be an algebraic functions field over an algebraically closed field \overline{K} of characteristic 0, given by f(x, y) = 0. For simplicity we suppose f(x, y) to be monic in y. Let us denote by n = [F : \overline{K}(x)] the degree of F over \overline{K}(x). Then with every element ξ ∈ K there are associated r = r(ξ) ≤ n natural number e_{i} = e_{i}(ξ) whose sum is

e_{1} + ⋯ + e_{r} = n;

similar numbers e_{i}(∞) are associated with the symbol ξ = ∞. These numbers have the following meaning: Setting

z_{ξ} = x − ξ, z_{∞} = 1/x,  (3.15)

the irreducible equation f(x, y) = 0 satisfied by an arbitrary function y of F over \overline{K} has for solutions the r = r(ξ) power series

y_{i} = ∑_{k=v_{i}}^{∞} a_{ik}( ^{e_{i}(ξ)}\sqrt{z_{ξ}})^{k}, a_{iv_{i}} ≠ 0, i = 1, 2, …, r(ξ).  (3.16)

With a primitive e_{i}th root of unity ζ form

y_{ij} = ∑_{k} a_{ik}ζ^{jk}( ^{e_{i}(ξ)}\sqrt{z_{ξ}})^{k}, j = 0, …, e_{i}(ξ) − 1;  (3.17)

then the left side of f(x, y) = 0 is identical with

f(x, y) = ∏_{j,i}(y − y_{ij}).  (3.18)

The coefficients a_{ik} are elements of a finite field extension K′ of K, and their images under isomorphisms of K′ give permutations of the y_{ij} in (3.18). The power series have respective radii of convergence ≠ 0.

Let us mention that the theory of Puiseux expansions is equivalent to the valuation theory. The numbers e_{i}(ξ), i = 1, …, r(ξ) are called ramifications indices related to the place generated by z_{ξ} = x − ξ, respectively z_{∞} = 1/x in the rational function field \overline{K}(x). They have the following meaning in valuation theory: The place generated by z_{ξ} in the rational function field can be extended to r(ξ) places P_{1}, …, P_{r} in the extension field F. For the valuation of the extended place we have

v_{P_{i}} = e_{i}v_{z_{ξ}},

for all i = 1, …, r.

We also want to state an explicit form of the last theorem, which enables us to derive estimates for the coefficients of the Puiseux expansions of an algebraic function and which is due to Coates (cf. [17]).

We introduce the following notation first. If α is an algebraic number, then deg α, δ(α), h(α) denote respectively the degree of α, the least positive rational integer such that δ(α)α is an algebraic integer, and the maximum of the absolute values of the conjugates of α, and we put σ(α) = max{deg α, δ(α), h(α)}. Let f(x, y) be as above and let the maximum of the absolute values of the conjugates of the coefficients of f(x, y) be at most f, where f ≥ 2, and let f(x, y) have degree m and n in x and y, respectively. Put N = max{n, m, 3}.

Theorem 3.4. (Explicit Puiseux’s Theorem, Coates) Let F be an algebraic functions field over an algebraically closed field \overline{K} of characteristic 0, given by f(x, y) = 0. Let ξ ∈ \overline{K} and A_{i} (1 ≤ i ≤ r = r(ξ)) be the valuations of F extending the valuation of \overline{K}(x) defined by x − ξ, and let e_{i} be the ramification index of A_{i}. We write

y = ∑_{k=0}^{∞} w_{ik}(x − ξ)^{k/e_{i}}

for the Puiseux expansion of y at A_{i}. Then the coefficients w_{ik} (1 ≤ i ≤ r, k = 0, 1, …) are algebraic numbers, and the number field K′ obtained by adjoining ξ and these coefficients to K has degree at most (N deg ξ)^{N} over K. Further, K′ is generated over K by ξ and w_{ik} (1 ≤ i ≤ r, 0 ≤ k ≤ 2N^{4}). Finally, there exists a positive rational integer Δ such that Δ^{k+1}w_{ik} (1 ≤ i ≤ r, k = 0, 1, …) is an algebraic integer with

h(w_{ik}) ≤ (\frac{Λ}{Δ})^{k+1},  (3.19)

where Λ = (fσ(ξ))^{μ}, μ = (N^{4}n deg ξ)^{3N^{4}}.

Let Q_{i} (1 ≤ i ≤ r(∞)) be the valuations of F extending the valuation of \overline{K}(x) defined by 1/x. Let e_{i} be the ramification index of Q_{i}, and let

y = (\frac{1}{x})^{−e_{i}} ∑_{k=0}^{∞} w_{ik} (\frac{1}{x})^{k/e_{i}}

be the expansion of y at Q_{i}. Then the coefficients are algebraic numbers, and the number field K′ obtained by adjoining them to K has degree at most N^{N} over K. Further K′ is generated over K by the w_{ik} (1 ≤ i ≤ r, 0 ≤ k ≤ 2N^{4}). Finally, there exists a positive rational integer Δ such that Δ^{k+1}w_{ik} (1 ≤ i ≤ r, k = 0, 1, …) is an algebraic integer with

h(w_{ik}) ≤ (\frac{Λ}{Δ})^{k+1},  (3.20)

where Λ = f^{μ}, μ = (N^{4}n)^{3N^{4}}.

We want to remark that the proof of the last theorem yields an algorithm for the actual determination of the coefficients of the Puiseux expansion of an algebraic function. In fact, for the proof one constructs polynomials p_{i}(w) with coefficients in the field obtained by adjoining the first i coefficients of the Puiseux expansion in question, such that the (i + 1)st coefficient is a root of p_{i}(w). From this sequence of polynomials everything follows (see [17]). We want to show this construction by looking at the following example.

Example. We want to derive the Puiseux expansions of

g(x, y) = y^{2} − xy + 1 ∈ ℚ(x)[y],

at all infinite places of the functional field F = \overline{ℚ}(x, y) generated by g(x, y) = 0. First of all, we have

y_{1,2} = \frac{x}{2} ± \frac{\sqrt{x^{2} − 4}}{2}

for the roots of the equation g(x, y) = 0. Thus, we have F = \overline{ℚ}(x)(\sqrt{x^{2} − 4}). From this we see (see e.g. [90]) that F is a so called Kummer extension (which is a special Galois extension). Therefore we know that there are r = 2 places P_{1}, P_{2} lying over the place generated by 1/x in the rational function field and the corresponding ramification indices e_{1}, e_{2} are both equal, say e, and the common value is e = 1.

First we set

G(x, y) = x^{δe}g(x^{−e}, x^{−e}y) = x^{2} − x + y^{2},

where δ is the total degree of g(x, y), so G(x, y) is a polynomial in x, y. Next we write G(x, y) = x^{r_{0}}G_{0}(x, y) = x^{2} − x + y^{2}, where G_{0} is not divisible by x, then we choose p_{0}(w) = g_{0}(0, w). So we have p_{0}(w) = w(w − 1), which has the zeros 0, 1. So the first coefficient of each expansion is know. Next we set y = xw, y = 1 + xw respectively and substitute it into G(x, y) and repeat the procedure. We do it only for the second case. There we have

G(x, 1 + xw) = (1 + xw)^{2} − 1 − xw + x^{2} = x(xw^{2} − 3w + x),

which gives p_{1}(w) = −3w, which yields that the next coefficient is 0. Next we consider

G(x, 1 + x^{2}w) = x^{2}(w + x^{2}w^{2} + 1),

which implies p_{2}(w) = w + 1 and therefore the coefficient 1. The next step is

G(x, 1 − x^{2} + x^{3}w) = x^{3}(w + x − 2x^{2}w + x^{3}w^{2}),

which gives p_{3}(w) = w and thus again the coefficient 0. Continuing in this way we get

y_{1} = x − \frac{1}{x} − \frac{1}{x^{3}} − 2\frac{1}{x^{5}} + O(\frac{1}{x^{7}} ),

y_{2} = \frac{1}{x} + \frac{1}{x^{3}} + 2\frac{1}{x^{5}} + O (\frac{1}{x^{7}} ) .

So we have constructed the first terms of the requested Puiseux expansions.

3.4 Proof of the Main Theorem

First of all we can assume that f(z, x) = 0 depends on z and x, otherwise the assertion of our Theorem 3.1 would be trivially false. We can also suppose without loss of generality that f(z, x) is absolutely irreducible. Otherwise we can find a finite extension field L of K such that f(z, x) splits into a product of absolutely irreducible factors in L[z, x]. Then we can proceed with each of those factors as below and sum up the number of solutions to get the final result. So let us denote by F the functions field obtained by adjoining a root of f(z, x) = 0 to \overline{K}(z), where \overline{K} denotes the algebraic closure of K.

We work only in the case |α_{1}| > 1 and consider the Puiseux expansion at z = ∞ of the solution x = x(z) of f(z, x) = 0. The arguments in the case |α_{1}| < 1 are completely analogous and use the expansion at z = 0.

In the sequel C_{1}, C_{2}, … will denote positive numbers depending only on f(z, x) and on λ_{1} and on the P_{i}, α_{i}.

According to Theorem 1.9 and Remark 3.2 the number of solutions of (3.5) of the form (n, 0), n ∈ ℕ can be estimated by

C_{2} = e^{7C_{1}^{8C_{1}}}.

Observe that this follows from the fact that the α_{1}, …, α_{t} are multiplicatively independent. Therefore also G_{n}^{2}, G_{n}^{3}, … are nondegenerate recurring sequences. Consequently, we can restrict ourselves to solutions of the form (n, x) ∈ ℕ × K with x ≠ 0. These solutions are denoted by (n, x_{n}) ∈ ℕ×K with n ∈ Σ, where Σ is a set of positive integers.

Now by Puiseux’s Theorem 3.3 we can conclude that

f(z, x) = ∏_{j,i}(x − x_{ij}),

where

x_{ij} = ∑_{k=v_{i}}^{∞} a_{ik}ζ^{jk}(\frac{1}{z})^{\frac{k}{e_{i}}} ,

for j = 0, …, e_{i} − 1, i = 1, 2, …, r and where e_{1}, …, e_{r} are the ramification indices of the valuations extending 1/z to the function field F. Furthermore by the Explicit Puiseux’s Theorem 3.4 we get that all coefficients lie in a fixed finite extension field K′ of K and we have

h(a_{ik}ζ^{jk}) ≤ C_{3}^{k+1},

for j = 0, …, e_{i} − 1, i = 1, 2, …, r. Therefore for each solution (n, x_{n}) of (3.5) we get

x_{n} = ∑_{k=v}^{∞} β_{k}G_{n}^{− \frac{k}{e}} ,  (3.21)

for some v, e and β_{k} with

|β_{k}| ≤ C_{3}^{k+1}

for all k = 1, 2, …, which lie in a fixed finite extension of K. In what follows we will only consider those n, lying in a subsequence R ⊆ Σ, for which the same expansion occurs. The final number is just the sum of all numbers obtained by all those expansions.

Let us remark that for n > C_{4} the above series converges absolutely; this is because

|G_{n}| = |λ_{1}α_{1}^{n} + … + P_{t}(n)α_{t}^{n}| = |λ_{1}||α_{1}|^{n}|1 + ∑_{j=2}^{t} \frac{P_{j}(n)}{λ_{1}} (\frac{α_{j}}{α_{1}} )^{n} | ≥

≥ |λ_{1}||α_{1}|^{n}|1 − \underbrace{∑_{j=2}^{t} |\frac{P_{j}(n)}{λ_{1}} | |\frac{α_{j}}{α_{1}} |^{n}}_{≤1/2 for n>C_{5}} | ≥ C_{6}C_{7}^{n} ⟶ ∞,

because C_{7} = |α_{1}| > 1. Thus

∑_{k=v}^{∞} |β_{k}||G_{n}|^{−k/e} ≤ ∑_{k=v}^{∞} C_{3}^{k+1} [C_{6}C_{7}^{n}]^{−k/e} = C_{3} ∑_{k=v}^{∞} [C_{3} [C_{6}C_{7}^{n}]^{−1/e}]^{k} < ∞,

if n > C_{4}.

Since |α_{1}| > |α_{i}| for i = 2, …, t, we have binomial expansions

G_{n}^{− \frac{k}{e}} = λ_{1}^{− \frac{k}{e}} α_{1}^{− \frac{kn}{e}} (1 + ∑_{i=2}^{t} \frac{P_{i}(n)}{λ_{1}} (\frac{α_{i}}{α_{1}} )^{n})^{− \frac{k}{e}} =

= λ_{1}^{− \frac{k}{e}} α_{1}^{− \frac{kn}{e}} ∑_{r=0}^{∞} \binom{−\frac{k}{e}}{r} (∑_{i=2}^{t} \frac{P_{i}(n)}{λ_{1}} (\frac{α_{i}}{α_{1}} )^{n})^{r} ,

for some choice of the eth roots of λ_{1} and α_{1}, which we may assume to be fixed for all n ∈ R. Because of the fact that

|∑_{i=2}^{t} \frac{P_{i}(n)}{λ_{1}} (\frac{α_{i}}{α_{1}} )^{n}| ≤ C_{8}n^{C_{9}}C_{10}^{n} < 1,

because C_{10} = |α_{2}/α_{1}| < 1, if n > C_{11}, the expansion converges again absolutely for large n.

Next we are going to approximate x_{n} by a finite sum extracted from the Puiseux expansion (3.21). We define

H_{n} := ∑_{k=v}^{H} β_{k}λ_{1}^{− \frac{k}{e}} α_{1}^{− \frac{kn}{e}} ∑_{r=0}^{H} \binom{−\frac{k}{e}}{r} (∑_{i=2}^{t} \frac{P_{i}(n)}{λ_{1}} (\frac{α_{i}}{α_{1}} )^{n})^{r} ,

where H ≥ 1 is an integer to be chosen later. We may write

H_{n} = ∑_{j=1}^{h} τ_{j}(n)γ_{j}^{n}, n ∈ R,

where the τ_{j}(n) ∈ \overline{K}[n] and the γ_{j} are distinct and lie in the multiplicative group generated by α_{1}^{1/e} and α_{2}, …, α_{t}. Clearly H_{n} is nondegenerate, in fact the roots γ_{j} are again multiplicatively independent. Moreover, we have

h ≤ C_{12}(H),  (3.22)

where C_{12}(H) means that the constant depends also on H.

We enlarge K at once and assume that it contains all the α_{i}^{1/e} and all the coefficients β_{j} in the Puiseux series. In particular, we may assume that K contains all the coefficients of τ_{j} and the γ_{j}.

Next we estimate the approximation error we make, when we approximate x_{n} through H_{n}. We have

|x_{n} − H_{n}| = |x_{n} − ∑_{k=v}^{H} β_{k}λ_{1}^{− \frac{k}{e}} α_{1}^{− \frac{kn}{e}} ∑_{r=0}^{H} \binom{−\frac{k}{e}}{r} (∑_{i=2}^{t} \frac{P_{i}(n)}{λ_{1}} (\frac{α_{i}}{α_{1}} )^{n})^{r}| ≤

≤ |∑_{k=v}^{H} β_{k}λ_{1}^{− \frac{k}{e}} α_{1}^{− \frac{kn}{e}} ∑_{r=H+1}^{∞} \binom{−\frac{k}{e}}{r} (∑_{i=2}^{t} \frac{P_{i}(n)}{λ_{1}} (\frac{α_{i}}{α_{1}} )^{n})^{r}| +  (3.23)

+| ∑_{k=H+1}^{∞} β_{k}λ_{1}^{− \frac{k}{e}} α_{1}^{− \frac{kn}{e}} \underbrace{∑_{r=0}^{∞} \binom{−\frac{k}{e}}{r} (∑_{i=2}^{t} \frac{P_{i}(n)}{λ_{1}} (\frac{α_{i}}{α_{1}} )^{n})^{r}}_{=G_{n}^{−k/e}} | ≤

≤ ∑_{k=v}^{H} C_{13}^{k+1}C_{14}^{kn}C_{15}(k, H) |∑_{i=2}^{t} \frac{P_{i}(n)}{λ_{1}} (\frac{α_{i}}{α_{1}} )^{n}|^{H+1} +  (3.24)

+ ∑_{k=H+1}^{∞} C_{13}^{k+1}C_{14}^{kn} [C_{6}C_{7}^{n}]^{−k/e} ≤

≤ ∑_{k=v}^{H} C_{13}^{k+1}C_{14}^{kn}C_{15}(k, H) [C_{8}n^{C_{9}}C_{10}^{n}]^{H+1} +  (3.25)

+C_{13} \frac{[C_{13}C_{14}^{n} (C_{6}C_{7}^{n})^{−1/e}]^{H+1}}{1 − \underbrace{C_{13}C_{14}^{n} (C_{6}C_{7}^{n})^{−1/e}}_{≤1/2 for n>C_{16}}} ≤

≤ C_{17}(H) [C_{8}n^{C_{9}}C_{10}^{n}]^{H+1} \underbrace{C_{19}^{n}}_{≤1} +C_{20} [C_{21}n^{C_{22}}C_{23}^{n}]^{H+1} ≤  (3.26)

≤ [C_{24}n^{C_{25}}C_{26}^{n}]^{H+1} (C_{17}(H) + C_{20}) =  (3.27)
= C_{27}(H)n^{C_{25}(H+1)}C_{26}^{n(H+1)},  (3.28)

where C_{10}, C_{14} < 1, C_{7} > 1 thus C_{23} < 1 and therefore also C_{26} < 1. Observe that we have used

| ∑_{r=H+1}^{∞} \binom{d}{r}w^{r}| ≤ C(d, H)|w|^{H+1}

and

| ∑_{k=H+1}^{∞} q^{k}| ≤ \frac{q^{H+1}}{1 − q}

to estimate the tails of the above series.

For later purposes we need an estimate of |x_{n}|. For n larger than a constant C_{27} we obtain

|x_{n}| = |∑_{k=v}^{∞} β_{k}G_{n}^{− \frac{k}{e}} | ≤ C_{3} \frac{[C_{3} [C_{6}C_{7}^{n}]^{−1/e}]^{v}}{1 − \underbrace{C_{3}[C_{6}C_{7}^{n}]^{−1/e}}_{≤1/2 for n>C_{27}}} ≤ C_{28}C_{29}^{n}  (3.29)

Observe that v ∈ ℤ, so that we cannot conclude that C_{29} < 1 holds.

We choose H so that

C_{26}^{H+1}C_{29} < 1.  (3.30)

To get this, we must have

H > max \{1, \frac{log C_{29}}{log C_{26}^{−1}} − 1\} .

Observe that from now on H is fixed and therefore also h, τ_{i}(n), γ_{i}, i = 1, …, h are fixed. Also, we choose a finite set S so that it contains all infinite absolute values of K. Moreover we require that all the α_{j}, λ_{1} and all coefficients of the P_{j}(n), all the nonzero coefficients of f(z, x) and of τ_{1}(n), …, τ_{h}(n) are S-units, which means that the | · |_{v} of those values = 1 for each v ∉ S. In particular, with this choice all γ_{j} are S-units. Also, the G_{n} are S-integer, that is |G_{n}|_{v} ≤ 1 for each v ∉ S, and f(z, x) is monic in x; therefore, the x_{n} too are S-integers, in view of the equations f(G_{n}, x_{n}) = 0.

We shall apply Theorem 1.5, so let us define, for every s ∈ S, h + 1 independent linear forms in X := (X_{0}, …, X_{h}) as follows: put

L_{0,∞}(X) = X_{0} + X_{1} + ⋯ + X_{h}

and for v ∈ S, 0 ≤ i ≤ h, (i, v) ≠ (0, ∞) put

L_{i,v}(X) = X_{i}.

Here ∞ denotes the infinite absolute value, which coincides with the complex absolute value in the embedding of K in ℂ. We have

H(L_{i,v}) ≤ C_{30}

for v ∈ S, i = 0, …, h. Furthermore K(L_{i,v}) = K and therefore

[K(L_{i,v}) : K] = 1 ∀v ∈ S, i = 0, …, h.

Moreover, we have

det(L_{0,v}, …, L_{h,v}) = | \begin{matrix}1 & 0 & 0 & … & 0 \\ 1 & 1 & 0 & … & 0 \\ 1 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ 1 & 0 & 0 & … & 1 \end{matrix} | = 1,

which yields

| det(L_{0,v}, …, L_{h,v})|_{v} = 1 ∀ v ∈ S.

For n ∈ P define the vectors x_{n} = (−x_{n}, τ_{1}(n)γ_{1}^{n}, …, τ_{h}(n)γ_{h}^{n}) ∈ K^{h+1} and consider the double product

∏_{v∈S} ∏_{i=0}^{h} \frac{|L_{i,v}(x_{n})|_{v}}{|x_{n}|_{v}} .

By putting

σ = −x_{n} + τ_{1}(n)γ_{1}^{n} + … + τ_{h}(n)γ_{h}^{n} = L_{0,∞}(x_{n}),

we can rewrite the double product as

|σ|_{∞} · ( ∏_{v∈S\{∞}} |x_{n}|_{v}) (∏_{v∈S} ∏_{i=1}^{h} |τ_{i}(n)γ_{i}^{n}|_{v}) (∏_{v∈S} |x_{n}|_{v})^{−(h+1)} .

Observe that x_{n} is an S-integer and that, due to our choice of S, the τ_{i}(n)γ_{i}^{n} are S-units for i ≥ 1. In particular, this implies

(∏_{v∈S} ∏_{i=1}^{h} |τ_{i}(n)γ_{i}^{n}|_{v}) = 1  (3.31)

and

( ∏_{v∈S\{∞}} |x_{n}|_{v}) = ∏_{v∉S} |x_{n}|_{v} · |x_{n}|_{∞} ≤ |x_{n}|_{∞} ≤ C_{28}C_{29}^{n},  (3.32)

where we have used the product formula (1.3) and (3.29). Therefore we get using (3.28), (3.31) and (3.32)

∏_{v∈S} ∏_{i=0}^{h} \frac{|L_{i,v}(x_{n})|_{v}}{|x_{n}|_{v}} ≤ C_{27}(H)n^{C_{25}(H+1)}C_{26}^{n(H+1)}C_{28}C_{29}^{n} (∏_{v∈S} |x_{n}|_{v})^{−(h+1)} .

Last we need an upper bound for H(x_{n}). We have

H(x_{n}) ≤ ∏_{v∈S} |x_{n}|_{v} ≤ C_{31} ∏_{v∈S} max{|x_{n}|_{v}, |τ_{1}(n)γ_{1}^{n}|_{v}, …, |τ_{h}(n)γ_{h}^{n}|_{v}},

where we have used again our choice of S and the fact the two norms on K^{h+1} are equivalent. Observe that C_{31} does only depend on h and on [K : ℚ]. We need an estimate for |x_{n}|_{v} and we derive it from the equation f(G_{n}, x_{n}) = 0. Observe that we trivially have an estimate

|G_{n}|_{v} ≤ C_{32}n^{C_{33}}C_{34}^{n}.

On the other hand, we can estimate the absolute value of the roots of an equation in terms of the absolute value of the coefficients. We finally obtain

|x_{n}|_{v} ≤ C_{35}n^{C_{36}}C_{37}^{n}.

Moreover we have

|τ_{i}(n)γ_{i}^{n}|_{v} ≤ C_{38}n^{C_{39}}C_{40}^{n},

for all i = 1, …, h. Consequently we get

H(x_{n}) ≤ C_{41}n^{C_{42}}C_{43}^{n}.  (3.33)

Let us point out that these constants depend not on n.

We now choose 0 < δ < 1 so that

C_{26}^{H+1}C_{29}C_{43}^{δ} < 1.  (3.34)

This will be possible for small δ in view of (3.30).

In view of the bound for the double product we derived and (3.33), the verification of (1.4) of the Quantitative Subspace Theorem 1.5 will follow from

C_{27}(H)C_{28}n^{C_{25}(H+1)} (C_{26}C_{29})^{n} < (C_{41}n^{C_{42}}C_{43}^{n})^{−δ} ,

which is the same as

n^{C_{25}(H+1)+δC_{42}} (C_{26}^{H+1}C_{29}C_{43}^{δ})^{n} < (C_{27}(H)C_{28}C_{41}^{δ})^{−1} .

However, this latter inequality follows from (3.34) for n > C_{44}.

Therefore, by the Quantitative Subspace Theorem 1.5, there exist finitely many nonzero linear forms Λ_{1}(X), …, Λ_{g}(X) with coefficients in \overline{K} and with

g ≤ C_{45},

such that each vector x_{n} is a zero of some Λ_{j}.

Suppose first Λ_{j} does not depend on X_{0}. Then, if Λ_{j}(x_{n}) = 0, we have a nontrivial relation

∑_{i=1}^{h} u_{i}τ_{i}(n)γ_{i}^{n} = 0, u_{i} ∈ \overline{K}, i = 1, …, h.

By Theorem 1.9 this can hold for at most a finite number of n. More precisely, we can conclude that the number of those solutions can be bounded by a constant C_{46}, since the γ_{i} are nondegenerate.

Suppose that Λ_{j} depends on X_{0} and that Λ_{j}(x_{n}) = 0. Then we have

x_{n} = ∑_{i=1}^{h} v_{i}τ_{i}(n)γ_{i}^{n}, v_{i} ∈ \overline{K}, i = 1, …, h.  (3.35)

Substituting this into f(z, x) = 0 we get

f(G_{n}, ∑_{i=1}^{h} v_{i}τ_{i}(n)γ_{i}^{n}) = 0.  (3.36)

If the above equation does not hold identically and we have assumed that this is not the case, we can conclude that

|{n|n satisfies (3.36)}| < C_{47}

also in this case, because the left hand side of (3.36) defines a nondegenerate linear recurring sequence and the conclusion follows again by Theorem 1.9. Observe that from the assumption that the α_{1}, …, α_{t} are multiplicatively independent, we conclude the nondegeneracy.

Then the number of solutions of (3.5) can be bounded by

C_{2} + ∑_{i=1}^{r} ∑_{j=0}^{e_{i}−1} [C_{45}(C_{46} + C_{47}) + max{C_{4}, C_{11}, C_{16}, C_{27}, C_{44}}] ,

where the constants in the sum clearly can depend on i, j. This completes the proof. □

3.5 Proof of Theorem 3.2 and the corollaries

Proof of Theorem 3.2.

The proof is the same as the proof of Theorem 3.1, except that we can conclude the degeneracy of H_{n} and the recurring sequence defined by the left hand side of (3.36) by the fact that the roots α_{1}, …, α_{t} of G_{n} are different positive integers. Therefore possibly the order of the recurrence becomes smaller, but it cannot happen that we get roots which differ by a root of unity different from 1. □

Proof of Corollary 3.2.

This follows readily from Theorem 3.1 in form of Corollary 3.1 and a result, mentioned in the introduction, which is due to Shorey and Stewart [84, Theorem 3]. Let us remark that by Eisenstein’s criterion for absolutely irreducibility the polynomials f(z, x) = x^{q} − z are absolutely irreducible for all q. Moreover, observe that G_{n} − λ_{1}α_{1}^{n} ≠ 0, because of our assumption t ≥ 2. □

Proof of Corollary 3.3.

Let d be the degree of α_{1} over ℚ. Using a result of Shorey and Stewart [84, Corollary 1], also mentioned in the introduction, we can conclude that the number of solutions n, x and q with n ≥ 0, |x| > 1, and

q > max (\frac{d log |α_{1}|}{log(|α_{1}|/ max(1, |α_{2}|))}, d + r)

of the equation

G_{n} = x^{q} + T (x)

is finite. It remains to show that the number of solutions n, x and q with n ≥ 0, |x| > 1 and

1 ≤ q ≤ max (\frac{d log |α_{1}|}{log(|α_{1}|/ max(1, |α_{2}|))}, d + r)

is also finite. But this follows now from our Theorem 3.1. Observe that only for the solutions with small q, an upper bound for the number of solutions can be given. □

3.6 Example

In this section we consider the very special equation

2^{n}x = x^{2} + 1, n ≠ 0, x ∈ ℤ,  (3.37)

and we calculate an upper bound for the number of solutions. We have

f(z, x) = x^{2} − zx + 1 and G_{n} = 2^{n}.

This example does not fit perfectly, because t = 1, but after all it will show the important steps used in our method. We will use the notations of the proof of Theorem 3.1.

We have already calculated the Puiseux expansions we need. Let us repeat them:

x_{1} = z − \frac{1}{z} − \frac{1}{z^{3}} − 2\frac{1}{z^{5}} + O(\frac{1}{z^{7}} ),  (3.38)
x_{2} = \frac{1}{z} + \frac{1}{z^{3}} + 2\frac{1}{z^{5}} + O (\frac{1}{z^{7}} ) .  (3.39)

By the Explicit Puiseux’s Theorem 3.4 we get the following bound for the coefficients of the above series

Λ = C_{3} = 2^{(3^{4})^{3·2^{4}}}.

Moreover we get C_{2} = 0, C_{5} = 1, C_{6} = 1, C_{7} = 2.

Let us now consider the solutions (n, x_{n}), which come from (3.38). We approximate x_{n} by H_{n} = 2^{n} − 2^{−n} with error C_{27}(H) = 2Λ^{H+2}, C_{25} = 0, C_{26} = 1/2 and H = 1, if

n > C_{16} = \frac{log Λ}{log 2} + 1.

Next we have to choose S = \{∞, 2\} and we get C_{30} = \sqrt{3}, C_{41} = 2\sqrt{3}, C_{43} = 0, C_{39} = 4. Choosing δ = 1/4 we get, if

n > C_{44} = \begin{matrix}log & ((2\sqrt{3})^{1/4} · 4Λ^{3}) \\ log & (2 · 4^{−1/4}) \end{matrix},

for the number of relevant subspaces

g ≤ (2^{540} · 4^{21})^{2} log 4 log log 4 + (150 · 81 · 4)^{7}(2 + log log 2\sqrt{3}) = C_{45}.

Last we get

C_{46} = e^{14^{16}} and C_{47} = e^{28^{32}}.

In the same way we handle the solutions (n, x_{n}), which come from (3.39). Here we approximate x_{n} by H_{n} = 2^{−n} and therefore choose H = 1 as before. Again we have to choose S = \{∞, 2\} and we get C_{30} = \sqrt{2}, C_{41} = 2\sqrt{2}, C_{42} = 0, C_{43} = 2, provided that

n > \frac{2 log Λ}{log 2}.

We choose δ = 1/2 and get for

n > \frac{log (4Λ^{5}\sqrt{2\sqrt{2}})}{log(8\sqrt{2})}

an upper bound for the number of relevant subspaces. Because of the fact that H_{n} = 2^{−n}, we can choose C_{46} = C_{47} = 0.

At last we get the following upper bound for the number of solutions (n, x) of (3.37)

(2^{540} · 4^{21})^{2} log 4 log log 4 + (150 · 81 · 4)^{7}(2 + log log 2\sqrt{3})(e^{14^{16}} + e^{28^{32}})+

+ max \{\frac{log Λ}{log 2} + 1, \begin{matrix}log & ((2\sqrt{3})^{1/4} · 4Λ^{3}) \\ log & (2 · 4^{−1/4}) \end{matrix}, \frac{2 log Λ}{log 2}, \frac{log (4Λ^{5}\sqrt{2\sqrt{2}})}{log(8\sqrt{2})}\} < e^{10^{47}},

where Λ = 2^{(3^{4})^{3·2^{4}}}.

We can see from this example that we cannot expect the bound to be small. This is also clear from the point of view, that we use quite general estimates for the zero multiplicity of nondegenerate recurring sequences and for the coefficients of Puiseux expansions. Let us remark, that it is easy to show that our equation (3.37) has only the trivial solution (n, x) = (1, 1).

On the Diophantine equation G_{n}(x) = G_{m}(P (x))

Let K be a field of characteristic 0 and let p, q, G_{0}, G_{1}, P ∈ K[x], deg P ≥ 1. Further let the sequence of polynomials (G_{n}(x))_{n=0}^{∞} be defined by the second order linear recurring sequence

G_{n+2}(x) = p(x)G_{n+1}(x) + q(x)G_{n}(x), for n ≥ 0.

In this chapter we give conditions under which the Diophantine equation G_{n}(x) = G_{m}(P (x)) has at most exp(10^{18}) many solutions (n, m) ∈ ℤ^{2}, n, m ≥ 0. The proof uses a very recent result on S-unit equations over fields of characteristic 0 due to J.-H. Evertse, H. P. Schlickewei and W. M. Schmidt (cf. Theorem 1.7 and [45]). Under the same conditions we present also bounds for the cardinality of the set

{(m, n) ∈ ℕ | m ≠ n, ∃ c ∈ K\{0} such that G_{n}(x) = c G_{m}(P (x))}.

In the last part we specialize our results to certain families of orthogonal polynomials.

This chapter is identically equal to a joint paper with A. Pethő and R. F. Tichy which is submitted for publication in Monatsh. Math. (cf. [49]).

4.1 Introduction

Let K denote a field of characteristic 0. There is no loss of generality in assuming that this field is algebraically closed and we will assume this for the rest of the chapter. Let p, q, G_{0}, G_{1} ∈ K[x] and let the sequence of polynomials (G_{n}(x))_{n=0}^{∞} be defined by the second order linear recurring sequence

G_{n+2}(x) = p(x)G_{n+1}(x) + q(x)G_{n}(x), for n ≥ 0.  (4.1)

By α(x), \overline{α}(x) we denote the roots of the corresponding characteristic polynomial

T^{2} − p(x)T − q(x).  (4.2)

CHAPTER 4. ON THE EQUATION G_{n}(x) = G_{m}(P (x)) 56

Let Δ(x) = p(x)^{2} + 4q(x) be the discriminant of the characteristic polynomial of the recurring sequence (G_{n})_{n=0}^{∞}. Then we have

α(x) = \frac{p(x) + \sqrt{Δ(x)}}{2}, \overline{α}(x) = \frac{p(x) − \sqrt{Δ(x)}}{2}.

We will always assume that the recurring sequence is simple, which means that Δ(x) ≠ 0. Then for n ≥ 0

G_{n}(x) = g_{1}(x)α(x)^{n} + g_{2}(x)\overline{α}(x)^{n},  (4.3)

where

g_{1}(x) = \frac{G_{1}(x) − G_{0}(x)\overline{α}(x)}{α(x) − \overline{α}(x)} and g_{2}(x) = \frac{G_{1}(x) − G_{0}(x)α(x)}{α(x) − \overline{α}(x)}.  (4.4)

Notice that

g_{1}, g_{2} ∈ K(x, \sqrt{p(x)^{2} + 4q(x)}) = K(x, \sqrt{Δ}).

In fact we have

G_{n}(x) = \frac{G_{1}(x) − G_{0}(x)\overline{α}(x)}{α(x) − \overline{α}(x)}α(x)^{n} + \frac{G_{1}(x) − G_{0}(x)α(x)}{α(x) − \overline{α}(x)}\overline{α}(x)^{n}.

(G_{n}(x))_{n=0}^{∞} is called nondegenerate, if the quotient \overline{α}(x)/α(x) is not a root of unity.

Many Diophantine equations involving the recurrence (G_{n}(x))_{n=0}^{∞} were studied previously. For example let us consider the equation

G_{n}(x) = s(x),  (4.5)

where s(x) ∈ K[x] is given. We denote by N(s(x)) the number of integers n for which (4.5) holds. Schlickewei [78] established an absolute bound for N(s(x)), provided that the sequence is nondegenerate and that also α, \overline{α} are not equal to a root of unity. His bound was substantially improved by Beukers and Schlickewei [9] who showed that

N(s(x)) ≤ 61.

In the particular case that not all algebraic functions g_{1}(x)/s(x), g_{2}(x)/s(x), α(x), \overline{α}(x) are constants (which is always the case here), Beukers and Tijdemann (cf. Theorem 2 on p. 206 in [10]) showed that

N(s(x)) ≤ 3.

Very recently, Schmidt [82] obtained the remarkable result that for arbitrary nondegenerate complex recurring sequences of order q one has N(a) ≤ C(q), where a ∈ ℂ and C(q) depends only (and in fact triply exponentially) on q (cf. Theorem 1.9).

Another kind of result is due to Glass, Loxton and van der Poorten [53]. They showed that, if (G_{n}(x))_{n=0}^{∞} is nonperiodic and nondegenerate, then there are only finitely many pairs of integers m, n with m > n ≥ 0 and

G_{n}(x) = G_{m}(x).  (4.6)

In a recent paper Dujella and Tichy [35] showed for linear recurring sequences G_{n+1}(x) = xG_{n}(x)+BG_{n−1}(x), G_{0}(x) = 0, G_{1}(x) = 1 of polynomials with B ∈ ℤ\{0} that there does not exist a polynomial P (x) ∈ ℂ[x] satisfying

G_{n}(x) = G_{m}(P (x))  (4.7)

(for all m, n ≥ 3, m ≠ n). Applying a general theorem of Bilu and Tichy [12], this result was used to show that the Diophantine equation G_{n}(x) = G_{m}(y) has only finitely many solutions in integers n, m, x, y, with n ≠ m.

It is the aim of this chapter to present suitable extensions of the results (4.5) and (4.7).

4.2 General results

Our first main result is a generalization of (4.6) to the Diophantine equation

G_{n}(x) = G_{m}(P (x)),  (4.8)

where P ∈ K[x], deg P ≥ 1 is arbitrary.

Theorem 4.1. Let p, q, G_{0}, G_{1}, P ∈ K[x], deg P ≥ 1 and (G_{n}(x))_{n=0}^{∞} be defined as above. Assume that the following conditions are satisfied: 2 deg p > deg q ≥ 0 and

deg G_{1} > deg G_{0} + deg p ≥ 0, or
deg G_{1} < deg G_{0} + deg q − deg p.

Then there are at most min{exp(18^{10}), C(p, q, P)} pairs of integers (n, m) with n, m ≥ 0 with n ≠ m such that

G_{n}(x) = G_{m}(P (x))

holds. We have

C(p, q, P) = 10^{28} · log(2C_{1} deg p) · (4e)^{8C_{1} deg q} · 7^{4C_{1} deg q},

where C_{1} = 2(deg P + 1).

CHAPTER 4. ON THE EQUATION G_{n}(x) = G_{m}(P (x)) 58 Remark 4.1. It is clear that for m = n Theorem 4.1 cannot hold. Remark 4.2. Let us consider the following example, which shows that all sequences must be excluded, where deg G_{n}(x) = 0 for all n:

G_{0}(x) = 1, G_{1}(x) = 1,

G_{n+2}(x) = \frac{1}{2}G_{n+1}(x) + \frac{1}{2}G_{n}(x).

This means

2 deg p = deg q = 0 and deg G_{0} = deg G_{1} = 0.

We have

G_{n}(x) = 1 ∀n.

Clearly (4.8) is satisfied for all m, n and all polynomials P ∈ K[x]. Remark 4.3. This example shows that also in the polynomial case the condition 2 deg p > deg q is needed. Assume that

G_{0}(x) = 1, G_{1}(x) = x,

G_{n+2}(x) = \frac{x}{2}G_{n+1}(x) + \frac{x^{2}}{2} G_{n}(x).

Here we have

2 deg p = deg q = 2 and deg G_{0} = 0, deg G_{1} = 1.

It follows that

G_{n}(x) = x^{n} ∀n.

In this case the following equation holds

G_{2n}(x) = G_{n}(x^{2})

for all integers n. Remark 4.4. Let us consider the Chebyshev polynomials of the first kind, which are defined by

T_{n}(x) = cos(n arccos x).

It is well known that they satisfy the following second order recurring relation: T_{0}(x) = 1, T_{1}(x) = x,

T_{n+2}(x) = 2xT_{n+1}(x) − T_{n}(x).

In this case we have

2 deg p > deg q and deg T_{1} = deg T_{0} + deg p = 1.

It is also well known and in fact easy to prove that

T_{2n}(x) = T_{n}(2x^{2} − 1)

holds for all integers n. This example shows that also the second assumption in Theorem 4.1 is needed.

Actually, it is also possible to give an upper bound for the number of pairs (m, n) with G_{n}(x) = c G_{m}(P (x)), c ∈ K^{∗} = K\{0} variable. This means that we can give an upper bound for the cardinality of the set

{(m, n) ∈ ℕ | m ≠ n, ∃ c ∈ K^{∗} such that G_{n}(x) = c G_{m}(P (x))}.

(Here c may vary with m, n). In fact, the second part of the upper bound in the last theorem, which depends on the degrees of the polynomials involved, follows from this more general theorem. The bound has the advantage that it is only exponential whereas the absolute bound of Theorem 4.1 is double exponential.

Theorem 4.2. Let p, q, G_{0}, G_{1}, P ∈ K[x], deg P ≥ 1 and (G_{n}(x))_{n=0}^{∞} be defined as above. Assume that the following conditions are satisfied: 2 deg p > deg q ≥ 0 and

deg G_{1} > deg G_{0} + deg p ≥ 0, or
deg G_{1} < deg G_{0} + deg q − deg p.

Then the number of pairs of integers (n, m) with n, m ≥ 0, n ≠ m for which there exists c ∈ K^{∗} with

G_{n}(x) = c G_{m}(P (x))

is at most C(p, q, P). We have

C(p, q, P) = 10^{28} · log(2C_{1} deg p) · (4e)^{8C_{1} deg q} · 7^{4C_{1} deg q},

where C_{1} = 2(deg P + 1).

It is also possible to replace the conditions concerning the degree by algebraic conditions.

Theorem 4.3. Let p, q, G_{0}, G_{1}, P ∈ K[x] and (G_{n}(x))_{n=0}^{∞} be defined as above. Assume that

(1) deg Δ ≠ 0, (2) deg P ≥ 2,

(3) gcd(p, q) = 1 and

(4) gcd(2G_{1} − G_{0}p, Δ) = 1.

Then there are at most min{exp(10^{18}), C̃(p, q, P)} pairs of integers (n, m) with n, m ≥ 0 such that

G_{n}(x) = G_{m}(P (x))

holds. We have

C̃(p, q, P) = 10^{28} · log(C_{1} max{2 deg p, deg q}) · (4e)^{8C_{1} deg q} · 7^{4C_{1} deg q},

where C_{1} = 2(deg P + 1).

Remark 4.5. The reason for this different kind of assumptions lie in the fact that the infinite valuation in the rational function field K(x) leads to degree assumptions, whereas by looking at finite valuations one gets divisibility conditions as in the above theorem.

Remark 4.6. It is obvious that for deg P = 1 Theorem 4.3 cannot hold in full generality. For example: if G_{n}(x) is a polynomial in x^{2} for all n we get

G_{n}(x) = G_{n}(−x)

for all n.

Remark 4.7. By looking at the proof, it is clear that Theorem 4.3 also holds, if we assume instead of (2)

(2′) There is no c ∈ K such that Δ(P (x)) = cΔ(x) holds.

To our knowledge this is the weakest condition under which our proof works. It is clear that (2′) holds if deg P ≥ 2 or if P is a constant. If P (x) = x then Δ(P (x)) = cΔ(x) holds with c = 1. Suppose that P (x) = ax + b with a, b ∈ K and a ≠ 0, (a, b) ≠ (1, 0). Denote by P^{(k)} the k-th iterate of P. Let Δ_{0} be the leading coefficient of Δ(x). It is left to the reader to show that (2′) does not hold if and only if a^{deg Δ} = c, a is a root of unity of order k > 1 and

Δ(x) = Δ_{0}(x + \frac{b}{a − 1})^{s} ∏_{i=1}^{r} ∏_{j=0}^{k−1}(x − P^{(j)}(x_{i})),

where r, s are non-negative integers with rk + s = deg Δ and −\frac{b}{a−1}, x_{1}, …, x_{r} are distinct elements of K.

Again we can handle the case G_{n}(x) = c G_{m}(P (x)) with c ∈ K^{∗} variable under this conditions and actually the second part of the bound of the last theorem follows from this theorem.

Theorem 4.4. Let p, q, G_{0}, G_{1}, P ∈ K[x] and (G_{n}(x))_{n=0}^{∞} be defined as above. Assume that the conditions (1)-(4) of Theorem 4.3 are satisfied. Then the number of pairs of integers (n, m) with n, m ≥ 0 for which there exists c ∈ K^{∗} with

G_{n}(x) = c G_{m}(P (x))

is at most C̃(p, q, P). We have

C̃(p, q, P) = 10^{28} · log(C_{1} max{2 deg p, deg q}) · (4e)^{8C_{1} deg q} · 7^{4C_{1} deg q},

where C_{1} = 2(deg P + 1).

4.3 Results for families of orthogonal polynomials

We will turn now our discussion to sequences of certain orthogonal polynomials satisfying (4.1). The following results can be found in the monograph of Borwein and Erdélyi [13, Chapter 2.3], Chihara [16, Chapter I and II] or Szegő [91, Chapter III]. Let (μ_{n})_{n=0}^{∞} be a sequence of complex numbers and let L : ℂ[x] ⟶ ℂ be linear functional defined by

L[x^{n}] = μ_{n}, n = 0, 1, 2, ….

Then L is called the moment functional determined by the formal moment sequence (μ_{n}). The number μ_{n} is called the moment of order n. A sequence (P_{n}(x))_{n=0}^{∞} is called an orthogonal polynomial sequence (OPS) with respect to a moment functional L provided that for all nonnegative integers m and n the following conditions are satisfied:

(i) deg P_{n}(x) = n,

(ii) L[P_{m}(x)P_{n}(x)] = 0 for m ≠ n,

(iii) L[P_{n}^{2}(x)] ≠ 0.

If there exists an OPS for L, then each P_{n}(x) is uniquely determined up to an arbitrary nonzero factor. An OPS in which each P_{n}(x) is monic will be referred to as a monic OPS; it is indeed unique.

It is well known that a necessary and sufficient condition for the existence of an OPS for a moment functional L with moment sequence (μ_{n}) is that for the determinants defined by

Δ_{n} = det(μ_{i+j})_{i,j=0}^{n} = | \begin{matrix}μ_{0} & μ_{1} & … & μ_{n} \\ μ_{1} & μ_{2} & … & μ_{n+1} \\ ⋮ & ⋮ & ... & ⋮ \\ μ_{n} & μ_{n+1} & … & μ_{2n} \end{matrix} |

the following conditions hold

Δ_{n} ≠ 0, n = 0, 1, 2, ….

In this case L is called quasi-definite.

A moment functional L is called positive-definite if L[π(x)] > 0 for every π(x) ∈ ℂ[x] that is not identically zero and which satisfies π(x) ≥ 0 for all real x. The following holds: L is positive-definite if and only if its moments are all real and Δ_{n} > 0 for all n ≥ 0. Furthermore, using the Gram-Schmidt process, a corresponding OPS consisting of real polynomials exists. Moreover, L is positive-definite if and only if a bounded, non-decreasing function ψ exists, whose moments

μ_{n} = ∫_{−∞}^{∞} x^{n}dψ(x), n = 0, 1, 2, …,

are all finite and the set

𝔖(ψ) = {x | ψ(x + δ) − ψ(x − δ) > 0 for all δ > 0}

is infinite. Further for the function ψ

∫_{−∞}^{∞} x^{n}dψ(x) = μ_{n} = L[x^{n}], n = 0, 1, 2, …

is valid. This is known as the representation theorem for positive-definite moment functionals or as the solution to the Hamburger moment problem.

Thus, an OPS with respect to a positive-definite moment functional L induces an inner product defined by

⟨p, q⟩ = L[p(x)\overline{q(x)}], p, q ∈ ℂ[x],

where \overline{q(x)} is obtained by taking the complex conjugates of the coefficients of q(x), on the linear space of polynomials with complex coefficients. In particular, we have ⟨P_{m}, P_{n}⟩ = L[P_{m}(x)\overline{P_{n}(x)}] = 0, m ≠ n. Thus our definition of orthogonality for the OPS is consistent with the usual definition of orthogonality in an inner product space.

One of the most important characteristics of an OPS is the fact that any three consecutive polynomials are connected by a very simply relation: Let L be a quasidefinite moment functional and let (P_{n}(x))_{n=0}^{∞} be the corresponding monic OPS. Then there exist constants c_{n} and λ_{n} ≠ 0 such that

P_{n}(x) = (x − c_{n})P_{n−1}(x) − λ_{n}P_{n−2}(x), n = 1, 2, 3, …,  (4.9)

where we define P_{−1}(x) = 0. Moreover, if L is positive-definite, then c_{n} is real and λ_{n+1} > 0 for n ≥ 1 (λ_{1} is arbitrary).

The converse is also true and it is referred to as Favard’s theorem: Let (c_{n})_{n=0}^{∞} and (λ_{n})_{n=0}^{∞} be arbitrary sequences of complex numbers and let (P_{n}(x))_{n=0}^{∞} be defined by the recurring formula

P_{n}(x) = (x − c_{n})P_{n−1}(x) − λ_{n}P_{n−2}(x), n = 1, 2, 3, …,  (4.10)
P_{−1}(x) = 0, P_{0}(x) = 1.  (4.11)

Then there is a unique moment functional L such that

L[1] = λ_{1}, L[P_{m}(x)P_{n}(x)] = 0 for m ≠ n, m, n = 0, 1, 2, ….

L is quasi-definite and (P_{n}(x))_{n=0}^{∞} is the corresponding monic OPS if and only if λ_{n} ≠ 0 for all n ≥ 1, while L is positive-definite if and only if c_{n} is real and λ_{n} > 0 for all n ≥ 1.

More generally, let (P_{n}(x))_{n=0}^{∞} be a sequence of polynomials in ℂ[x] satisfying

P_{n}(x) = (A_{n}x + B_{n})P_{n−1}(x) + D_{n}P_{n−2}(x) (n ≥ 1)
P_{−1}(x) = 0, P_{0}(x) = g ≠ 0,

where A_{n}, B_{n}, D_{n} are complex numbers with A_{n} ≠ 0, D_{n} ≠ 0 for every n ≥ 1. It follows easily by induction on n that P_{n} has degree n and that P_{n} has leading coefficient k_{n} = gA_{1} ⋯ A_{n} for n ≥ 0. Let k_{−1} := 1. For n ≥ −1 write P_{n}(x) = k_{n}P̂_{n}(x). Thus P̂_{n}(x) is monic for n ≥ 0. Further, P̂_{−1}(x) = 0, P̂_{1}(x) = 1 and the sequence (P̂_{n}(x))_{n=0}^{∞} satisfies (8) with c_{n} = −B_{n}k_{n−1}/k_{n} = −B_{n}/A_{n}, λ_{1} arbitrary and λ_{n} = −D_{n}k_{n−2}/k_{n} = −D_{n}/A_{n−1}A_{n} for n ≥ 2. So by Favard’s Theorem, (P̂_{n}(x))_{n=0}^{∞} is a monic OPS and therefore, (P_{n}(x))_{n=0}^{∞} is an OPS for some quasi-definite moment functional L. Moreover, L is positive definite if B_{n}/A_{n} ∈ ℝ and D_{n}/A_{n−1}A_{n} < 0 for n ≥ 2.

We now consider the special case that A_{1} = e/g, B_{1} = f/g, D_{1} = 0 where g ≠ 0 and A_{n} = a, B_{n} = b, D_{n} = d do not depend on n for n ≥ 2, that is, we consider the sequence of polynomials (P_{n}(x))_{n=0}^{∞} with P_{n}(x) ∈ ℂ[x] given by

P_{n+1}(x) = (ax + b)P_{n}(x) + dP_{n−1}(x), n ≥ 1,  (4.12)
P_{0}(x) = g, P_{1}(x) = ex + f,  (4.13)

where a, b, d, e, f, g are complex numbers with adeg ≠ 0. By the comments just made this sequence is an OPS for some quasi-definite moment functional L.

In the view of Remark 4.2 it is clear that Theorem 4.1 and Theorem 4.2 cannot hold for all OPS (P_{n}(x))_{n=0}^{∞}, because the Chebyshev polynomials of the first kind are orthogonal with respect to the positive-definite moment functional

L[π(x)] = ∫_{−1}^{1} π(x)(1 − x^{2})^{−1/2}dx.

Using the same methods as above we will prove the following analogues of Theorems 4.1 and 4.2.

Theorem 4.5. Let a, b, d, e, f, g ∈ ℂ, adeg ≠ 0 and (P_{n}(x))_{n=0}^{∞} a sequence of polynomials in ℂ[x] defined by (4.12) and (4.13). Let S(x) ∈ ℂ[x], deg S ≥ 1. If we assume that e = ag, then there are at most min{exp(18^{10}), C(S)} pairs of integers (n, m) with n, m ≥ 0 with n ≠ m such that

P_{n}(x) = P_{m}(S(x))

holds. We have

C(S) = 10^{28} · log(4(deg S + 1)).

Again we can prove the following result concerning the more general equation P_{n}(x) = c P_{m}(S(x)) with some c ∈ ℂ^{∗} variable and again the second part of the last theorem follows from this theorem.

Theorem 4.6. Let a, b, d, e, f, g ∈ ℂ, adeg ≠ 0 and (P_{n}(x))_{n=0}^{∞} defined by (4.12) and (4.13). Let S(x) ∈ ℂ[x], deg S ≥ 1 and e = ag. Then the number of pairs of integers (n, m) with n ≠ m for which there exists c ∈ ℂ^{∗} with

P_{n}(x) = c P_{m}(S(x))

is at most C(S). We have

C(S) = 10^{28} · log(4(deg S + 1)).

Remark 4.8. We want to mention that Theorem 4.3 and therefore also Theorem 4.4 can be applied to this situation. The conditions (1) and (3) are trivially satisfied in this case. Condition (4) holds, if Δ(x) = p(x)^{2} + 4q(x) = (ax + b)^{2} + 4d and 2P_{1}(x) − P_{0}(x)p(x) = 2(ex + f) − g(ax + b) = (2e − ag)x + 2f − bg have no common roots. This means that, if 2e = ag, 2f = bg does not hold or

x = \frac{bg − 2f}{2e − ag}

is not a root of Δ(x), then we get our assertion for all S(x) ∈ ℂ[x], deg S ≥ 2. This is satisfied for example if we consider sequences P_{n}(x) ∈ ℂ[x] defined by

P_{n+1}(x) = (ax + b)P_{n}(x) + dP_{n−1}(x), n ≥ 1,
P_{−1}(x) = 0, P_{0}(x) = g ≠ 0,

where a, b, d, g are complex numbers with adg ≠ 0.

4.4 Auxiliary results

In this section we collect some important theorems which we will need in our proofs. We need some results from the theory of algebraic function fields, which can be found for example in the monograph of Stichtenoth [90]. We will need the following estimates for the genus of a function field F/K (cf. [90], page 130 and 131).

Theorem 4.7. (Castelnuovo’s Inequality) Let F/K be a function field with constant field K. Suppose there are given two subfields F_{1}/K and F_{2}/K of F/K satisfying

(1) F = F_{1}F_{2} is the compositum of F_{1} and F_{2}.

(2) [F : F_{i}] = n_{i}, and F_{i}/K has genus g_{i} (i = 1, 2).

Then the genus g of F/K is bounded by

g ≤ n_{1}g_{1} + n_{2}g_{2} + (n_{1} − 1)(n_{2} − 1).

In the special case F_{1} = K(x) and F_{2} = K(y), Castelnuovo’s Inequality yields:

Theorem 4.8. (Riemann’s Inequality) Let φ be a non-constant irreducible polynomial in two variables with coefficients in K and suppose that F = K(x, y) with φ(x, y) = 0. Then we have the following estimate for the genus g of F/K:

g ≤ ([F : K(x)] − 1) · ([F : K(y)] − 1).

Observe that Riemann’s Inequality (and therefore also Castelnuovo’s Inequality) is often sharp, and it cannot be improved in general.

Let K be an algebraically closed field of characteristic 0. Let K be a finite extension of K(x) where x is transcendental over K. For ξ ∈ K define the valuation ν_{ξ} such that for Q ∈ K(x) we have Q(x) = (x − ξ)^{ν_{ξ}(Q)}A(x)/B(x) where A, B are polynomials with A(ξ)B(ξ) ≠ 0. Further, for Q = A/B with A, B ∈ K[x] we put deg Q := deg A−deg B; thus ν_{∞} := − deg is a discrete valuation on K(x). Each of the valuations ν_{ξ}, ν_{∞} can be extended in at most [K : K(x)] ways to a discrete valuation on K and in this way one obtains all discrete valuations on K. A valuation on K is called finite if it extends ν_{ξ} for some ξ ∈ K and infinite if it extends ν_{∞}. We choose one of the extensions of ν_{∞} to L and denote this by −ord. Thus ord is a function from K to ℚ having the properties

(a) ord(Q) = deg Q for Q ∈ K[x],
(b) ord(AB) = ord(A) + ord(B) for A, B ∈ K,
(c) ord(A + B) ≤ max{ord(A), ord(B)} for A, B ∈ K,
(d) ord(A + B) = max{ord(A), ord(B)} for A, B ∈ K

with ord(A) ≠ ord(B).

4.5 Proof of Theorem 4.1

First we reduce the solvability of (4.8) to the solvability of three systems of exponential equations in n, m. We start with a sequence of polynomials (P_{n}(x))_{n=0}^{∞} defined by (4.1). Then, in the sequel α(x), \overline{α}(x), g_{1}(x), g_{2}(x) are always be given by (4.3).

Lemma 4.1. Let (G_{n}(x))_{n=0}^{∞} be a sequence of polynomials defined by (4.1) and let P ∈ K[x], deg P ≥ 1. Then (4.8) has at most exp(18^{9} · 3) solutions m, n ∈ ℤ, m ≠ n, if g_{1}(x), g_{2}(x) ≠ 0 and which do not satisfy any of the systems:

\{ \begin{matrix}g_{1}(x)α(x)^{n} + g_{2}(x)\overline{α}(x)^{n} = 0 \\ g_{1}(P (x))α(P (x))^{m} + g_{2}(P (x))\overline{α}(P (x))^{m} = 0 \end{matrix}  (4.14)
\{ \begin{matrix}g_{1}(x)α(x)^{n} = g_{1}(P (x))α(P (x))^{m} \\ g_{2}(x)\overline{α}(x)^{n} = g_{2}(P (x))\overline{α}(P (x))^{m} \end{matrix}  (4.15)
\{ \begin{matrix}g_{2}(x)\overline{α}(x)^{n} = g_{1}(P (x))α(P (x))^{m} \\ g_{1}(x)α(x)^{n} = g_{2}(P (x))\overline{α}(P (x))^{m} \end{matrix}  (4.16)

Proof. First we define

K = K(x, \sqrt{p(x)^{2} + 4q(x)}, \sqrt{p(P (x))^{2} + 4q(P (x))}).

Clearly, K is finitely generated extension field of ℚ. Furthermore, let Γ be the multiplicative subgroup of (K^{∗})^{3} generated by

(α(x), \overline{α}(x), 1) and (\overline{α}(P (x))^{−1}, \overline{α}(P (x))^{−1}, α(P (x))/\overline{α}(P (x))).

We consider now for n ≠ m the equation G_{n}(x) = G_{m}(P (x)) and obtain

g_{1}(x)α(x)^{n} + g_{2}(x)\overline{α}(x)^{n} − g_{1}(P (x))α(P (x))^{m} − g_{2}(P (x))\overline{α}(P (x))^{m} = 0.

This yields

\frac{g_{1}(x)}{g_{2}(P (x))} \frac{α(x)^{n}}{\overline{α}(P (x))^{m}} + \frac{g_{2}(x)}{g_{2}(P (x))} \frac{\overline{α}(x)^{n}}{\overline{α}(P (x))^{m}} − \frac{g_{1}(P (x))}{g_{2}(P (x))} \frac{α(P (x))^{m}}{\overline{α}(P (x))^{m}} = 1.  (4.17)

Now we consider the weighted unit equation

\frac{g_{1}(x)}{g_{2}(P (x))}x_{1} + \frac{g_{2}(x)}{g_{2}(P (x))}x_{2} − \frac{g_{1}(P (x))}{g_{2}(P (x))}x_{3} = 1 in (x_{1}, x_{2}, x_{3}) ∈ Γ.  (4.18)

According to Theorem 1.7, equation (4.18) has at most exp(18^{9} · 3) solutions if no non-trivial subsum vanishes. By observing that g_{1}(x), g_{2}(x) ≠ 0 this means that (4.17) has at most exp(18^{9} · 3) solutions m, n not satisfying (4.14), (4.15) and (4.16). □

In the next lemma we calculate the order of α(x) and \overline{α}(x) respectively in the function field K/K, where K is defined as in the previous proof. We will assume that

ord(α) ≥ ord(\overline{α}).

If this is not satisfied we can achieve this by interchanging α(x) and \overline{α}(x). Then we have:

Lemma 4.2. Let (G_{n}(x))_{n=0}^{∞} be a sequence of polynomials defined by (4.1) and assume that 2 deg p > deg q ≥ 0. Then

ord(α) = deg p,  (4.19)
ord(\overline{α}) = deg q − deg p < deg p.  (4.20)

Proof. First assume ord(α) = ord(\overline{α}). Then by (a), (c), (b) we have

deg p = ord(α + \overline{α}) ≤ ord(α) = \frac{1}{2} deg q

which is against our assumption. Therefore, ord(α) > ord(\overline{α}). Now it follows from (a),

(d) that deg p = ord(α + \overline{α}) = ord(α). Using (a), (b) and α(x)\overline{α}(x) = −q(x) we then obtain

ord(\overline{α}) = deg q − deg p < deg p.

Therefore the proof is finished. □

Next we prove the following lemma.

Lemma 4.3. Let (G_{n}(x))_{n=0}^{∞} be a sequence of polynomials defined by (4.1) and let P ∈ K[x], deg P ≥ 1. Assume that neither α(x)/\overline{α}(x), nor α(P (x))/\overline{α}(P (x)) is a root of unity. We consider the systems of equations

\{ \begin{matrix}g_{1}(x)α(x)^{n} + g_{2}(x)\overline{α}(x)^{n} = 0 \\ g_{1}(P (x))α(P (x))^{m} + g_{2}(P (x))\overline{α}(P (x))^{m} = 0 \end{matrix}

The first equation has at most one solution in n, and the second one at most one solution in m.

Proof. This follows from the fact that neither α(x)/\overline{α}(x), nor α(P (x))/\overline{α}(P (x)) are roots of unity. In particular, assume that we have two solutions n_{1}, n_{2}. Then we obtain

−\frac{g_{1}(x)}{g_{2}(x)} = (\frac{\overline{α}(x)}{α(x)})^{n_{1}} = (\frac{\overline{α}(x)}{α(x)})^{n_{2}} ,

which implies that n_{1} = n_{2}. □

Proof of Theorem 4.1.

First, it is clear that we have

ord(α − \overline{α}) = deg p.

Moreover, the following relations hold

ord(α(P)) = deg p deg P,
ord(\overline{α}(P)) = (deg q − deg p) deg P,
ord(α(P) − \overline{α}(P)) = deg p deg P.

The important relations

g_{1}(x)(α(x) − \overline{α}(x)) = G_{1}(x) − G_{0}(x)\overline{α}(x),  (4.21)
g_{2}(x)(\overline{α}(x) − α(x)) = G_{1}(x) − G_{0}(x)α(x)  (4.22)

are consequences of (4.4). Observe that under the condition 2 deg p > deg q ≥ 0 our sequence (G_{n}(x))_{n=0}^{∞} is nondegenerate. This follows from the fact that α(x)^{n} = \overline{α}(x)^{n} implies ord(α)=ord(\overline{α}), which by Lemma 4.2 yields a contradiction. The same is true for the quotient α(P (x))/\overline{α}(P (x)).

In order to finish our proof, we want to show that g_{1}(x), g_{2}(x) ≠ 0 and that (4.15) and (4.16) have no solutions.

Case 1. deg G_{1} > deg G_{0} + deg p. In this case we have

ord(G_{1} − G_{0}\overline{α}) = deg G_{1},
ord(G_{1} − G_{0}α) = deg G_{1}.

This implies

ord(G_{1}(P) − G_{0}(P)\overline{α}(P)) = deg G_{1} deg P,
ord(G_{1}(P) − G_{0}(P)α(P)) = deg G_{1} deg P.

Therefore we get

ord(g_{1}) = deg G_{1} − deg p,
ord(g_{2}) = deg G_{1} − deg p.

Observe that from this we can conclude that g_{1}(x), g_{2}(x) ≠ 0. Now assume that m, n is a solution of (4.15). Then we obtain by calculating the order of both sides of the equations and by using Lemma 4.2

(deg G_{1} − deg p) + n deg p = (deg G_{1} − deg p) deg P + m deg p deg P, (4.23)
(deg G_{1} − deg p) + n(deg q − deg p) = (deg G_{1} − deg p) deg P +  (4.24)
+m(deg q − deg p) deg P.

Subtraction yields

n(2 deg p − deg q) = m(2 deg p − deg q) deg P.

By our assumption 2 deg p > deg q we derive

n = m deg P ,  (4.25)

and substituting this in (4.23) implies

(deg G_{1} − deg p) = (deg G_{1} − deg p) deg P.

But this yields deg P = 1, which by (4.25) gives m = n, or deg G_{1} = deg p, leading to a contradiction.

In the same way we conclude that a solution m, n of (4.16) implies

(deg G_{1} − deg p) + n deg p = (deg G_{1} − deg p) deg P +
+m(deg q − deg p) deg P,
(deg G_{1} − deg p) + n(deg q − deg p) = (deg G_{1} − deg p) deg P +
+m deg p deg P.

Again subtraction yields

n(2 deg p − deg q) = m(deg q − 2 deg p) deg P ,

and therefore we get n = −m deg P, which contradicts deg P ≠ 0.

Case 2. deg G_{1} < deg G_{0} + deg q − deg p. Here we have

ord(G_{1} − G_{0}\overline{α}) = deg G_{0} + deg q − deg p,
ord(G_{1} − G_{0}α) = deg G_{0} + deg p.

Thus

ord(g_{1}) = deg G_{0} + deg q − 2 deg p,
ord(g_{2}) = deg G_{0}.

From this we derive g_{1}(x), g_{2}(x) ≠ 0. We again can conclude by Lemma 4.2 that a solution of (4.15) would imply

(deg G_{0} + deg q − 2 deg p) + n deg p = (deg G_{0} + deg q −  (4.26)
−2 deg p) deg P + +m deg p deg P,
deg G_{0} + n(deg q − deg p) = deg G_{0} deg P + m(deg q − deg p) deg P. (4.27)

Subtraction yields

(n − 1)(2 deg p − deg q) = (m − 1) deg P (2 deg p − deg q)

and therefore

(n − 1) = (m − 1) deg P.

By (4.26) we obtain

(deg G_{0} + deg q − deg p)(1 − deg P) = 0.

This yields deg P = 1, which again implies n = m, or deg G_{0} + deg q − deg p = 0, which gives deg G_{1} < 0, in both cases a contradiction.

Again we get from (4.16)

(deg G_{0} + deg q − 2 deg p) + n deg p = deg G_{0} deg P +
+m(deg q − deg p) deg P,
deg G_{0} + n(deg q − deg p) = (deg G_{0} + deg q − 2 deg p) deg P +
+m deg p deg P.

Subtraction gives

(n − 1) = −(m − 1) deg P,

which implies deg P = 0, a contradiction.

Now by Lemma 4.1 we get that (4.8) has at most

1 + exp(18^{9} · 3) ≤ exp(18^{10})

solutions n, m ∈ ℤ, n, m ≥ 0 with m ≠ n. The second part of our upper bound will follow from the proof of Theorem 4.2 where a different proof method is used and thus the proof is finished. □

4.6 Proof of Theorem 4.5

Here (P_{n}(x))_{n=0}^{∞} is an OPS and we have in the sense of (4.12) and (4.13)

p(x) = ax + b, q(x) = d.

Again we want to apply Lemma 4.1 to show that the equation

P_{n}(x) = P_{m}(S(x)),  (4.28)

where S(x) ∈ ℂ[x], deg S ≥ 1, has only finitely many solutions m, n ∈ ℤ, m ≠ n. Let α(x), \overline{α}(x), g_{1}(x), g_{2}(x) be given by (4.3).

As above we can assume without loss of generality that ord(α) ≥ ord(\overline{α}). By observing 2 deg p = 2 > 0 = deg q, we get by Lemma 4.2 that

ord(α) = 1 and ord(\overline{α}) = −1.

This yields

ord(α − \overline{α}) = 1,
ord(P_{1} − P_{0}\overline{α}) = 1.

We have to calculate ord(P_{1} − P_{0}α). Using that P_{0}(x) = g, P_{1}(x) = ex + f and the assumption that e = ag we get

(P_{1}(x) − P_{0}(x)α(x))(P_{1}(x) − P_{0}(x)\overline{α}(x)) =
= P_{1}(x)^{2} − P_{0}(x)P_{1}(x)p(x) − P_{0}(x)^{2}q(x) =
= (ex + f)^{2} − g(ex + f)(ax + b) − g^{2}d = sx + t

for certain s, t ∈ ℂ. By invoking (a), (b) we then obtain

ord(P_{1} − P_{0}α) =: w ≤ 1 − ord(P_{1} − P_{0}\overline{α}) = 0.

Observe that Lemma 4.3 can be sharpened in this case. Because of the fact that deg P_{n} = n the number of solutions of (4.14) is zero. Furthermore it is clear that g_{1}(x), g_{2}(x) cannot be zero in this case, because the following relations hold

ord(g_{1}) = 0,
ord(g_{2}) = w − 1 < 0.

Now assume that m, n ∈ ℤ, m ≠ n is a solution of (4.15). Then we get

n = m deg S,
(w − 1) − n = [(w − 1) − m] deg S.

Consequently deg S = 1, and therefore m = n, a contradiction.

In the same way we get for a solution of (4.16), that

(w − 1) − n = m deg S,
n = [(w − 1) − m] deg S.

Adding the two equations gives deg S = 1, and thus n = (w − 1) − m, a contradiction because the left side of the equation is positive and the right side negative.

By Lemma 4.1 the theorem follows and the proof is finished as the second part of the bound will follow from Theorem 4.6. □

4.7 Proof of Theorem 4.3

We start our proof with some useful lemmas.

Lemma 4.4. Let A, B, P ∈ K[x]. If gcd(A, B) = 1 then gcd(A(P), B(P)) = 1.

This lemma is a special case of a lemma in the monograph of Schinzel [74], page 16. It was originally proved in [73].

We will use the same notations as introduced in the proof of Theorem 4.1. There we calculated the order which was defined as the negative value of some valuation extending 1/x from K(x) to the function field

K = K(x)(\sqrt{Δ(x)}, \sqrt{Δ(P (x))})

of the elements g_{1}(x), g_{2}(x) by using the equations (4.21) and (4.22). Here we want to calculate the valuations ν(g_{1}) and ν(g_{2}) where ν extends ν_{ξ} to K for some ξ ∈ K.

Lemma 4.5. Let (G_{n}(x))_{n=0}^{∞} be a sequence of polynomials defined by (4.1) and assume that gcd(2G_{1} − G_{0}p, Δ) = 1. If ν is a finite valuations on K with ν(Δ) > 0 then ν(g_{1}\sqrt{Δ}) = ν(g_{2}\sqrt{Δ}) = 0.

Proof. We have equation (4.21)

g_{1}(x)(α(x) − \overline{α}(x)) = G_{1}(x) − G_{0}(x)\overline{α}(x),

which we may rewrite in the form

2g_{1}(x)\sqrt{Δ(x)} = 2G_{1}(x) − G_{0}(x)p(x) + G_{0}(x)\sqrt{Δ(x)}.

By our assumption that gcd(2G_{1} − G_{0}p, Δ) = 1 we have that

ν(2G_{1} − G_{0}p) = 0.

Because of the fact that

ν(G_{0}\sqrt{Δ}) = ν(G_{0}) + \frac{1}{2}ν(Δ) > 0

we get that

ν(g_{1}\sqrt{Δ}) = ν(2g_{1}\sqrt{Δ}) = min\{ν(2G_{1} − G_{0}p\}, ν(G_{0}\sqrt{Δ})) = 0

which was our assumption.

The same holds for g_{2}(x) and therefore the proof is finished. □

Assumption (4) of Theorem 4.3 together with Lemma 4.4 imply

gcd(2G_{1}(P) − G_{0}(P)p(P), Δ(P)) = 1.

As

2g_{1}(P (x))\sqrt{Δ(P (x))} =
= 2G_{1}(P (x)) − G_{0}(P (x))p(P (x)) + G_{0}(P (x))\sqrt{Δ(P (x))}

we have again, as in the proof of the previous lemma, that for a finite valuation ν on K with ν(Δ(P)) > 0 we have ν(g_{1}\sqrt{Δ(P)}) = ν(g_{2}\sqrt{Δ(P)}) = 0.

Proof of Theorem 4.3.

We are intended to prove that the systems of equations (4.15) and (4.16) are not solvable.

Consider for example the equation

g_{1}(x)α(x)^{n} = g_{1}(P (x))α(P (x))^{m}.  (4.29)

The other equations can be handled analogously.

We have deg Δ(P) = deg Δ deg P > deg Δ > 0, as deg P > 1 by assumption (2). Hence Δ(P) has a zero ξ such that

ν_{ξ}(Δ(P)) > ν_{ξ}(Δ) ≥ 0.

This implies that there is a finite valuation ν on K such that

ν(g_{1}(P)) = −ν(Δ(P)).

Next we want to show that ν(α(P)) = 0. Indeed, as ν(Δ(P)) > 0 and

α(P (x)) = \frac{p(P (x)) + \sqrt{Δ(P (x))}}{2}

we have

ν (α(P) − \frac{1}{2}p(P)) > 0.  (4.30)

By assumption (3) of Theorem 4.3 and Lemma 4.4 we have gcd(p(P), q(P)) = 1 which implies min{ν(p(P)), ν(q(P))} = 0. If ν(p(P)) > 0 then from

ν(Δ(P)) = ν(p(P)^{2} + 4q(P)) > 0

it follows that then also ν(q(P)) > 0 which is impossible. Therefore, ν(p(P)) = 0. Consequently we have ν(α(P)) = 0. In a similar fashion it follows that ν(\overline{α}(P)) = 0.

Thus equation (4.29) implies

ν(g_{1}) + nν(α) = ν(g_{1}(P)),

which yields

nν(α) = ν(g_{1}(P)) − ν(g_{1}) < 0,

hence (4.29) has no solution in n, if ν(α) ≥ 0 and at most one, if ν(α) < 0.

Studying the second equation of (4.15) we may conclude in the same way that this equation has no solution in n, if ν(\overline{α}) ≥ 0 and at most one if ν(\overline{α}) < 0. Thus the system of equations (4.15) may have a solution only if ν(α), ν(\overline{α}) < 0. Observe that this is impossible since α(x), \overline{α}(x) are integral over K[x], as they are zeros of the monic equation T^{2} − p(x)T − q(x) = 0 with coefficients in K[x]. The integral closure of K[x] in K consists of those elements f such that ν(f) ≥ 0 for every finite valuation ν of K. So in particular, ν(α) ≥ 0, ν(\overline{α}) ≥ 0. Hence (4.15) has no solution.

The proof of the unsolvability of (4.16) is analogous. It is clear that g_{1}(x), g_{2}(x) ≠ 0 holds, because from assumption (1) we can conclude that there is a zero ζ of Δ(x), for which we can derive using Lemma 4.5 that ν(g_{1}) < 0 and ν(g_{2}) < 0, where ν is a finite valuation extending ν_{ζ} to K. Consequently they must be different from zero. Since Lemma 4.3 is true also in this case, (because α(x)/\overline{α}(x) and α(P (x))/\overline{α}(P (x)) are not roots of unity), we get the assertion of Theorem 4.3 by Lemma 4.1. The second part of the bound will follow from Theorem 4.4. □

4.8 Proof of the Theorems 4.2, 4.4 and 4.6

Again we will use all notations from above. Especially, let K/K be the algebraic function field in one variable defined by

K = K(x, \sqrt{p(x)^{2} + 4q(x)}, \sqrt{p(P (x))^{2} + 4q(P (x))}).

Moreover, we consider

Γ = ⟨α(x), \overline{α}(x), α(P (x)), \overline{α}(P (x))⟩_{(K^{∗},·)}

which means that Γ is the subgroup of (K^{∗}, ·) the multiplicative group of K generated by α(x), \overline{α}(x), α(P (x)), \overline{α}(P (x)). Now we have the following lemma.

Lemma 4.6. There exists a finite subset S ⊂ M_{K} of valuations of the function field K such that Γ is contained in the group of S-units U_{S} and such that

|S| ≤ 4 deg q(deg P + 1) + 4.

Proof. Let S_{∞} be the set of infinite valuations of K and S_{0} the set of finite valuations of K. Note that for every ν ∈ S_{0} we have ν(α) ≥ 0, ν(\overline{α}) ≥ 0, ν(α(P)) ≥ 0, ν(\overline{α}(P)) ≥ 0 since these functions are integral over K[x]. Take S = S_{∞} ∪ S_{1} ∪ S_{2} ∪ S_{3} ∪ S_{4}, where

S_{1} = {ν ∈ S_{0}| ν(α) > 0},
S_{2} = \{ν ∈ S_{0}| ν(\overline{α}) > 0\},
S_{3} = {ν ∈ S_{0}| ν(α(P)) > 0},
S_{4} = \{ν ∈ S_{0}| ν(\overline{α}(P)) > 0\}.

Then clearly Γ is a subgroup of U_{S}. Since [K : K(x)] ≤ 4, we have |S_{∞}| ≤ 4. Further, α(x) · \overline{α}(x) · α(P (x)) · \overline{α}(P (x)) = q(x) · q(P (x)) =: Q(x). Therefore, S_{1} ∪ S_{2} ∪ S_{3} ∪ S_{4} =: S_{5} := {ν ∈ S_{0} : ν(Q) > 0}. Each of the valuations in S_{5} is an extension to K of some valuation ν_{ξ} on K(x) corresponding to a zero ξ of Q(x). The polynomial Q(x) has at most deg Q = deg q(deg P + 1) zeros, and for each of these zeros ξ, the valuation ν_{ξ} can be extended in at most four ways to a valuation on K. Therefore, |S_{5}| ≤ 4 deg q(deg P + 1). This implies Lemma 8. □

Next we want to estimate the genus of the function field K/K. This can be done using Castelnuovo’s Inequality (Theorem 4.7).

Lemma 4.7. We denote by g the genus of the function field K/K. Then we have

g ≤ 2 max{2 deg p, deg q}(deg P + 1) − 3.

Proof. First observe that we have

F = K(x, \sqrt{Δ(x)}, \sqrt{Δ(P (x))}) = K(x, \sqrt{Δ(x)}) · K(x, \sqrt{Δ(P (x))}).

Let us denote F_{1} = K(x, \sqrt{Δ(x)}), F_{2} = K(x, \sqrt{Δ(P (x))}). Thus we have

F_{1} = K(x, y), φ_{1}(x, y) = y^{2} − Δ(x) = 0

and

F_{2} = K(x, y), φ_{2}(x, y) = y^{2} − Δ(P (x)) = 0.

Furthermore we denote by g_{i} the genus of F_{i}/K (i = 1, 2). We have

n_{1} = [F : F_{1}] ≤ 2 and n_{2} = [F : F_{2}] ≤ 2.

By Riemann’s Inequality (Theorem 4.8) we get the following estimates:

g_{1} ≤ ([F_{1} : K(x)] − 1) · ([F_{1} : K(y)] − 1) ≤ deg Δ − 1,
g_{2} ≤ ([F_{2} : K(x)] − 1) · ([F_{2} : K(y)] − 1) ≤ deg Δ deg P − 1.

Since Δ(x) = p(x)^{2} + 4q(x) we have deg Δ ≤ max{2 deg p, deg q}. Now using Castelnuovo’s Inequality (Theorem 4.7) we get

g ≤ 2(deg Δ − 1) + 2(deg Δ deg P − 1) + 1 = 2 deg Δ(deg P + 1) − 3,

and therefore our proof is finished. □

Finally, we need the following lemma.

Lemma 4.8. Assume 2 deg p > deg q ≥ 0 or gcd(p, q) = 1 and p, q not both in K. Let γ_{1}, γ_{2} be nonzero elements of K. Then there is at most one pair of integers n, m such that

γ_{1} \frac{α(x)^{n}}{\overline{α}(P (x))^{m}} ∈ K^{∗} and γ_{2} \frac{\overline{α}(x)^{n}}{\overline{α}(P (x))^{m}} ∈ K^{∗}  (4.31)

or

γ_{1} \frac{α(x)^{n}}{\overline{α}(P (x))^{m}} ∈ K^{∗} and γ_{2} \frac{α(P (x))^{m}}{\overline{α}(P (x))^{m}} ∈ K^{∗}  (4.32)

or

γ_{1} \frac{\overline{α}(x)^{n}}{\overline{α}(P (x))^{m}} ∈ K^{∗} and γ_{2} \frac{α(P (x))^{m}}{\overline{α}(P (x))^{m}} ∈ K^{∗},  (4.33)

respectively.

Proof. First we prove equation (4.31). Suppose there are two such pairs (n_{1}, m_{1}), (n_{2}, m_{2}). Let n = n_{1} − n_{2}, m = m_{1} − m_{2}. Then (γ_{1}/γ_{2})(α(x)/\overline{α}(x))^{n_{i}} ∈ K^{∗} for i = 1, 2, hence (α(x)/\overline{α}(x))^{n} ∈ K^{∗}. Suppose n ≠ 0. Then α(x)/\overline{α}(x) ∈ K^{∗}. Using p(x) = α(x) + \overline{α}(x), q(x) = α(x) · \overline{α}(x) it follows that p(x) = c_{1}α(x), q(x) = c_{2}α(x)^{2} with c_{1}, c_{2} ∈ K^{∗} and so p(x)^{2} = c_{3}q(x) with c_{3} ∈ K^{∗}. But this contradicts both 2 deg p > deg q ≥ 0 and gcd(p, q) = 1. It follows that n = 0, whence n_{1} = n_{2} and so (n_{1}, m_{1}) = (n_{2}, m_{2}). This proves the first part of the lemma.

Now we consider equation (4.32). As above we assume that there are two such pairs. Hence we get (α(P (x))/\overline{α}(P (x)))^{m} ∈ K^{∗}. This implies that either α(P (x))/\overline{α}(P (x)) ∈ K^{∗} which is impossible by the same arguments as above or m = 0. But now, using the other expression in (4.32) we get that also n = 0 must hold. Consequently we have (n_{1}, m_{1}) = (n_{2}, m_{2}). So we proved the second part of our lemma.

The arguments for (4.33) are the same as for (4.32). This proves the lemma also in the third case. □

Assume that n, m are integers satisfying G_{n}(x) = c G_{m}(P (x)) for some c ∈ K^{∗}. It follows that

β_{1}x_{1} + β_{2}x_{2} + β_{3}x_{3} = 1

where

β_{1} := \frac{g_{1}(x)}{g_{2}(P (x))}, β_{2} := \frac{g_{2}(x)}{g_{2}(P (x))}, β_{3} := −\frac{g_{1}(P (x))}{g_{2}(P (x))},  (4.34)

x_{1} = c^{−1} \frac{α(x)^{n}}{\overline{α}(P (x))^{m}} , x_{2} = c^{−1} \frac{\overline{α}(x)^{n}}{\overline{α}(P (x))^{m}} , x_{3} = \frac{α(P (x))^{m}}{\overline{α}(P (x))^{m}} .

Observe that x_{1}, x_{2}, x_{3} are elements of the set U_{S} which exists by Lemma 4.6. This is because of fact that Γ is contained in U_{S} and c ∈ K^{∗}. Lemma 4.8 implies that any given pair of elements (x_{i}, x_{j}) gives rise to at most one pair (n, m), especially any triple (x_{1}, x_{2}, x_{3}) induces at most one solution (n, m) of the equation in consideration.

By Theorem 1.15, either β_{1}x_{1}, β_{2}x_{2}, β_{3}x_{3} all belong to K^{∗}, which by Lemma 4.8 is possible for at most one pair (n, m), or (x_{1}, x_{2}, x_{3}) lies in one of at most log(g + 2) · (4e)^{4s+2} proper linear subspaces of K^{3}, where s denotes the cardinality of the set of valuations S introduced by Lemma 4.8. That is, (x_{1}, x_{2}, x_{3}) satisfies one of at most log(g + 2)(4e)^{4s+2} relations of the shape

γ_{1}x_{1} + γ_{2}x_{2} + γ_{3}x_{3} = 0  (4.35)

with (γ_{1}, γ_{2}, γ_{3}) a nonzero triple in K^{3}. Assume for the moment that γ_{i} ≠ 0 for i = 1, 2, 3 and write Δ_{ij} = γ_{j}^{−1}(β_{i}γ_{j} −β_{j}γ_{i}). Assume for the moment also that all Δ_{ij} are nonzero. Then we have

Δ_{13}x_{1} + Δ_{23}x_{2} = 1, Δ_{12}x_{1} + Δ_{32}x_{3} = 1, Δ_{21}x_{2} + Δ_{31}x_{3} = 1.

In fact it suffices to consider one of these equation for example the first one

Δ_{13}x_{1} + Δ_{23}x_{2} = 1.

Lemma 4.8 implies that there is at most one pair (n, m) such that both quantities Δ_{13}x_{1} and Δ_{23}x_{2} belong to K. Theorem 1.16 implies that there are at most 2 · 7^{2s} pairs (x_{1}, x_{2}) such that at least one of these quantities does not belong to K. It follows that there are at most 1 + 2 · 7^{2s} pairs (n, m) for which γ_{1}x_{1} + γ_{2}x_{2} + γ_{3}x_{3} = 0.

Next, we handle the case that one of the Δ_{ij}, where (i, j) = (1, 3) or (2, 3), is zero. We assume that all γ_{i} ≠ 0 for i = 1, 2, 3. It is clear that Δ_{ij} = 0 implies also Δ_{ji} = 0. Moreover, we remark that if Δ_{ij} = 0 then neither Δ_{ik} = 0 nor Δ_{jk} = 0 where {i, j, k} = {1, 2, 3} can hold. Because assume that Δ_{ij} = 0 and Δ_{ik} = 0. This implies that also Δ_{jk} = 0 which means that all the quantities Δ_{13}, …, Δ_{31} are zero. Hence we would get

β_{1}x_{1} + β_{2}x_{2} + β_{3}x_{3} = (γ_{1}x_{1} + γ_{2}x_{2} + γ_{3}x_{3}) (\frac{β_{3}}{γ_{3}} ) = 1,

a contradiction to equation (4.35). From this discussion it follows that Δ_{ij} = 0 for (i, j) = (1, 3) or (2, 3) implies that we have

Δ_{ik}x_{i} + Δ_{jk}x_{j} = 1,

with nonzero Δ_{ik}, Δ_{jk} and {i, j, k} = {1, 2, 3}. As above, Theorem 1.16 implies that there are at most 2 · 7^{2s} pairs (x_{i}, x_{j}) such that at least one of these quantities does not belong to K which can happen for at most one pair (n, m) by Lemma 4.8.

Finally, we handle the case that γ_{i} = 0 for some i = 1, 2, 3. Observe that at most one of the γ_{i} can be zero. Now we assume that γ_{1} = 0. Then (4.35) becomes

γ_{2}x_{2} + γ_{3}x_{3} = 0.

Therefore we have

β_{1}x_{1} + (β_{3} − \frac{γ_{3}}{γ_{2}} β_{2}) x_{3} = 1.

From here we can deduce that there are at most 1 + 2 · 7^{2s} pairs of solutions (n, m) under the condition that β_{3} − (γ_{3}/γ_{2})β_{2} ≠ 0. If this condition is not satisfied then we have

β_{1}x_{1} = 1 and β_{2}x_{2} + β_{3}x_{3} = 0

which means that (4.34) has a vanishing subsum. The cases γ_{2} = 0 and γ_{3} = 0 are totally analogous. We get in both cases that there are at most 1 + 2 · 7^{2s} pairs of solutions (n, m), if we assume that (4.34) has no vanishing subsum.

The cases for vanishing subsums of equation (4.34) can be rewritten in the following form:

\{ \begin{matrix}g_{1}(x)α(x)^{n} = c g_{2}(P (x))\overline{α}(P (x))^{m} \\ g_{2}(x)\overline{α}(x)^{n} = c g_{1}(P (x))α(P (x))^{m} \end{matrix}
\{ \begin{matrix}g_{2}(x)\overline{α}(x)^{n} = c g_{2}(P (x))\overline{α}(P (x))^{m} \\ g_{1}(x)α(x)^{n} = c g_{1}(P (x))α(P (x))^{m} \end{matrix}
\{ \begin{matrix}g_{1}(x)α(x)^{n} + g_{2}(x)\overline{α}(x)^{n} = 0 \\ g_{1}(P (x))α(P (x))^{m} + g_{2}(P (x))\overline{α}(P (x))^{m} = 0 \end{matrix}

But these three equations are up to the constant c totally the same as in the proof of Theorem 4.1, 4.3 and 4.5 . Because of the fact that we have shown the unsolvability of the first and second of the above systems by calculating the valuation at some finite or infinite valuation in our function field K and the fact that for every valuation on K we have

ν(c) = 0,

we get that the third system has at most one solution in (n, m), where the second and the third systems do not have a solution at all.

Altogether we get for the number of pairs (n, m) of integers with n ≠ m such that there exists a c ∈ K^{∗} with G_{n}(x) = c G_{m}(P (x)) the upper bound

1 + log(g + 2)(4e)^{4s+2} · [7 + 12 · 7^{2s}] ≤ log(g + 2)(4e)^{4s+2}7^{2s+2}.

Now using the estimation for the genus of our function field (Lemma 4.7) and the estimate for the cardinality of the set S (Lemma 4.8) we get that the number of solutions can be bounded by

C(p, q, P) = C̃(p, q, P) = log(2 max{2 deg p, deg q}(deg P + 1)) ·
·(4e)^{16 deg q(deg P +1)+18}7^{8 deg q(deg P +1)+10}.

Last observe that in Theorem 4.2 we have assumed that 2 deg p > deg q and in Theorem 4.6 we have

p(x) = ax + b and q(x) = d,

consequently we know that deg q = 0 and deg p = 1. This proves the bounds as claimed. □

On the equation G_{n}(x) = G_{m}(P (x)) for third order linear recurring sequences

Let K be a field of characteristic 0 and let a, b, c, G_{0}, G_{1}, G_{2}, P ∈ K[x], deg P ≥ 1. Further let the sequence of polynomials (G_{n}(x))_{n=0}^{∞} be defined by the third order linear recurring sequence

G_{n+3}(x) = a(x)G_{n+2}(x) + b(x)G_{n+2}(x) + c(x)G_{n}(x), for n ≥ 0.

In this chapter we give conditions under which the Diophantine equation

G_{n}(x) = G_{m}(P (x))

has at most exp(10^{24}) many solutions (n, m) ∈ ℤ^{2}, n, m ≥ 0. The proof uses a very recent result on S-unit equations over fields of characteristic 0 due to J.-H. Evertse, H. P. Schlickewei and W. M. Schmidt (cf. Theorem 1.7 and [45]).

This chapter is identically equal to a manuscript which is a joint work with A. Pethő and R. F. Tichy [50]. This paper is a continuation of the work of the authors on this equation in the case of second order linear recurring sequences (cf. Chapter 4 and [49]).

5.1 Introduction

Let K denote a field of characteristic 0. There is no loss of generality in assuming that this field is algebraically closed and we will assume this for the rest of the chapter. Let a, b, c, G_{0}, G_{1}, G_{2} ∈ K[x] and let the sequence of polynomials (G_{n}(x))_{n=0}^{∞} be defined by the third order linear recurring sequence

G_{n+3}(x) = a(x)G_{n+2}(x) + b(x)G_{n+1}(x) + c(x)G_{n}(x), for n ≥ 0.  (5.1)

CHAPTER 5. ON G_{n}(x) = G_{m}(P (x)) FOR ORDER THREE 81

By α_{1}(x), α_{2}(x), α_{3}(x) we denote the roots of the corresponding characteristic polynomial

T^{3} − a(x)T^{2} − b(x)T − c(x).  (5.2)

Setting S = T − \frac{1}{3}a(x) the characteristic polynomial becomes

S^{3} − p(x)S − q(x),

where

p(x) = \frac{1}{3}a(x)^{2} + b(x), q(x) = \frac{2}{27}a(x)^{3} + \frac{1}{3}a(x)b(x) + c(x).

Let

D(x) = (\frac{q(x)}{2})^{2} − (\frac{p(x)}{3})^{3} =

= \frac{1}{27}a(x)^{3}c(x) − \frac{1}{108}a(x)^{2}b(x)^{2} + \frac{1}{6}a(x)b(x)c(x) + \frac{1}{4}c(x)^{2} − \frac{1}{27}b(x)^{3}.

Moreover, let

u(x) = ^{3}\sqrt{\frac{q(x)}{2} + \sqrt{D(x)}}, v(x) = ^{3}\sqrt{\frac{q(x)}{2} − \sqrt{D(x)}}.

Then we have by Cardano’s formulae

α_{1}(x) = u(x) + v(x) + \frac{1}{3}a(x),  (5.3)
α_{2}(x) = −\frac{u(x) + v(x)}{2} + i\sqrt{3}\frac{u(x) − v(x)}{2} + \frac{1}{3}a(x) and  (5.4)
α_{3}(x) = −\frac{u(x) + v(x)}{2} − i\sqrt{3}\frac{u(x) − v(x)}{2} + \frac{1}{3}a(x).  (5.5)

We will always assume that the sequence (G_{n}(x))_{n=0}^{∞} is simple which means D(x) ≠ 0. Then, for n ≥ 0

G_{n}(x) = g_{1}(x)α_{1}(x)^{n} + g_{2}(x)α_{2}(x)^{n} + g_{3}(x)α_{3}(x)^{n},  (5.6)

where

g_{1}(x), g_{2}(x), g_{3}(x) ∈ K(i\sqrt{3})(x, \sqrt{D(x)}, u(x), v(x)).

(G_{n}(x))_{n=0}^{∞} is called nondegenerate, if no quotient α_{i}(x)/α_{j}(x), 1 ≤ i < j ≤ 3 is equal to a root of unity and degenerate otherwise.

Many Diophantine equations involving the recurrence (G_{n}(x))_{n=0}^{∞} were studied previously. For example let us consider the equation

G_{n}(x) = s(x),  (5.7)

where s(x) ∈ K[x] is given. We denote by N(s(x)) the number of integers n for which (5.7) holds. Schlickewei [78] established an absolute bound for N(s(x)), provided that the sequence is nondegenerate and that also α_{1}α_{2}, α_{3} are not equal to a root of unity. His bound was substantially improved by Beukers and Schlickewei [9] who showed that

N(s(x)) ≤ 61.

Very recently, Schmidt [80] obtained the remarkable result that for arbitrary nondegenerate complex recurrence sequences of order q one has N(a) ≤ C(q), where a ∈ ℂ and C(q) depends only (and in fact triply exponentially) on q (cf. Theorem 1.9).

Recently, the Pethő, Tichy and the author used new developments on S-unit equations over fields of characteristic 0 due to Evertse, Schlickewei and Schmidt (cf. Theorem 1.7 and [45]) to handle the equation G_{n}(x) = G_{m}(P (x)) for sequences (G_{n}(x))_{n=0}^{∞} of polynomials satisfying a second order linear recurring sequence (see Chapter 4). The result was: Let p, q, G_{0}, G_{1}, P ∈ K[x], deg P ≥ 1 and (G_{n}(x))_{n=0}^{∞} be defined by the second order linear recurrence

G_{n+2}(x) = p(x)G_{n+1}(x) + q(x)G_{n}(x), n ≥ 0.

Assume that the following conditions are satisfied: 2 deg p > deg q ≥ 0 and

deg G_{1} > deg G_{0} + deg p ≥ 0, or
deg G_{1} < deg G_{0} + deg q − deg p.

Then there are at most exp(10^{18}) pairs of integers (n, m) with n, m ≥ 0 with n ≠ m such that

G_{n}(x) = G_{m}(P (x))

holds. They showed a second result in their paper: Let Δ(x) = p(x)^{2} + 4q(x). Assume that

(1) deg Δ ≠ 0,

(2) deg P ≥ 2,

(3) gcd(p, q) = 1 and

(4) gcd(2G_{1} − G_{0}p, Δ) = 1.

Then there are at most exp(10^{18}) pairs of integers (n, m) with n, m ≥ 0 such that

G_{n}(x) = G_{m}(P (x))

holds.

The motivation for this equation was the following observation which shows that the problem is non-trivial: Consider the Chebyshev polynomials of the first kind, which are defined by

T_{n}(x) = cos(n arccos x).

It is well known that they satisfy the following second order recurring relation:

T_{0}(x) = 1, T_{1}(x) = x,
T_{n+2}(x) = 2xT_{n+1}(x) − T_{n}(x).

It is also well known and in fact easy to prove that

T_{2n}(x) = T_{n}(2x^{2} − 1).

This example shows that at least some conditions are needed to exclude this case.

It is the aim of this chapter to present suitable extensions of the above results for third order linear recurrences.

5.2 General results

Our first main result is a suitable analog of Theorem 1 in [49] (cf. Theorem 4.1) for the number of solutions of

G_{n}(x) = G_{m}(P (x))  (5.8)

for third order linear recurring sequence (G_{n}(x))_{n=0}^{∞}.

Theorem 5.1. Let a, b, c, G_{0}, G_{1}, G_{2}, P ∈ K[x], deg P ≥ 1 and (G_{n}(x))_{n=0}^{∞} be defined as above. Assume that the following conditions are satisfied: 3 deg a > deg c ≥ 0, 2 deg a > deg b and deg a + deg c > 2 deg b. Moreover, assume

deg G_{2} > deg G_{1} + deg a ≥ 0, and
deg G_{1} > deg G_{0} + \frac{1}{2}(deg c − deg a).

Then there are at most exp(10^{24}) pairs of integers (n, m) with n, m ≥ 0 with n ≠ m such that

G_{n}(x) = G_{m}(P (x))

holds.

Remark 5.1. We can also assume that

deg G_{2} < deg G_{1} + deg a, and
deg G_{1} < deg G_{0} + \frac{1}{2}(deg c − deg a).

instead of the conditions concerning the initial polynomials of the recurrence in the above theorem.

The case a(x) = 0 is excluded by the conditions in Theorem 5.1. This special case is handled in the following theorem.

Theorem 5.2. Let b, c, G_{0}, G_{1}, G_{2}, P ∈ K[x], deg P ≥ 1 and (G_{n}(x))_{n=0}^{∞} be defined by

G_{n+3}(x) = b(x)G_{n+1}(x) + c(x)G_{n}(x), for n ≥ 0.

Assume that the following conditions are satisfied: 3 deg b > 2 deg c ≥ 0 and

deg G_{2} > deg G_{1} + 2 deg b ≥ 0, and
deg G_{1} > deg G_{0} + 2 deg b − deg c.

Then there are at most exp(10^{24}) pairs of integers (n, m) with n, m ≥ 0 with n ≠ m such that

G_{n}(x) = G_{m}(P (x))

holds.

Remark 5.2. Observe that the conditions in this special case are quite similar to those for second order linear recurring sequences proved in [49] and mentioned in the introduction.

It is also possible to replace the conditions concerning the degree by algebraic conditions.

Theorem 5.3. Let a, b, c, G_{0}, G_{1}, G_{2}, P ∈ K[x] and (G_{n}(x))_{n=0}^{∞} be defined as above. Assume that

(1) deg D ≠ 0, deg q ≠ 0

(2) deg P ≥ 2,

(3) gcd(c, D) = 1, gcd(p, q) = 1,

(4) gcd(G_{2}−\frac{2}{3}aG_{1}−\frac{2}{9}a^{2}G_{0}−bG_{0}, q) = 1, gcd(G_{2}^{2}−\frac{4}{3}bG_{2}G_{0}−\frac{1}{3}bG_{1}^{2}+\frac{4}{9}b^{2}G_{0}^{2}, D) = 1 and

(5) gcd(a, 27c^{2} − 4b^{3}) > 1.

Then there are at most exp(10^{24}) pairs of integers (n, m) with n, m ≥ 0 such that

G_{n}(x) = G_{m}(P (x))

holds. Remark 5.3. The reason for this different kind of assumptions lie in the fact that the infinite valuation in the rational function field K(x) leads to degree assumptions, whereas by looking at finite valuations one gets divisibility conditions as in the above theorem.

In this case a(x) = 0 is included in the above theorem. Let us mention it as a corollary. Corollary 5.1. Let b, c, G_{0}, G_{1}, G_{2}, P ∈ K[x] and (G_{n}(x))_{n=0}^{∞} be defined by

G_{n+3}(x) = b(x)G_{n+1}(x) + c(x)G_{n}(x), for n ≥ 0.

Assume that

(1) deg D ≠ 0, deg c ≠ 0

(2) deg P > 1,

(3) gcd(b, c) = 1,

(4) gcd(G_{2} − bG_{0}, c) = 1, and gcd(G_{2}^{2} − \frac{4}{3}bG_{2}G_{0} − \frac{1}{3}bG_{1}^{2} + \frac{4}{9}b^{2}G_{0}^{2}, D) = 1, where D(x) = (c(x)/2)^{2} − (b(x)/3)^{3}. Then there are at most exp(10^{24}) pairs of integers (n, m) with n, m ≥ 0 such that

G_{n}(x) = G_{m}(P (x))

holds.

Again we want to remark that this condition are quite similar to those obtained in the case of second order linear recurring sequences [49] (see Theorem 4.3).

5.3 Auxiliary results

In this section we collect some important theorems which we will need in our proofs.

Let K be an algebraically closed field of characteristic 0, n ≥ 1 an integer, α_{1}, …, α_{n} elements of K^{∗} = K\{0} and Γ a finitely generated multiplicative subgroup of K^{∗}. A solution (x_{1}, …, x_{n}) of the so called weighted unit equation

α_{1}x_{1} + ⋯ + α_{n}x_{n} = 1 in x_{1}, …, x_{n} ∈ Γ  (5.9)

is called nondegenerate if

∑_{j∈J} α_{j}x_{j} ≠ 0 for each non-empty subset J of {1, …, n}  (5.10)

and degenerate otherwise. It is clear that if Γ is infinite and if (5.9) has a degenerate solution then (5.9) has infinitely many degenerate solutions. For the nondegenerate solutions we have Theorem 1.7, which is due to Evertse, Schlickewei and Schmidt [45]. It is sufficient for us to state their result for finite type subgroups of ℂ^{∗} (cf. [45] and [43, Theorem 2] for the following version). First, we remark that Γ is called a finite type subgroup of ℂ^{∗} = ℂ\{0} if it has a free subgroup Γ_{0} of finite rank such that Γ/Γ_{0} is a torsion group; the rank of Γ is then defined as the rank of Γ_{0}.

Theorem 5.4. (Evertse, Schlickewei and Schmidt) Let Γ be a finite type subgroup of ℂ^{∗} of rank r and α_{1}, …, α_{n} ∈ ℂ^{∗}. Then the number of nondegenerate solutions of the equation

α_{1}x_{1} + … + α_{n}x_{n} = 1 in x_{1}, …, x_{n} ∈ Γ

is at most

exp((6n)^{3n}(r + 1)).

This theorem is a special case of the Main Theorem on S-unit equations over fields with characteristic 0. It is a generalization of earlier results due to Evertse and Győry [41], Evertse [38] and van der Poorten and Schlickewei [72] on the finiteness of the number of nondegenerate solutions of (5.9). For a general survey on these equations and their applications we refer to Chapter 1.2.

In the special case n = 2 a much better result is known due to Baker [5] and to Beukers and Schlickewei (cf. [9] and [43, Theorem F]).

Theorem 5.5. (Beukers and Schlickewei) Let Γ be a finite type subgroup of ℂ^{∗} of rank r and a, b ∈ ℂ^{∗}. Then the equation

ax + by = 1 in x, y ∈ Γ

has at most

2^{16(r+1)}

solutions.

This result is comparable to Evertse’s upper bound 3 × 7^{4s} for the case Γ = O_{S}^{∗} the ring of S-integers, where S has cardinality s (cf. [40]).

5.4 Reduction to a system of equations

We start with a sequence of polynomials (P_{n}(x))_{n=0}^{∞} defined by (5.1). Then, in the sequel α_{1}(x), α_{2}(x), α_{3}(x), g_{1}(x), g_{2}(x), g_{3}(x), u(x), v(x), D(x) are always be given by (3),

(4) and (5) (see introduction).

First we remark that in fact G_{n}(x) ∈ K[x] for all n ∈ ℕ where K is finitely generated over ℚ. We may take

K = ℚ(coefficients of a, b, c, G_{0}, G_{1}, G_{2}).

Let us define

F = K(x\sqrt{D(x}, \sqrt{P (x))}, u(x), u(P (x)), v(x), v(P (x))).

Clearly, F is a finitely generated extension field of ℚ. In fact F is an algebraic function field in one variable over the constant field K. Furthermore, we set

Γ = ⟨α_{1}(x), α_{2}(x), α_{3}(x), α_{1}(P (x)), α_{2}(P (x)), α_{3}(P (x))⟩_{(F^{∗},·)},

so Γ is the subgroup of the multiplicative group of F generated by the characteristic roots of (G_{n}(x))_{n=0}^{∞} and (G_{n}(P (x)))_{n=0}^{∞}.

It is obvious that Γ can be seen as a finitely generated subgroup of ℂ^{∗}, because we can embed K^{∗} into ℂ^{∗} by sending the transcendental elements which appear in the coefficients of a, b, c, G_{0}, G_{1}, G_{2} and the variable x to linearly independent transcendental elements of ℂ. Moreover, it is clear that the rank r of Γ is at most 6.

First we reduce the solvability of (5.8) to the solvability of seven types of systems of exponential equations in n, m.

We consider for n ≠ m the equation G_{n}(x) = G_{m}(P (x)) and obtain

g_{1}(x)α_{1}(x)^{n} + g_{2}(x)α_{2}(x)^{n} + g_{3}(x)α_{3}(x)^{n}−

−g_{1}(P (x))α_{1}(P (x))^{m} − g_{2}(P (x))α_{2}(P (x))^{m} − g_{3}(P (x))α_{3}(P (x))^{m} = 0.

This can be rewritten as

\frac{g_{1}(x)}{g_{3}(P (x))}x_{1} + \frac{g_{2}(x)}{g_{3}(P (x))}x_{2} + \frac{g_{3}(x)}{g_{3}(P (x))}x_{3} − \frac{g_{1}(P (x))}{g_{3}(P (x))}x_{4} − \frac{g_{2}(P (x))}{g_{3}(P (x))}x_{5} = 1

in x_{1}, …, x_{5} ∈ Γ.

According to the theorem of Evertse, Schlickewei and Schmidt (Theorem 5.4) we conclude that, if g_{1}(x), g_{2}(x), g_{3}(x) ≠ 0 and the following systems have only finitely many solutions (m, n) ∈ ℤ^{2} with n, m ≥ 0 which can be estimated by C say, then our original equation (5.8) has only finitely many solutions which can be bounded by

C + exp(30^{15} · 7).

The systems which correspond to the non-trivial vanishing subsums of the above weighted unit equation are:

\{ \begin{matrix}g_{1}(x)α_{1}(x)^{n} + g_{2}(x)α_{2}(x)^{n} + g_{3}(x)α_{3}(x)^{n} = g_{k}(P (x))α_{k}(P (x))^{m} \\ g_{i}(P (x))α_{i}(P (x))^{m} + g_{j}(P (x))α_{j}(P (x))^{m} = 0 \end{matrix} (5.11)

\{ \begin{matrix}g_{i}(x)α_{i}(x)^{n} + g_{j}(x)α_{j}(x)^{n} = 0 \\ g_{1}(P (x))α_{1}(P (x))^{m} + g_{2}(P (x))α_{2}(P (x))^{m}+ \\ +g_{3}(P (x))α_{3}(P (x))^{m} = g_{k}(x)α_{k}(x)^{n} \end{matrix}  (5.12)

\{ \begin{matrix}g_{i}(x)α_{i}(x)^{n} = g_{j}(P (x))α_{j}(P (x))^{m} \\ g_{j}(x)α_{j}(x)^{n} + g_{k}(x)α_{k}(x)^{n} = g_{i}(P (x))α_{i}(P (x))^{m}+ \\ +g_{k}(P (x))α_{k}(P (x))^{m} \end{matrix}  (5.13)

\{ \begin{matrix}g_{i}(x)α_{i}(x)^{n} = g_{i}(P (x))α_{i}(P (x))^{m} \\ g_{j}(x)α_{j}(x)^{n} + g_{k}(x)α_{k}(x)^{n} = g_{j}(P (x))α_{j}(P (x))^{m}+ \\ +g_{k}(P (x))α_{k}(P (x))^{m} \end{matrix}  (5.14)

\{ \begin{matrix}g_{1}(x)α_{1}(x)^{n} + g_{2}(x)α_{2}(x)^{n} + g_{3}(x)α_{3}(x)^{n} = 0 \\ g_{1}(P (x))α_{1}(P (x))^{m} + g_{2}(P (x))α_{2}(P (x))^{m}+ \\ +g_{3}(P (x))α_{3}(P (x))^{m} = 0 \end{matrix}  (5.15)

\{ \begin{matrix}g_{i}(x)α_{i}(x)^{n} + g_{j}(x)α_{j}(x)^{n} = g_{i}(P (x))α_{i}(P (x))^{m} \\ g_{j}(P (x))α_{j}(P (x))^{m} + g_{k}(P (x))α_{k}(P (x))^{m} = g_{k}(x)α_{k}(x)^{n} \end{matrix}  (5.16)
\{ \begin{matrix}g_{i}(x)α_{i}(x)^{n} + g_{j}(x)α_{j}(x)^{n} = g_{k}(P (x))α_{k}(P (x))^{m} \\ g_{i}(P (x))α_{i}(P (x))^{m} + g_{j}(P (x))α_{j}(P (x))^{m} = g_{k}(x)α_{k}(x)^{n} \end{matrix}  (5.17)

where i, j, k are always such that {i, j, k} = {1, 2, 3}. Now we have the following lemma.

Lemma 5.1. Let g_{1}(x), g_{2}(x), g_{3}(x), g_{1}(P (x)), g_{2}(P (x)), g_{3}(P (x)) ≠ 0 and assume that both (G_{n}(x))_{n=0}^{∞} and (G_{n}(P (x)))_{n=0}^{∞} are nondegenerate. Then for every choice of {i, j, k} = {1, 2, 3} we have

(5.11) and (5.12) have at most 3 + exp(18^{9} · 4), (5.15) has at most 3721, (5.16) and (5.17) have at most 2^{64}

solutions (n, m) ∈ ℤ^{2} with n, m ≥ 0, n ≠ m.

Proof. First observe that an equation of the type

h_{1}(x)α(x)^{n} + h_{2}(x)β(x)^{n} = 0  (5.18)

with h_{1}, h_{2}, α, β ∈ F^{∗} and α(x)/β(x) not equal to a root of unity has at most one solution in n ∈ ℤ. In particular, assume that we have two solutions n_{1}, n_{2}. Then we obtain

−\frac{h_{1}(x)}{h_{2}(x)} = (\frac{β(x)}{α(x)})^{n_{1}} = (\frac{β(x)}{α(x)})^{n_{2}} ,

which implies that n_{1} = n_{2}.

Let us first look at (5.11) with some choice of {i, j, k} = {1, 2, 3}. The second equation is of the above type (5.18) and therefore it has at most one solution m ∈ ℕ. Now the first equation in this system becomes

b_{1}(x)α_{1}(x)^{n} + b_{2}(x)α_{2}(x)^{n} + b_{3}(x)α_{3}(x)^{n} = 1,

with

b_{i}(x) = \frac{g_{i}(x)}{g_{k}(P (x))α_{k}(P (x))^{m}} , i = 1, 2, 3,

which can be seen as a 3-dimensional weighted unit equation over the field F of characteristic 0 where we search for solutions in the finitely generated subgroup which is generated by α_{1}(x), α_{2}(x), α_{3}(x). By our assumptions we have b_{i}(x) ≠ 0 for i = 1, 2, 3. Moreover, each of the three non-trivial subsums vanishes for at most one n ∈ ℕ as this subsums are again of the type (5.18). By using Theorem 5.4 again, we can conclude that there are at most

3 + exp(18^{9} · 4)

pairs of solutions (n, m). The second system (5.12) is completely analogous.

Now for the equations in (5.15) we can calculate the number of solutions by using the bound for the zero multiplicity of nondegenerate third order linear recurring sequences (see introduction). Therefore the first equation has at most 61 solutions in n and the second at most 61 solutions in m. Consequently, there are at most 61·61 = 3721 pairs (n, m) for which (5.15) holds.

Each of the equations in the system (5.16) can be seen as a 2-dimensional weighted unit equations where we are interested in solutions which lie in the group generated by the three characteristic roots which are involved in the equation. Therefore by Theorem 5.5 we can conclude that the first and the second equation has at most 2^{16·4} solutions. Altogether the systems has at most 2^{16·4} solutions as claimed in the lemma. □

Lemma 5.2. Let g_{1}(x), g_{2}(x), g_{3}(x), g_{1}(P (x)), g_{2}(P (x)), g_{3}(P (x)) ≠ 0 and assume that both (G_{n}(x))_{n=0}^{∞} and (G_{n}(P (x)))_{n=0}^{∞} are nondegenerate. Then (5.12) and (5.13) have at most

1 + exp(18^{9} · 7)

solutions (n, m) ∈ ℤ^{2} with n, m ≥ 0, n ≠ m respectively, provided that none of the following systems has a solution:

\{ \begin{matrix}g_{i}(x)α_{i}(x)^{n} = g_{i}(P (x))α_{i}(P (x))^{m} \\ g_{j}(x)α_{j}(x)^{n} = g_{j}(P (x))α_{j}(P (x))^{m} \\ g_{k}(x)α_{k}(x)^{n} = g_{k}(P (x))α_{k}(P (x))^{m} \end{matrix}  (5.19)

\{ \begin{matrix}g_{i}(x)α_{i}(x)^{n} = g_{i}(P (x))α_{i}(P (x))^{m} \\ g_{j}(x)α_{j}(x)^{n} = g_{k}(P (x))α_{k}(P (x))^{m} \\ g_{k}(x)α_{k}(x)^{n} = g_{j}(P (x))α_{j}(P (x))^{m} \end{matrix}  (5.20)

\{ \begin{matrix}g_{i}(x)α_{i}(x)^{n} = g_{j}(P (x))α_{j}(P (x))^{m} \\ g_{j}(x)α_{j}(x)^{n} = g_{k}(P (x))α_{k}(P (x))^{m} \\ g_{k}(x)α_{k}(x)^{n} = g_{i}(P (x))α_{i}(P (x))^{m} \end{matrix}  (5.21)

where i, j, k are such that {i, j, k} = {1, 2, 3}.

Proof. We handle only the system (5.12) since (5.13) is completely analogous. Let {i, j, k} = {1, 2, 3} be fixed. The second equation in both systems can be seen as a 3-dimensional weighted unit equation

\frac{g_{j}(x)}{g_{k}(P (x))}x_{1} + \frac{g_{k}(x)}{g_{k}(P (x))}x_{2} − \frac{g_{i}(P (x))}{g_{k}(P (x))}x_{3} = 1 in x_{1}, x_{2}, x_{3} ∈ Γ.

According to Theorem 5.4 this equation has at most

exp(18^{9} · 7)

solutions in Γ for which no non-trivial subsum vanishes. But the vanishing subsums are

\{ \begin{matrix}g_{j}(x)α_{j}(x)^{n} + g_{k}(x)α_{k}(x)^{n} = 0 \\ g_{i}(P (x))α_{i}(P (x))^{m} + g_{k}(P (x))α_{k}(P (x))^{m} = 0 \end{matrix}

which has at most one pair of solutions (n, m) by the proof of Lemma 5.1, and the first and the last system in our assumptions, which are assumed to have no solutions in (n, m) at all. Therefore we have proved the upper bound for the number of solutions (n, m) ∈ ℤ^{2} with n, m ≥ 0, n ≠ m as claimed in the lemma. □

From this discussion we see that it suffices to prove that g_{1}(x), g_{2}(x), g_{3}(x), g_{1}(P (x)), g_{2}(P (x)), g_{3}(P (x)) ≠ 0, that α_{i}(x)/α_{j}(x) and α_{i}(P (x))/α_{j}(P (x)) is not equal to a root of unity for 1 ≤ i < j ≤ 3 and that the systems (5.19), (5.20) and (5.21) do not have a solution (n, m) ∈ ℤ^{2} with n, m ≥ 0, n ≠ m. We will show this for each of our theorems separately in the following sections.

5.5 Proof of Theorem 5.1

In the next lemma we calculate the order of α_{1}(x), α_{2}(x) and α_{3}(x) respectively in the function field F/K, where F and K are defined as in the previous section. Then we have:

Lemma 5.3. Let (G_{n}(x))_{n=0}^{∞} be a sequence of polynomials defined by (5.1) and assume that 3 deg a > deg c, 2 deg a > deg b and deg a + deg c > 2 deg b. Then

ord(α_{1}) = deg a,  (5.22)
ord(α_{2}) = ord(α_{3}) = \frac{1}{2}(deg c − deg a) < deg a.  (5.23)

Proof. First of all, observe that we have

deg q = 3 deg a and deg p = 2 deg a.

Moreover, we have by our assumptions

deg D = 3 deg a + deg c.

Therefore, we trivially have

ord(u) = ord(v) = deg a

and the leading coefficients of the Puiseux expansions of u(x) and v(x) at the absolute value which corresponds to ord are equal to 1/3 times the leading coefficient of a(x). Consequently, we have ord(α_{1}) = deg a. Now it follows from (b) and from the following equation

u(x)^{3} − v(x)^{3} = 2\sqrt{D(x)}

that

ord(u^{3} − v^{3}) = \frac{1}{2}(3 deg a + deg c).

But using

u(x)^{3} − v(x)^{3} = (u(x) − v(x)) (u(x)^{2} + u(x)v(x) + v(x)^{2})

and the observation that ord(u^{2} + uv + v^{2}) = 2 deg a, which follows again from the fact that all the summands have the same leading coefficient in their Puiseux expansion, we get

ord(u − v) = \frac{1}{2}(deg c − deg a).

We want to remark that we have

α_{1}(x)α_{2}(x)α_{3}(x) = −c(x).  (5.24)

Now assume that ord(α_{2}) ≠ ord(α_{3}). Furthermore, we may assume without loss of generality that ord(α_{2}) > ord(α_{3}). But then we have using (d)

ord(α_{2}) = ord(α_{2} − α_{3}) = ord(u − v) = \frac{1}{2}(deg c − deg a)

which yields by (5.24)

ord(α_{3}) = \frac{1}{2}(deg c − deg a) = ord(α_{2}),

a contradiction. Therefore we conclude again using (5.24) that

ord(α_{2}) = ord(α_{3}) = \frac{1}{2}(deg c − deg a).

This means that the proof is finished. □

It is clear that

ord(α_{1} − α_{3}) = ord(α_{1} − α_{2}) = deg a,

ord(α_{2} − α_{3}) = ord (2i\sqrt{3}\frac{u − v}{2} ) = \frac{1}{2}(deg c − deg a).

To finish our proof, we want to calculate the order of g_{1}(x), g_{2}(x) and g_{3}(x). From the initial conditions

G_{0}(x) = g_{1}(x) + g_{2}(x) + g_{3}(x),
G_{1}(x) = g_{1}(x)α_{1}(x) + g_{2}(x)α_{2}(x) + g_{3}(x)α_{3}(x),
G_{2}(x) = g_{1}(x)α_{1}(x)^{2} + g_{2}(x)α_{2}(x)^{2} + g_{3}(x)α_{3}(x)^{2},

we get

g_{1}(x)Δ(x) = G_{2}(x) (α_{3}(x) − α_{2}(x)) + G_{1}(x) (α_{2}(x)^{2} − α_{3}(x)^{2}) + (5.25)
+G_{0}(x)α_{2}(x)α_{3}(x) (α_{3}(x) − α_{2}(x)) ,
g_{2}(x)Δ(x) = G_{2}(x)(α_{1}(x) − α_{3}(x)) + G_{1}(x) (α_{3}(x)^{2} − α_{1}(x)^{2}) + (5.26)
+G_{0}(x)α_{1}(x)α_{3}(x)(α_{1}(x) − α_{3}(x)),
g_{3}(x)Δ(x) = G_{2}(x)(α_{2}(x) − α_{1}(x)) + G_{1}(x) (α_{1}(x)^{2} − α_{2}(x)^{2}) + (5.27)
+G_{0}(x)α_{1}(x)α_{2}(x)(α_{2}(x) − α_{1}(x)),

where

Δ(x) = α_{1}(x)α_{2}(x)(α_{2}(x) − α_{1}(x)) + α_{1}(x)α_{3}(x)(α_{1}(x) − α_{3}(x)) +
+α_{2}(x)α_{3}(x)(α_{3}(x) − α_{2}(x)) =
= −6i\sqrt{3}\sqrt{D(x)}.

In the proof of Lemma 5.3 we have already seen that a(x)^{3}c(x) is the dominant term in D(x). Consequently, we have

ord(Δ) = \frac{1}{2}(3 deg a + deg c).

Therefore, we can conclude

ord(g_{1}) = ord(G_{2}) + \frac{1}{2}(deg c − deg a) − \frac{1}{2}(3 deg a + deg c) =

= deg G_{2} − 2 deg a,

ord(g_{2}) = ord(g_{3}) = deg G_{2} + deg a − \frac{1}{2}(3 deg a + deg c) =

= deg G_{2} − \frac{1}{2} deg a − \frac{1}{2} deg c.

Thus, we deduce that g_{1}(x), g_{2}(x), g_{3}(x) and therefore also g_{1}(P (x)), g_{2}(P (x)), g_{3}(P (x)) are different from zero.

Next we are intended to show that α_{i}(x)/α_{j}(x) is not equal to a root of unity for 1 ≤ i < j ≤ 3. First observe that

α_{1}(x) = ζα_{2}(x) or α_{1}(x) = ζα_{2}(x)

with ζ a root of unity is impossible because of the different order. Namely this would imply

ord(α_{1}) = ord(α_{2}) or ord(α_{1}) = ord(α_{3})

respectively, a contradiction. Now assume that we have

α_{2}(x) = ζα_{3}(x)

with ζ a root of unity. Observe that the leading coefficients in the Puiseux expansion of α_{2}(x), α_{3}(x) are conjugate complex numbers. This follows from the fact that

ord(α_{2}) = ord(α_{3}) = ord(u − v)

and u(x) − v(x) is one of the summands in the definition of those characteristic roots. Thus the only possibilities are ζ = 1 or −1 which both lead to a contradiction since

ord(α_{2} − α_{3}) = \frac{1}{2}(deg c − deg a)

and α_{2}(x) + α_{3}(x) = a(x) − α_{1}(x) ≠ 0.

The proof that the sequence (G_{n}(P (x)))_{n=0}^{∞} is nondegenerate is completely analogous to the above case since we are only considering the order of the elements.

It remains to show the unsolvability of (5.19), (5.20) and (5.21). Because of

ord(α_{2}) = ord(α_{3}) and ord(g_{2}) = ord(g_{3})

it suffices to consider the following two cases:

\{ \begin{matrix}g_{1}(x)α_{1}(x)^{n} = g_{1}(P (x))α_{1}(P (x))^{m} \\ g_{2}(x)α_{2}(x)^{n} = g_{2}(P (x))α_{2}(P (x))^{m} \end{matrix}  (5.28)

\{ \begin{matrix}g_{1}(x)α_{1}(x)^{n} = g_{2}(P (x))α_{2}(P (x))^{m} \\ g_{2}(x)α_{2}(x)^{n} = g_{1}(P (x))α_{1}(P (x))^{m} \\ g_{3}(x)α_{3}(x)^{n} = g_{3}(P (x))α_{3}(P (x))^{m} \end{matrix}  (5.29)

Calculating orders we get

deg G_{2} − 2 deg a + n deg a = (deg G_{2} − 2 deg a)(deg P + m deg a deg P)
(deg G_{2} − \frac{deg a}{2} − \frac{deg c}{2}) (1 − deg P) = (m deg P − n)\frac{deg c − deg a}{2}

or

(deg G_{2} − 2 deg a)(1 − deg P) = (m deg P − n) deg a
(2 deg G_{2} − deg a − deg c)(1 − deg P) = (m deg P − n)(deg c − deg a)

This yields

(m − 1) deg P = n − 1.

Substituting this into the first equation leads to

(deg G_{2} − deg a)(1 − deg P) = 0,

which implies deg P = 1 and therefore n = m or deg G_{2} = deg a and therefore deg G_{1} < 0, in both cases a contradiction.

The second system leads to

deg G_{2} − 2 deg a + n deg a = (deg G_{2} − \frac{1}{2} deg a − \frac{1}{2} deg c) deg P +
+m deg P \frac{1}{2}(deg c − deg a)
deg G_{2} − \frac{1}{2} deg c − \frac{1}{2} deg a + n\frac{1}{2}(deg c − deg a) =
= (deg G_{2} − \frac{1}{2} deg a − \frac{1}{2} deg c) deg P + m deg P \frac{1}{2}(deg c − deg a)
deg G_{2} − \frac{1}{2} deg c − \frac{1}{2} deg a + n\frac{1}{2}(deg c − deg a) =
= (deg G_{2} − 2 deg a) deg P + m deg a deg P,

which yields

0 ≤ \frac{1}{2}(3 deg a − deg c) = −m\frac{1}{2}(deg a + deg c) < −m deg b < 0,

a contradiction.

So the proof of Theorem 5.1 is finished. By Lemma 5.1 and Lemma 5.2 we get by counting how often each system can appear, the following bound:

exp (30^{15} · 7) + 2 · 3 · [3 + exp (18^{9} · 4)] + 3721 + 9 · 2^{64} + 9 · (1 + exp (18^{9} · 7))

which can be estimated by

exp(10^{24}).

This was the claim of Theorem 5.1. □

5.6 Proof of Theorem 5.2

First we want to mention that a(x) = 0 means that we have

p(x) = b(x), q(x) = c(x) and D(x) = \frac{1}{4}c(x)^{2} − \frac{1}{27}b^{3}.

By our assumption that 3 deg b > 2 deg c we get

ord(u) = ord(v) = \frac{1}{2} deg b

and the leading coefficients of the relevant Puiseux expansions are equal to i\sqrt{3} and −i\sqrt{3} times the square root of the leading coefficient of b(x). Therefore we can conclude

ord(u − v) = \frac{1}{2} deg b and ord(u + v) = deg c − deg b.

Thus we get

ord(α_{1}) = deg c − deg b,

ord(α_{2}) = ord(α_{3}) = \frac{1}{2} deg b,

ord(α_{1} − α_{2}) = ord(α_{1} − α_{3}) = \frac{1}{2} deg b,

ord(α_{2} − α_{3}) = \frac{1}{2} deg b

ord(α_{1} + α_{2}) = ord(α_{1} + α_{3}) = deg c − deg b,

ord(α_{2} + α_{3}) = \frac{1}{2} deg b.

Using (5.25), (5.26) and (5.27) we get from our assumptions concerning the degrees of the initial polynomials

ord(g_{1}) = deg G_{2} − 2 deg b,
ord(g_{2}) = ord(g_{3}) = deg G_{2} − 2 deg b.

Therefore we can conclude that g_{1}(x), g_{2}(x), g_{3}(x), g_{1}(P (x)), g_{2}(P (x)), g_{3}(P (x)) are nonzero. The proof that (G_{n}(x))_{n=0}^{∞} and (G_{n}(P (x)))_{n=0}^{∞} are nondegenerate is analogous to the proof of this fact in Theorem 5.1.

As in the proof of Theorem 5.1 it suffices to prove the unsolvability of (5.28) and (5.29). By calculating orders we get

deg G_{2} − 2 deg b + n(deg c − deg b) = (deg G_{2} − 2 deg b) deg P +
+m deg P (deg c − deg b)
deg G_{2} − 2 deg b + n\frac{deg b}{2} = (deg G_{2} − 2 deg b) deg P + m deg P \frac{deg b}{2}

This yields n = m deg P and by substituting this into one of the equations above we get deg P = 1 which implies n = m or deg G_{2} = 2 deg b from which we get deg G_{1} < 0, in both cases a contradiction. The second system (5.28) can be handled analogously.

By Lemma 5.1 and 5.2 the theorem follows and the proof is finished. □

5.7 Proof of Theorem 5.3

We start our proof with some useful lemmas.

Lemma 5.4. Let A, B, P ∈ K[x]. Then gcd(A, B) = 1 if and only if gcd(A(P), B(P)) = 1.

Proof. Let us assume that gcd(A(P), B(P)) = 1 and that gcd(A, B) > 1. Then there exists a common root of A(x) and B(x) which we denote by ξ ∈ K (observe that K is algebraically closed). Now let ζ ∈ K be a root of the polynomial P (x) − ξ with coefficients in K. Thus we have A(P (ζ)) = B(P (ζ)) = 0, contradicting our assumption. The proof of the converse can be found in [49, Lemma 4]. □

We will use the same notations as introduced in the proof of Theorem 5.1.

First of all we have because of deg D ≠ 0 and gcd(c, D) = 1 that c(x) ≠ 0. Therefore, from

α_{1}(x)α_{2}(x)α_{3}(x) = −c(x),

it follows that α_{1}(x), α_{2}(x), α_{3}(x) ≠ 0. Next we show that α_{1}(x), α_{2}(x), α_{3}(x) are nondegenerate. We take ξ ∈ K such that a(ξ) = 27c(ξ)^{2} − 4b(ξ)^{3} = 0. This implies that D(ξ) = 0. From this we can conclude

u(ξ) = ^{3}\sqrt{\frac{q(ξ)}{2}} = \sqrt{\frac{p(ξ)}{2}} = \sqrt{\frac{b(ξ)}{3}} and v(ξ) = u(ξ).

Therefore we have

α_{1}(ξ) = 2\sqrt{\frac{b(ξ)}{3}}.

On the other hand we get

α_{2}(ξ) = α_{3}(ξ) = −\sqrt{\frac{b(ξ)}{3}},

which implies that α_{1}(x) differs from α_{2}(x) and α_{3}(x) by more than a root of unity, because b(ξ) ≠ 0 by condition (3) in the theorem.

Now assume that we have

α_{2}(x) = ζα_{3}(x), ζ ∈ K.

This yields

(1 + ζ)i\sqrt{3}\frac{u(x) − v(x)}{2} = \frac{1 − ζ}{2} (u(x) + v(x)) − \frac{1 − ζ}{2} a(x).

As above we derive a contradiction unless ζ = 1. But assuming ζ = 1 yields

2i\sqrt{3}\frac{u(x) − v(x)}{2} = 0,

contradicting the fact that u(x) = v(x) ⟺ D(x) = 0.

Because of Lemma 5.4 we can conclude in the same way as above that the same holds for α_{1}(P (x)), α_{2}(P (x)), α_{3}(P (x)).

Next we want to proof the g_{1}(x), g_{2}(x), g_{3}(x) ≠ 0 holds. Observe that they are given by (5.25), (5.26) and (5.27) respectively.

First observe that for ξ ∈ K we have: Δ(ξ) = 0 ⟺ α_{2}(ξ) = α_{3}(ξ) and Δ(ξ) = 0 ⇒ α_{1}(ξ) ≠ α_{2}(ξ), α_{3}(ξ). We will need

g_{1}(x) = \frac{α_{3}(x) − α_{2}(x)}{Δ(x)}(G_{2}(x) − G_{1}(x)[a(x) − α_{1}(x)] + G_{0}(x)α_{2}(x)α_{3}(x))

and

G_{2}(x) − G_{1}(x)[a(x) − α_{1}(x)] + G_{0}(x)α_{2}(x)α_{3}(x) =

= G_{2}(x) − a(x)G_{1}(x) + \frac{a(x)}{3}G_{1}(x) + G_{1}(x)[u(x) + v(x)] + G_{2}(x)[u(x)^{2}

+v(x)^{2}] − G_{2}(x)\frac{a(x)}{3}[u(x) + v(x)] + G_{2}(x)\frac{a(x)^{2}}{9} − G_{2}(x)u(x)v(x).

Observe that 3u(x)v(x) = p(x). Let ξ ∈ K with q(ξ) = 0. This implies

u(ξ) = \frac{i}{\sqrt{3}}\sqrt{p(ξ)} and v(ξ) = −\frac{i}{\sqrt{3}}\sqrt{p(ξ)}

and therefore u(ξ) + v(ξ) = 0. Because of the above equation and condition (4) from the theorem we get

g_{1}(ξ) ≠ 0.

To handle g_{2}(x), g_{3}(x) we prove the following lemma which will also enable us to calculate ν(g_{2}) and ν(g_{2}) where ν extends ν_{ξ} to F for some ξ ∈ K.

Lemma 5.5. Let (G_{n}(x))_{n=0}^{∞} be a sequence of polynomials defined by (5.1) and assume that gcd (G_{2}^{2} − \frac{4}{3}bG_{2}G_{0} − \frac{1}{3}bG_{1}^{2} + \frac{4}{9}b^{2}G_{0}^{2}, D) = 1. Let ξ ∈ K be a common root of a(x) and D(x) and let ν be an extension of ν_{ξ} to F. Then ν(g_{1}Δ) = ν(g_{2}Δ) = 0.

Proof. Since D(ξ) = 0 we have α_{2}(ξ) = α_{3}(ξ) and by equation (5.26) we have to show that

ν(G_{2} − G_{1}(a − α_{2}) + G_{0}α_{1}α_{3}) = 0.

Observe that it is clear that we have ≥ 0 since the α_{i}(x), i = 1, 2, 3 are integral over K[x] and the integral closure is a ring. Therefore it suffices to show that

(G_{2} − G_{1}(a − α_{2}) + G_{0}α_{1}α_{3})(ξ) ≠ 0,

but this follows from our condition: We have

(G_{2} + G_{1}α_{2} + G_{0}α_{1}α_{3})(ξ) = (G_{2} − \frac{1}{3}G_{1}\sqrt{3b} − \frac{2}{3}bG_{0}) (ξ).

Assume this value to be zero. Then

[(G_{2} − \frac{2}{3}bG_{0})^{2} − \frac{1}{3}bG_{1}^{2}](ξ) = 0,

contradicting the assumption in our Lemma.

The same holds for g_{2}(x) and therefore the proof is finished. □

We are intended to prove that the systems of equations (5.19), (5.20) and (5.21) are not solvable. Observe that each of this systems contain at least one equation of the form

g_{i}(x)α_{i}(x)^{n} = g_{k}(P (x))α_{k}(P (x))^{m}  (5.30)

with i, k ∈ {2, 3} not necessarily different. We will show that already this equation cannot have a solution.

We have deg D(P) = deg D deg P > deg D > 0, as deg P > 1 by assumption (2). Hence D(P (x)) has a zero ξ ∈ K such that

ν_{ξ}(D(P)) > ν_{ξ}(D) ≥ 0

which is also a zero of a(P (x)) which means ν_{ξ}(a(P)) > 0. This implies that there is a finite valuation ν on F such that by Lemma 5.5

ν(g_{1}(P)) = ν(g_{2}(P)) = −ν(Δ(P)) = −\frac{1}{2}ν(D(P)).

Moreover, we can conclude that ν(α_{2}(P)) = ν(α_{3}(P)) = 0, because otherwise we would get a contradiction to condition (2) of our theorem.

Thus equation (5.30) implies

ν(g_{i}) + nν(α_{i}) = ν(g_{k}(P)),

which yields

nν(α_{i}) = ν(g_{k}(P)) − ν(g_{i}) ≤ −ν(Δ(P)) + ν(Δ) < 0,

hence (5.30) has no solution in n, if ν(α_{i}) ≥ 0 and at most one, if ν(α_{i}) < 0, which is impossible since α_{1}(x), α_{2}(x), α_{3}(x) are integral over K[x], as they are zeros of the monic equation T^{3} − a(x)T^{2} − b(x)T − c(x) = 0 with coefficients in K[x]. Therefore, we have ν(α_{i}) ≥ 0. Consequently (5.30) has no solution.

So, we have shown that (5.19), (5.20) and (5.21) have no solutions (n, m) ∈ ℤ^{2} with n, m ≥ 0, n ≠ m. It is clear that we get the same bound as in Theorem 5.1. □

A polynomial variant of a problem of Diophantus and Euler

In this chapter, we prove that there does not exist a set of four polynomials with integer coefficients, which are not all constant, such that the product of any two of them is one greater than a square of a polynomial with integer coefficients.

This chapter is identically equal to a joint paper with A. Dujella which is to appear in Rocky Mount. J. Math. (cf. [33]).

6.1 Introduction

Let n be an integer. A set of m positive integers is called a Diophantine m-tuple with the property D(n) or simply D(n)-m-tuple, if the product of any two of them increased by n is a perfect square. The first D(1)-quadruple, the set {1, 3, 8, 120}, was found by Fermat. The folklore conjecture is that there does not exist a D(1)-quintuple. In 1969, Baker and Davenport [6] proved that the Fermat’s set cannot be extended to a D(1)- quintuple. Recently, Dujella proved that there does not exist a D(1)-sextuple and there are only finitely many D(1)-quintuples (see [32]).

In the case n = −1, the conjecture is that there does not exist a D(−1)-quadruple (see [27]). It is known that some particular D(−1)-triples cannot be extended to D(−1)- quadruples (see [14], [28], [65], [57]). Let us mention that from [29, Theorem 4] it follows that there does not exist a D(−1)-33-tuple.

This n = −1 case is closely connected with an old problem of Diophantus and Euler. Namely, Diophantus studied the problem of finding numbers such that the product of any two increased by the sum of these two gives a square. He found two triples {4, 9, 28} and \{\frac{3}{10}, \frac{21}{5}, \frac{7}{10}\} satisfying this property. Euler found a quadruple \{\frac{5}{2}, \frac{9}{56}, \frac{9}{224}, \frac{65}{224}\} (see [25], [24]). In [31] an infinite family of rational quintuples with the same property was given. Since

xy + x + y = (x + 1)(y + 1) − 1,

we see that the problem of finding integer m-tuples with the same property is equivalent to finding D(−1)-m-tuples.

A polynomial variant of the above problems was first studied by Jones [55], [56], and it was for the case n = 1.

Definition 6.1. Let n be an integer. A set {a_{1}, a_{2}, …, a_{m}} of m polynomials with integer coefficients, which are not all constant, is called a polynomial D(n)-m-tuple if for all 1 ≤ 1 < j ≤ m the following holds: a_{i} · a_{j} + n = b_{ij}^{2}, where b_{ij} ∈ ℤ[x].

A natural question is how large such sets can be. Let us define

P_{n} = sup{|S| : S is a polynomial D(n)-tuple }.

From [29, Theorem 1] it follows that P_{n} ≤ 22 for all n ∈ ℤ. The above mentioned result about the existence of only finitely many D(1)-quintuples implies that P_{1} = 4.

In the present chapter, we will prove that P_{−1} = 3. First of all, P_{−1} ≥ 3. More precisely, if a · b − 1 = r^{2}, then

{a, b, a + b + 2r}

is a polynomial D(−1)-triple. E.g.

{x^{2} + 1, x^{2} + 2x + 2, 4x^{2} + 4x + 5}

is a polynomial D(−1)-triple (see [14]). Therefore, we have to prove that P_{−1} < 4, and this is the statement of our main theorem.

Theorem 6.1. There does not exist a polynomial D(−1)-quadruple.

The proof of Theorem 6.1 is divided into several parts. In section 6.2, we transform our problem into a system of polynomial Pellian equations, which leads to finding intersections of some binary recursive sequences. We obtain some useful information about initial terms of these sequences.

In section 6.3, we show that there is no loss of generality in assuming that one element of our initial triple is equal to 1. This, together with results from section 6.2, allow us to completely determine initial terms of corresponding sequences.

In section 6.4, we prove Theorem 6.1 by showing that our sequences cannot have nontrivial common terms. This is done by comparing degrees and leading coefficients of corresponding polynomials.

6.2 Two sequences of polynomials

Let ℤ^{+}[x] denote the set of all polynomials with integer coefficients with positive leading coefficient. For a, b ∈ ℤ[x], a < b means that b − a ∈ ℤ^{+}[x]. The usual fundamental properties of inequality hold for this order. For a ∈ ℤ[x], we define |a| = a if a ≥ 0, and |a| = −a if a < 0.

If {a, b, c, d}, a < b < c < d is a polynomial D(−1)-quadruple, then d is nonconstant. Assume now that a and b are constant polynomials. Considering leading coefficients of ad − 1 and bd − 1 we conclude that ab is a perfect square, contradicting the assertion that ab − 1 is also a perfect square. Therefore, we proved that in a polynomial D(−1)-quadruple there is at most one constant polynomial. It is also clear that all leading coefficients of the polynomials in a polynomial D(−1)-m-tuple have the same sign. This implies that there is no loss of generality in assuming that they are all positive, i.e. that all polynomials are in ℤ^{+}[x].

Let {a, b, c}, where 0 < a < b < c, be a polynomial D(−1)-triple and let r, s, t ∈ ℤ^{+}[x] be defined by

ab − 1 = r^{2}, ac − 1 = s^{2}, bc − 1 = t^{2}.

In this chapter, the symbols r, s, t will always have this meaning. Assume that d ∈ ℤ^{+}[x], d > c, is a polynomial such that {a, b, c, d} is a polynomial D(−1)-quadruple. We have

ad − 1 = u^{2}, bd − 1 = y^{2}, cd − 1 = z^{2},  (6.1)

with u, y, z ∈ ℤ^{+}[x]. Eliminating d from (6.1) we obtain the following system of polynomial Pellian equations

az^{2} − cu^{2} = c − a,  (6.2)
bz^{2} − cy^{2} = c − b.  (6.3)

We will describe the sets of solutions of equations (6.2) and (6.3). We will follow the arguments in the classical case of Pellian equations in integers (cf. [30]).

Lemma 6.1. If (z, u) and (z, y), with u, y, z ∈ ℤ^{+}[x], are polynomial solutions of (6.2) and (6.3) respectively, then there exist z_{0}, u_{0} ∈ ℤ[x] and z_{1}, y_{1} ∈ ℤ[x] with

(i) (z_{0}, u_{0}) and (z_{1}, y_{1}) are solutions of (6.2) and (6.3) respectively,

(ii) the following inequalities are satisfied:

0 ≤ |u_{0}| < s,  (6.4)
0 < z_{0} < c,  (6.5)

0 ≤ |y_{1}| < t,  (6.6)
0 < z_{1} < c,  (6.7)

and there exist integers m, n ≥ 0 such that

z\sqrt{a} + u\sqrt{c} = (z_{0}\sqrt{a} + u_{0}\sqrt{c})(s + \sqrt{ac})^{2m},  (6.8)
z\sqrt{b} + y\sqrt{c} = (z_{1}\sqrt{b} + y_{1}\sqrt{c})(t + \sqrt{bc})^{2n},  (6.9)

where this means that the coefficients of \sqrt{a}, \sqrt{b} and \sqrt{c} respectively on both sides are equal.

Proof. It is clear that it suffices to prove the statement of the lemma for equation (6.2). First observe that

(s + \sqrt{ac})^{2m} = (s^{2} + ac + 2s\sqrt{ac})^{m} = (2ac − 1 + 2s\sqrt{ac})^{m}

and by multiplying with the conjugate (s − \sqrt{ac})^{2m} we see that

(s + \sqrt{ac})^{2m}(s − \sqrt{ac})^{2m} = (s^{2} − ac)^{2m} = (−1)^{2m} = 1.  (6.10)

Let now (z, u) be a solution of (6.2) in polynomials from ℤ^{+}[x]. Consider all pairs (z^{∗}, u^{∗}) of polynomials of the form

z^{∗}\sqrt{a} + u^{∗}\sqrt{c} = (z\sqrt{a} + u\sqrt{c})(s + \sqrt{ac})^{2m}, m ∈ ℤ.

By (6.10) it is clear that (z^{∗}, u^{∗}) satisfies (6.2).

We would like to show that z^{∗} > 0. We write (s + \sqrt{ac})^{2m} = A + B\sqrt{ac}, where A, B ∈ ℤ[x] satisfying A^{2} − acB^{2} = 1. Therefore we have

z^{∗}\sqrt{a} + u^{∗}\sqrt{c} = (z\sqrt{a} + u\sqrt{c})(A + B\sqrt{ac}) = (Az + cuB)\sqrt{a} + (Au + azB)\sqrt{c},

and this yields

z^{∗} = Az + cuB.

Now, if m ≥ 0 then we have A, B > 0 and thus z^{∗} > 0. On the other hand, if m < 0 we have A > 0, B < 0. If we assume that z^{∗} ≤ 0, we have Az ≤ −Bcu and both sides are > 0. Squaring yields A^{2}z^{2} ≤ B^{2}c^{2}z^{2}. Using the fact that A^{2} − acB^{2} = 1 we obtain z^{2}B^{2}ac + z^{2} ≤ B^{2}c^{2}u^{2} and therefore

z^{2} ≤ cB^{2}(cu^{2} − az^{2}) = cB^{2}(a − c) < 0,

a contradiction.

Among all pairs (z^{∗}, u^{∗}), we can now choose a pair with the property that z^{∗} is minimal, and we denote that pair by (z_{0}, u_{0}). Define polynomials z′ and u′ by

z′\sqrt{a} + u′\sqrt{c} = (z_{0}\sqrt{a} + u_{0}\sqrt{c})(2ac − 1 − 2sε\sqrt{ac}),

where ε = 1 if u_{0} > 0, and ε = −1 if u_{0} < 0. From the minimality of z_{0} we conclude that z′ = z_{0}(2ac − 1) − 2csu_{0}ε ≥ z_{0} and this leads to cs|u_{0}| ≤ z_{0}(ac − 1) and further to c|u_{0}| ≤ sz_{0}. Squaring this inequality we obtain

acz_{0}^{2} − c(c − a) = c^{2}u_{0}^{2} ≤ acz_{0}^{2} − z_{0}^{2}

and finally

z_{0}^{2} ≤ c(c − a) < c^{2},

which implies (6.5). Now we have

cu_{0}^{2} = az_{0}^{2} − c + a ≤ ac^{2} − a^{2}c − c + a < ac^{2} − c = cs^{2}  (6.11)

and therefore we obtain also (6.4). Hence, we have proved that there exists a solution (z_{0}, u_{0}) of (6.2), which satisfies (6.4) and (6.5), and an integer m ∈ ℤ such that

z\sqrt{a} + u\sqrt{c} = (z_{0}\sqrt{a} + u_{0}\sqrt{c})(s + \sqrt{ac})^{2m}.

It remains to show that m ≥ 0. Suppose that m < 0. Then, as above, we have (s + \sqrt{ac})^{2m} = A − B\sqrt{ac}, with A, B ∈ ℤ^{+}[x] satisfying A^{2} − acB^{2} = 1. We have u = Au_{0} − z_{0}Ba and from the condition u > 0 we obtain Au_{0} > z_{0}Ba and by squaring u_{0}^{2} > B^{2}a(c − a) ≥ ac − a^{2}, which by (6.11) implies

ac^{2} − a^{2}c ≤ cu_{0}^{2} ≤ ac^{2} − a^{2}c − c + a.

This implies −c + a ≥ 0, which is clearly a contradiction. □

The solutions z arising, for given (z_{0}, u_{0}), from formula (6.8) for varying m ≥ 0 form a binary recurrent sequence (v_{m})_{m≥0} whose initial terms are found by solving equation (6.8) for z when m = 0 and 1, and whose characteristic equation has the roots (s + \sqrt{ab})^{2} and (s − \sqrt{ab})^{2}. Therefore, we conclude that z = v_{m} for some (z_{0}, u_{0}) with the above properties and integer m ≥ 0, where

v_{0} = z_{0}, v_{1} = (2ac − 1)z_{0} + 2scu_{0}, v_{m+2} = (4ac − 2)v_{m+1} − v_{m}.  (6.12)

In the same manner, from (6.9), we conclude that z = w_{n} for some (z_{1}, y_{1}) with the above properties and integer n ≥ 0, where

w_{0} = z_{1}, w_{1} = (2bc − 1)z_{1} + 2tcy_{1}, w_{m+2} = (4bc − 2)w_{n+1} − w_{n}.  (6.13)

Now the following congruence relations follow easily from (6.12) and (6.13) by induction.

Lemma 6.2. Let the sequences (v_{m}) and (w_{n}) be given by (6.12) and (6.13). Then we have

v_{m} ≡ (−1)^{m}z_{0} (mod 2c), w_{n} ≡ (−1)^{n}z_{1} (mod 2c).

Proof. It suffices to prove the statement of the lemma for v_{m}. By looking on (6.12) we have

v_{0} = z_{0}, v_{1} ≡ −z_{0} (mod 2c).

Proceeding the induction step, we see using (6.12)

v_{m+2} ≡ −2(−1)^{m+1}z_{0} − (−1)^{m}z_{0} = (−1)^{m+2}z_{0} (mod 2c),

as stated. □

Now we can prove the following lemma, which says that a solution of v_{m} = w_{n} implies also a solution at the beginning of the sequences.

Lemma 6.3. If the equation v_{m} = w_{n} has a solution, then z_{0} = z_{1}.

Proof. Assume that v_{m} = w_{n} has a solution. By Lemma 6.2 we conclude

z_{0} ≡ ±z_{1} (mod 2c).

If we assume that z_{0} ≡ z_{1} (mod 2c), then we can conclude by using (6.4) and (6.7) from Lemma 6.1, namely

0 < z_{0} < c, 0 < z_{1} < c,

that z_{0} = z_{1} holds. If we assume that z_{0} ≡ −z_{1} (mod 2c), we have 2c|z_{0} + z_{1}, which contradicts the fact that z_{0} + z_{1} < 2c. This finishes the proof. □

6.3 Reduction to the case a = 1

In this section, we show that it suffices to prove that polynomial D(−1)-triples {a, b, c}, where a = 1, cannot be extended to a polynomial D(−1)-quadruple.

Lemma 6.4. Let {a, b, c, d} with 0 < a < b < c < d be a polynomial D(−1)-quadruple. Then there exists d_{0} ∈ ℤ^{+}[x] with d_{0} < c such that ad_{0} − 1, bd_{0} − 1, cd_{0} − 1 are perfect squares.

Proof. We are interested in sequences (v_{m}) (and (w_{n})) such that z^{2} = v_{m}^{2} = w_{n}^{2} = cd−1, where d ∈ ℤ^{+}[x]. This implies that v_{m}^{2} ≡ −1 (mod c). By Lemma 6.2 this means

z_{0}^{2} ≡ −1 (mod c).

In this case we define

d_{0} = \frac{z_{0}^{2} + 1}{c} ∈ ℤ^{+}[x].

For this d_{0} we have

cd_{0} − 1 = z_{0}^{2}.

By Lemma 6.3 we find

bd_{0} − 1 = b\frac{z_{1}^{2} + 1}{c} − 1 = \frac{cy_{1}^{2} + c − b + b}{c} − 1 = y_{1}^{2}

and finally also

ad_{0} − 1 = a\frac{z_{0}^{2} + 1}{c} − 1 = \frac{1}{c}(az_{0}^{2} + a − c) = \frac{1}{c}cu_{0}^{2} = u_{0}^{2}

holds. Furthermore, we have

cd_{0} = z_{0}^{2} + 1 < c^{2},

which implies

d_{0} < c.
□

Assume now that {a, b, c, d} is a polynomial D(−1)-quadruple with minimal d. We may use Lemma 4 to construct d_{0}. From the minimality of d, it follows that {a, b, c, d_{0}} is not a polynomial D(−1)-quadruple and this means that d_{0} ∈ {a, b}. But this implies that d_{0}^{2} − 1 is a perfect square, which can only hold in the case when d_{0} = 1. Since b > a ≥ 1, we conclude that a = 1.

Remark 6.1. It follows that it suffices to consider polynomial D(−1)-quadruples, which contain the constant polynomial 1.

Now let {1, b, c} with 1 < b < c be a polynomial D(−1)-triple. By the previous discussion, we have d_{0} = 1. This implies that z_{0}^{2} +1 = c and therefore we have z_{0} = ±s. Because of the fact that z_{0} > 0 we have z_{0} = s. In the same way, we can conclude that z_{1} = s. Now we have

cu_{0}^{2} = z_{0}^{2} − c + 1 = c − 1 − c + 1 = 0

and this yields u_{0} = 0. Finally we get

cy_{1}^{2} = bz_{1}^{2} − c + b = b(c − 1) − c + b = bc − c = cr^{2}

and therefore y_{1} = ±r. To sum up, it suffices to consider the following three sequences

v_{0} = s, v_{1} = (2c − 1)s, v_{m+2} = (4c − 2)v_{m+1} − v_{m},  (6.14)

and

w_{0} = s, w_{1} = (2bc − 1)s + 2tcr, w_{n+2} = (4bc − 2)w_{n+1} − w_{n},  (6.15)
w′_{0} = s, w′_{1} = (2bc − 1)s − 2tcr, w′_{n+2} = (4bc − 2)w′_{n+1} − w′_{n}.  (6.16)

6.4 Proof of Theorem 6.1

Let {1, b, c} be a polynomial D(−1)-triple. Let us repeat the defining equations:

b − 1 = r^{2}, c − 1 = s^{2}, bc − 1 = t^{2}.

In what follows, we need the leading coefficients of b and c. We know that b and c are non-constant, and thus their leading coefficients are perfect squares. Let us give them names:

lc(b) = β^{2}, lc(c) = γ^{2},

where β and γ are positive integers. Let v_{m} and w_{n}, w′_{n} be the remaining sequences from the last section. To finish the proof, we have to show that no nontrivial solution is obtained from these sequences. The trivial solution is always v_{0} = w_{0} = s, which leads to d = 1, which does not yield the extension of our triple {1, b, c}. We divide the proof in three cases. The first one is handled in the following lemma.

Lemma 6.5. The equation v_{m} = w_{n} has no nontrivial solution.

Proof. First let us mention that deg v_{m} < deg v_{m+1}, m = 0, 1, 2, . . .. To be precise we have

deg v_{m} = \frac{1}{2} deg c + m deg c, m ≥ 0.  (6.17)

This follows at once by induction using the recurring formula (6.14). The same is also true for the second sequence w_{n} with

deg w_{n} = \frac{1}{2} deg c + n (deg b + deg c) , n ≥ 0.  (6.18)

Again, by induction, we can now read off the leading coefficient of v_{m}, which is

2^{2m−1}γ^{2m+1}, m ≥ 1.

We have lc(v_{0}) = γ, lc(v_{1}) = 2γ^{3} and using the recursive formula (6.14) we get

lc(v_{m+1}) = 4γ^{2}lc(v_{m}) = 4γ^{2}2^{2m−1}γ^{2m+1} = 2^{2(m+1)−1}γ^{2(m+1)+1}.

In the same way, we find the leading coefficient of w_{n}, which is

2^{2n}β^{2n}γ^{2n+1}.

First we have lc(w_{0}) = γ, lc(w_{1}) = 2β^{2}γ^{2}γ + 2βγγ^{2}β = 4β^{2}γ^{3}. By using the recursive formula for w_{n}, one finds

lc(w_{n+1}) = 4β^{2}γ^{2}lc(w_{n}) = 2^{2n+2}β^{2n+2}γ^{2n+3}.

If the equation v_{m} = w_{n} has a solution, we must have equal leading coefficients, which means

2^{2m−1}γ^{2m+1} = 2^{2n}β^{2n}γ^{2n+1}.

This implies

(\frac{2^{m−n}γ^{m−n}}{β^{n}} )^{2} = 2,

which yields

\sqrt{2} = \frac{2^{m−n}γ^{m−n}}{β^{n}} ∈ ℚ,

a contradiction. Thus v_{m} = w_{n} cannot hold and the proof is finished. □

To handle the equation v_{m} = w′_{n}, we have to distinguish whether deg b < deg c or deg b = deg c holds.

Lemma 6.6. Assume that deg b < deg c. Then the equation v_{m} = w′_{n} has no nontrivial solution.

Proof. First we calculate

w′_{1}w_{1} = (2bc − 1)^{2}s^{2} − 4t^{2}c^{2}r^{2} =
= −4b^{2}c^{2} + 4bc + c − 1 + 4bc^{3} − 4c^{2}.

Because of our assumption deg b < deg c, we obtain that the dominating summand is 4bc^{3}. Therefore we get

lc(w′_{1}w_{1}) = 4β^{2}γ^{6}

and

deg w′_{1}w_{1} = 3 deg c + deg b.

On the other hand, we already know that

lc(w_{1}) = 4β^{2}γ^{3}

and

deg w_{1} = deg b + \frac{3}{2} deg c.

Hence, we can conclude that

lc(w′_{1}) = γ^{3} and deg w′_{1} = \frac{3}{2} deg c.

Now by induction and by the recursion (6.16) we get that deg w′_{n} < deg w′_{n+1} and that the leading coefficient of w′_{n} is given by

2^{2n−2}β^{2n−2}γ^{2n+1}, n ≥ 1.

Namely we have lc(w′_{0}) = γ, lc(w′_{1}) = γ^{3} and using (6.16) we obtain

lc(w′_{n+1}) = 4β^{2}γ^{2}lc(w′_{n}) = 2^{2n}β^{2n}γ^{2n+3}.

Again, if v_{m} = w′_{n} has a solution, we can conclude by comparing the leading coefficients that

2^{2m−1}γ^{2m+1} = 2^{2n−2}β^{2n−2}γ^{2n+1}.

As before we get

\sqrt{2} = 2^{n−m}γ^{n−m}β^{n−1} ∈ ℚ,

which is a contradiction. This yields that in this case no solution can exist. □

Before we can prove the remaining part, we need the following useful gap principle for the elements of a polynomial D(−1)-m-tuple. The principle is a direct modification from the integer case (see [29, Lemma 3]). The analogous statement for polynomial D(1)-triples was proved by Jones in [56].

Lemma 6.7. Let {a, b, c} be a polynomial D(−1)-triple. Then there exist polynomials e, u, y, z ∈ ℤ[x] such that

ae + 1 = u^{2}, be + 1 = y^{2}, ce + 1 = z^{2}

and

c = a + b − e + 2(abe + ruy).

Proof. Define

e = −(a + b + c) + 2abc − 2rst.

Then

(ae + 1) − (at − rs)^{2} = −a(a + b + c) + 2a^{2}bc − 2arst + 1 −
−a^{2}(bc − 1) + 2arst − (ab − 1)(ac − 1) = 0.

Hence, we may take u = at − rs, and analogously y = bs − rt, z = cr − st. We have

abe + ruy = −ab(a + b + c) + 2a^{2}b^{2}c − 2abrst + abrst −
−a(ab − 1)(bc − 1) − b(ab − 1)(ac − 1) + rst(ab − 1) =
= abc − (a + b) − rst,

and finally

a + b − e + 2(abe + ruy) = 2a + 2b + c − 2abc + 2rst + 2abc − 2a − 2b − 2rst = c.

□

Using this lemma we can finish our proof. Lemma 6.8. Assume that deg b = deg c. Then the equation v_{m} = w′_{n} has no nontrivial solution. Proof. First we conclude by Lemma 6.7 that there exist polynomials e, f, g, h such that

e + 1 = f^{2}, be + 1 = g^{2}, ce + 1 = h^{2}  (6.19)

and

c = 1 + b − e + 2(be + rfg).

By looking at the proof of Lemma 6.7, we see that we have

e = −1 − b − c + 2bc − 2rst.

We want to show that e = 0. Let us assume e ≠ 0 and define

ē = −1 − b − c + 2bc + 2rst.

Then

deg ē = deg b + deg c = 2 deg c = deg c^{2}.  (6.20)

Let us calculate

eē = (2bc − 1 − b − c)^{2} − 4r^{2}s^{2}t^{2} =
= (2bc − 1 − b − c)^{2} − 4(b − 1)(c − 1)(bc − 1) =
= 1 + b^{2} + c^{2} − 2b − 2bc − 2c + 4.

This yields

deg e + deg ē = deg eē ≤ deg c^{2}.

Using (6.20), we can conclude

deg e ≤ 0.

But looking at (6.19) we see that

e + 1 = φ^{2} and e = ψ^{2}

must hold with φ, ψ ∈ ℤ. This is only possible if e = 0.

This implies now that f = 1, g = 1 and c = 1 + b + 2r. Next let us express all polynomials in terms of the polynomial r. We have

b = r^{2} + 1,

and therefore

c = r^{2} + 2r + 2.

Next we calculate s^{2} = c − 1 = b + 2r = r^{2} + 2r + 1 = (r + 1)^{2}, thus

s = r + 1.

In the same way, we get via t^{2} = bc−1 = (r^{2}+1)(r^{2}+2r+2)−1 = r^{4}+2r^{3}+3r^{2}+2r+1 = (r^{2} + r + 1)^{2}, that

t = r^{2} + r + 1.

This gives us

w′_{1} = (2bc − 1)s − 2tcr =
= (2r^{4} + 4r^{3} + 6r^{2} + 4r + 3)(r + 1) − 2(r^{3} + r^{2} + r)(r^{2} + 2r + 2) =
= 2r^{2} + 3r + 3.

From this we conclude that

deg w′_{1} = deg c

and by induction, using the recurring formula (6.16), we get

deg w′_{n} = deg c + 2(n − 1) deg c, n ≥ 1.

Let us assume that v_{m} = w′_{n} has a solution. Then by comparing the degree of v_{m}, which is given by (6.17), and the degree of w′_{n}, we get

\frac{1}{2} deg c + m deg c = deg c + 2(n − 1) deg c

and

\frac{1}{2} + m = 2n − 1,

a contradiction. Therefore v_{m} = w′_{n} cannot have a solution and the proof is finished. □

Now Theorem 6.1 follows directly from Lemma 6.5, Lemma 6.6 and Lemma 6.8.

Diophantine m-tuples for linear polynomials

In this chapter, we prove that there does not exist a set with more than 26 polynomials with integer coefficients such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients.

This chapter is equal to a manuscript which is joint work with A. Dujella and R. F. Tichy (cf. [34]).

7.1 Introduction

Let n be a nonzero integer. A set of m positive integers {a_{1}, a_{2}, …, a_{m}} is called a Diophantine m-tuple with the property D(n) or simply D(n)-m-tuple, if the product of any two of them increased by n is a perfect square.

Diophantus [25] found the first quadruple {1, 33, 68, 105} with the property D(256). The first D(1)-quadruple, the set {1, 3, 8, 120}, was found by Fermat. The folklore conjecture is that there does not exist a D(1)-quintuple. In 1969, Baker and Davenport [6] proved that the Fermat’s set cannot be extended to a D(1)-quintuple. Recently, Dujella proved that there does not exist a D(1)-sextuple and there are only finitely many D(1)-quintuples (see [32]).

The natural question is how large such sets can be. We define

M_{n} = sup{|S| : S has the property D(n)},

CHAPTER 7. D(n)-m-TUPLES FOR LINEAR POLYNOMIALS 113

where |S| denotes the number of elements in the set S. Dujella proved that M_{n} is finite for all n ∈ ℤ\{0}. In his proof he estimated the number of “large” (greater than |n|^{3}), “small” (between n^{2} and |n|^{3}) and “very small” (less than n^{2}) elements of a set with the property D(n) by using a theorem of Bennett [8] on simultaneous approximations of algebraic numbers and a gap principle in the first, a weaker variant of the gap principle in the second and a large sieve method due to Gallagher [52] in the third case respectively (cf. [29]). Let us introduce the following notation:

A_{n} = sup{|S ∩ [|n|^{3}, ∞⟩| : S has the propertyD(n)},
B_{n} = sup{|S ∩ [n^{2}, |n|^{3}⟩| : S has the propertyD(n)},
C_{n} = sup{|S ∩ [1, n^{2}⟩| : S has the propertyD(n)}.

His result was (cf. [29, Theorem 1, 2, 3 and 4])

A_{n} ≤ 21,

B_{n} ≤ 0.65 log |n| + 2.24,

C_{n} ≤ \{ \begin{matrix}265.55 log |n|(log log |n|)^{2} + 9.01 log log |n| & for |n| > 400, \\ 5 & for |n| ≤ 400. \end{matrix}

Therefore

M_{n} ≤ 32 for |n| ≤ 400,
M_{n} < 267.81 log |n|(log log |n|)^{2} for |n| > 400.

A polynomial variant of the above problems was first studied by Jones [55], [56], and it was for the case n = 1.

Definition 7.1. Let n ∈ ℤ[x] and let {a_{1}, a_{2}, …, a_{m}} be a set of m nonzero polynomials with integer coefficients. We assume that there does not exist a polynomial p ∈ ℤ[x] such that a_{1}/p, …, a_{m}/p and n/p^{2} are integers. The set {a_{1}, a_{2}, …, a_{m}} is called a polynomial D(n)-m-tuple if for all 1 ≤ i < j ≤ m the following holds: a_{i} · a_{j} + n = b_{ij}^{2}, where b_{ij} ∈ ℤ[x].

Let us mention that the assumption that there does not exist a polynomial p such that a_{1}/p, …, a_{m}/p and n/p^{2} are integers means for constant n that not all elements a_{1}, …, a_{m} of a polynomial D(n)-m-tuple are allowed to be constant (compare with Definition 1 in [33] and with Definition 6.1). For linear n the condition under consideration is trivially always satisfied.

In analog to above we are interested in the size of

P_{n} = sup{|S| : S is a polynomial D(n)-tuple}.

From the result above (cf. [29, Theorem 1]) it follows that P_{n} ≤ 22 for all n ∈ ℤ. The above mentioned result about the existence of only finitely many D(1)-quintuples implies that P_{1} = 4. Recently, Dujella and the author proved that P_{−1} = 3 (cf. [33], see also Chapter 6) by successfully transferring the needed methods to the polynomial case.

The results of [29], by specialization, give a bound for P_{n} in terms of the degree and the maximum of the coefficients of n. We conjecture that there should exist a bound for P_{n}, which depends only on the degree of n. As we have seen, this is true for constant polynomials, and in the present chapter we will prove this conjecture for linear polynomials.

We want to handle the case for linear polynomials, i.e. n = ax + b, with integers a ≠ 0 and b. Let us define

L = sup{|S| : S is a polynomial D(ax + b)-tuple for some a ≠ 0 and b }.

We are intended to prove that L < ∞. More precisely, we want to find some good upper bound for L.

It is easy to prove that L ≥ 4. E.g. the set

{x, 16x + 8, 25x + 14, 36x + 20}

is a polynomial D(16x + 9)-quadruple and the set

{1, 9x^{2} + 8x + 1, 9x^{2} + 14x + 6, 36x^{2} + 44x + 13}

is a polynomial D(4x + 3)-quadruple (see [26]).

The idea is to estimate the number of polynomials in S with given degree and to consider separate cases whether the degree is “large” or “small”.

In analog to the classical case, we prove our result for “large” degree by using a theorem due to Mason [62] on the polynomial solutions of hyperelliptic equations over function fields in one variable and a gap principle. Let S be a polynomial D(ax + b)- m-tuple with integers a ≠ 0 and b. We prove

Theorem 7.1. There are at most 15 polynomials in S with degree ≥ 4.

We want to remark that a weaker result can be shown by applying the results from the classical integer case. From that it is possible to show that there are at most 21 polynomials in S with degree ≥ 4.

We have to estimate the number of constant, linear, quadratic and cubic polynomials in S. We denote these numbers by L_{0}, L_{1}, L_{2}, L_{3} respectively and we will consider them separately. First of all it is trivial to see that we have

L_{0} ≤ 1.

By using the mentioned gap principle once more, we get Theorem 7.2. There are at most three polynomials in S of degree 3. Therefore, we have

L_{3} ≤ 3.

Let us remark that in fact the proof gives us the following result: There is no polynomial D(ax + b)-quadruple which consists of polynomials all having the same degree μ ≥ 3. For the case n = 1 this was already proved by Jones in [56].

By more detailed analysis we get

Theorem 7.3. There are at most five polynomials in S of degree 2. Therefore, we have

L_{2} ≤ 5.

Theorem 7.4. There are at most eight linear polynomials in S. Therefore, we have

L_{1} ≤ 8.

Altogether, we can prove the following bound for the size of polynomial D(n)-m-tuples for linear polynomials n = ax + b.

Theorem 7.5.

L ≤ 26.

In section 7.2, we will collect auxiliary results which are needed to prove our theorems. In section 7.3, we handle Theorem 7.1 and 7.2 which are the cases of large degrees. In section 7.4, we prove the results for the small degrees, i.e. Theorem 7.3 and 7.4 and therefore finally get Theorem 7.5.

7.2 Auxiliary results

Before we can go to the proofs of the theorems, we need the following useful construction with the elements of a polynomial D(n)-triple where n is a polynomial with integer coefficients. The construction is a direct modification from the integer case (see [29, Lemma 3]). The analogous statement for polynomial D(1)-triples was proved by Jones in [56] and we did already use it in the case n = −1 (cf. [33] and Lemma 6.7 in section 6.4).

Lemma 7.1. Let {a, b, c} be a polynomial D(n)-triple and let ab + n = r^{2}, ac + n = s^{2}, bc + n = t^{2}. Then there exist polynomials e, u, v, w ∈ ℤ[x] such that

ae + n^{2} = u^{2}, be + n^{2} = v^{2}, ce + n^{2} = w^{2}.

More precisely,

e = n(a + b + c) + 2abc − 2rst.

Furthermore, it holds:

c = a + b + \frac{e}{n} + \frac{2}{n^{2}} (abe + ruv),

where u = at − rs, v = bs − rt.

Proof. We have

(ae + n^{2}) − (at − rs)^{2} = an(a + b + c) + 2a^{2}bc − 2arst + n^{2} −
−a^{2}(bc + n) + 2arst − (ab + n)(ac + n) = 0.

Hence, we may take u = at − rs, and analogously y = bs − rt, z = cr − st. We have

abe + ruv = abn(a + b + c) + 2a^{2}b^{2}c − 2abrst + abrst −
−a(ab + n)(bc + n) − b(ab + n)(ac + n) + rst(ab + n) =
= −abcn − n^{2}(a + b) + rstn,

and finally

a + b + \frac{e}{n} + \frac{2}{n^{2}} (abe + ruv) = 2a + 2b + c + \frac{2abc}{n} − \frac{2rst}{n} − \frac{2abc}{n} − 2a − 2b + \frac{2rst}{n} = c.

□

If we also define

\overline{e} = n(a + b + c) + 2abc + 2rst,  (7.1)

then easy computation shows that

e · \overline{e} = n^{2}(c − a − b − 2r)(c − a − b + 2r).  (7.2)

This equation will be very useful in the proof of Theorem 7.2, 7.3 and 7.4.

We conclude this section with the following definition: Let ℤ^{+}[x] denote the set of all polynomials with integer coefficients with positive leading coefficient. For a, b ∈ ℤ[x], a < b means that b − a ∈ ℤ^{+}[x]. The usual fundamental properties of inequality hold for this order. For a ∈ ℤ[x], we define |a| = a if a ≥ 0, and |a| = −a if a < 0.

Observe that it is clear that all leading coefficients of the nonconstant polynomials in a polynomial D(n)-m-tuple have the same sign. This implies that there is no loss of generality in assuming that they are all positive, i.e. that all polynomials are in ℤ^{+}[x].

7.3 Elements with large degree

Assume that the set {a, b, c, d} is a polynomial D(n)-quadruple with n ∈ ℤ[x]. Let ab + n = r^{2}, ac + n = s^{2}, bc + n = t^{2} where r, s, t ∈ ℤ^{+}[x]. In this chapter, the symbols r, s, r will always have this meaning. Moreover, we have

ad + n = u^{2}, bd + n = v^{2}, cd + n = w^{2},

with u, v, w ∈ ℤ^{+}[x]. Multiplying this equations we get the following hyperelliptic equation

(uvw)^{2} = (ad + n)(bd + n)(cd + n),

where we search for polynomial solutions d ∈ ℤ[x]. We will apply Mason’s Theorem 1.14 to this equation.

Lemma 7.2. Let {a, b, c, d}, 0 < a < b < c < d be a polynomial D(n)-quadruple with n ∈ ℤ[x]. Then

deg d ≤ 51(deg a + deg b + deg c) + 78 deg n.

Proof. Let us denote X = abcd and Y = abcuvw. Then by multiplying the above equation with a^{2}b^{2}c^{2} we get

Y^{2} = (X + nbc)(X + nac)(X + nab).

The polynomial on the left hand side becomes

(X + nbc)(X + nac)(X + nab) =
= X^{3} + n(ab + bc + ac)X^{2} + n^{2}abc(a + b + c)X + n^{3}a^{2}b^{2}c^{2}

so this polynomial has coefficients and roots in ℤ[x]. Let S be the set of coefficients of this polynomial, i.e.

S = {1, n(ab + bc + ac), n^{2}abc(a + b + c), n^{3}a^{2}b^{2}c^{2}}.

Since the elements of S are polynomials, we get for each ξ ∈ ℂ that

ν_{ξ}(S) = min_{s∈S}{0, ν_{ξ}(s)} = 0.

Moreover, we have

ν_{∞}(S) = min_{s∈S}{0, ν_{∞}(s)) = min_{s∈S}{− deg s} = − max_{s∈S} deg s,

and by comparing the degrees of the elements of S we get

ν_{∞}(S) = −2(deg a + deg b + deg c) − 3 deg n.

Therefore,

H(S) = − ∑_{ν} min{0, ν(S)} = − ∑_{ξ∈ℂ}min{0, ν_{ξ}(S)} − min{0, ν_{∞}(S)} =

= − min{0, ν_{∞}(S)} = 2(deg a + deg b + deg c) + 3 deg n.

Thus, we have for the height H of the polynomial on the right hand side of our hyperelliptic equation

H = 2(deg a + deg b + deg c) + 3 deg n.

By Mason’s Theorem 1.14 with L = ℂ(x) and O = ℂ[x] we therefore get

deg X ≤ 52(deg a + deg b + deg c) + 78 deg n,

where we have used that the genus of the rational function field ℂ(x) is zero (which can be found e.g. in [90], page 22) and that ℂ(x) has only one infinite valuation, namely ν_{∞}. But now by the definition of X = abcd we get

deg d ≤ 51(deg a + deg b + deg c) + 78 deg n

as claimed in our lemma. □

Observe that due to the sharpness of the fundamental inequality this bound is very good. Especially, it does not depend on a gap which has to appear between the elements of the quadruple as in the classical case (cf. [29, Lemma 2]).

We use Lemma 7.1 to prove the following gap principle. This is very similar to [29, Lemma 4] in the classical case for integers.

Lemma 7.3. If {a, b, c, d} is a polynomial D(n)-quadruple where n ∈ ℤ[x] and 2n^{2} < a < b < c < d, then

n^{2}d > 2bc.

Proof. We apply Lemma 7.1 to the triple {a, c, d}. Let e be defined as in Lemma 7.1. Since ce + n^{2} is a perfect square, we have that ce + n^{2} ≥ 0. Assume that e ≤ −1, then

ce + n^{2} < −2n^{2} + n^{2} = −n^{2} < 0,

a contradiction. Therefore, we have e ≥ 0. Observe that, if n > 0 we have

a^{2} < ac + r^{2} = ac + ab + n ⟺ na^{2} < na(b + c) + n^{2} ⟺
a^{2}t^{2} = a^{2}(n + bc) = na^{2} + a^{2}bc < a^{2}bc + na(b + c) + n^{2} =
= (ab + n)(ac + n) = r^{2}s^{2} ⟺
at < rs ⟺ u < 0,

and

b^{2} < bc + r^{2} = bc + ab + n ⟺ nb^{2} < nb(a + c) + n^{2} ⟺
b^{2}s^{2} = b^{2}(ac + n) = ab^{2}c + b^{2}n < ab^{2}c + nb(a + c) + n^{2} =
= (ab + n)(bc + n) = r^{2}t^{2} ⟺
bs < rt ⟺ v < 0.

In the same way one can show that n < 0 implies

u > 0 and v > 0.

If e = 0, then d = a + c + 2s. If e ≥ 1, then

n^{2}d = n^{2}(a + b) + en + 2(abe + ruv) > 2ab.

Note that we need here that uv > 0 which follows from the comments just made.

Analogously, we apply Lemma 7.1 to the triple {b, c, d} and obtain either d = b+c+2t or n^{2}d > 2bc. However, d = b+c+2t is impossible since s^{2} = ac+n < bc+n = t^{2} and therefore s < t which implies a + c + 2s < b + c + 2t and

n^{2}(b + c + 2t) < n^{2}4c < 2ac,

which follows from

t^{2} = bc + n ≤ (c − 1)c + n = c^{2} + n − c < c^{2} + n − n^{2} < c^{2}

since c > 2n^{2} and consequently t < c.

Hence, we proved

n^{2}d > 2bc,

as claimed in our lemma. □

Proof of Theorem 7.1.

Assume that {a, b, c, a_{4}, a_{5}, …, a_{16}} is a polynomial D(n)-16-tuple and |n|^{3} ≤ a < b < c < a_{4} < a_{5} < … < a_{16}. We apply Lemma 7.2 to the quadruple {a, b, c, a_{16}} and obtain

d_{16} < (abc)^{52}n^{78} < c^{156}n^{78} < c^{182},  (7.3)

since |n|^{3} < c.

Lemma 7.3 implies n^{2}a_{4} > bc > |n|^{3}c and a_{4} > c|n|. Furthermore, n^{2}a_{5} > a_{4}c > c^{2}|n| and |n|a_{5} > c^{2}. In the same manner, Lemma 7.3 gives

n^{2}a_{6} > a_{5}a_{4} > c^{3}, |n|^{5}a_{7} > |n|^{3}a_{6}a_{5} > c^{5}, |n|^{9}a_{8} > c^{8},
n^{16}a_{9} > c^{13},  |n|^{27}a_{10} > c^{21},  |n|^{45}a_{11} > c^{34},
n^{74}a_{12} > c^{55},  |n|^{121}a_{13} > c^{89},  |n|^{197}a_{14} > c^{144},
|n|^{320}a_{15} > c^{233}, |n|^{519}a_{16} > c^{377},

which implies (since |n|^{3} < c) that

c^{173}a_{16} > c^{377}

and therefore

a_{16} > c^{204},

a contradiction to (7.3). □

Proof of Theorem 7.2.

Let S = {a, b, c} with a < b < c be a polynomial D(n)-triple with linear n ∈ ℤ[x] and let deg a = deg b = deg c = 3. Then by (7.1) we get deg \overline{e} = 9. But from (7.2) it follows that deg e\overline{e} ≤ 8. Thus we have a contradiction unless e = 0, i.e. c = a + b + 2r. Consequently, if we fix a and b, then c is unique, which implies that S cannot be extended to a polynomial D(n)-quadruple. Therefore,

L_{3} ≤ 3,

as claimed in our theorem. □

Observe that Theorem 7.2 follows directly from Lemma 7.3, but the above proof gives more information on triples of cubic polynomials.

7.4 Elements with small degree

First we prove Theorem 7.3. Here the argument from the proof of Theorem 7.2 does not longer work. The polynomials e which are induced by a polynomial D(ax+b)-triple in Lemma 7.1 are constants. The proof uses the fact that u^{2} − n^{2} = (u − n)(u + n) is a complete factorization of the polynomial a up to the constant factor e.

Proof of Theorem 7.3.

Let {a, b, c} with a < b < c be a polynomial D(n)-triple with linear n ∈ ℤ[x] and let deg a = deg b = deg c = 2. Then by (7.1) we get deg \overline{e} = 6. Now (7.2) implies that e is a constant. Assume that two distinct e’s exist. We call them e and f. From ae + n^{2} = u^{2} we see that a is a product of two linear polynomials:

a = α(x − a_{0})(x − a_{1}).

Let us assume that we have

u − n = ε_{1}(x − a_{0}), u + n = ε_{2}(x − a_{1}),

where ε_{1}ε_{2} = αε. It implies

2n = x(ε_{2} − ε_{1}) + ε_{1}a_{0} − ε_{2}a_{1}.

In the same manner, we can conclude from af + n^{2} = u^{2} that

u − n = φ_{1}(x − a_{0}), u + n = φ_{2}(x − a_{1}),

or

u − n = φ_{1}(x − a_{1}), u + n = φ_{2}(x − a_{0})

holds. Let us first consider that the first of this equations holds. Then we get

2n = x(φ_{2} − φ_{1}) + φ_{1}a_{0} − φ_{2}a_{1},

where φ_{1}φ_{2} = αf. Hence, ε_{2} − ε_{1} = φ_{2} − φ_{1}, ε_{1}a_{0} − ε_{2}a_{1} = φ_{1}a_{0} − φ_{2}a_{1}. Consequently, we have a_{0}(ε_{1} − φ_{1}) = a_{1}(ε_{2} − φ_{2}) = a_{1}(ε_{1} − φ_{1}). We have two possibilities: ε_{1} = φ_{1} or a_{0} = a_{1}.

We first assume ε_{1} = φ_{1}. This implies also ε_{2} = φ_{2} and therefore e = f, a contradiction. Now we assume that a_{0} = a_{1} holds. Then x − a_{0}|n and together with ab + n = r^{2} this implies x − a_{0}|r. Therefore (x − a_{0})^{2}|n, and we obtained a contradiction since n is a linear polynomial.

Now let us consider the second case. So assume that we have

u − n = φ_{1}(x − a_{1}), u + n = φ_{2}(x − a_{0}),

where φ_{1}, φ_{2} are as above. It implies

2n = x(φ_{2} − φ_{1}) + φ_{1}a_{1} − φ_{2}a_{0}.

Hence, ε_{2} − ε_{1} = φ_{2} − φ_{1}, ε_{1}a_{0} − ε_{2}a_{1} = φ_{1}a_{1} − φ_{2}a_{0}. This yields, a_{0}(ε_{1} + φ_{2}) = a_{1}(φ_{1} + ε_{2}) = a_{0}(ε_{1} + φ_{2}). We have again two possibilities: ε_{1} = −φ_{2} which implies ε_{2} = −φ_{1} and therefore e = f, a contradiction, or a_{0} = a_{1}. But as above this yields a contradiction with the assumption that n is a linear polynomial.

Therefore, there is at most one such constant e. It follows that for fixed a and b, there are at most three c, namely c = a + b + 2r and two possible c(e) which come from

c = a + b + \frac{e}{n} + \frac{2}{n^{2}} (abe + 2ruv),

where u, v satisfy ae + n^{2} = u^{2}, be + n^{2} = v^{2}. This last equations fix u and v only up to the sign and therefore we get two possible c’s in this case. Consequently we get

L_{2} ≤ 5,

which was claimed in Theorem 7.3. □

As in the proof before we will see that also the proof of Theorem 7.4 heavily depends on the fact that we are considering linear polynomials. Especially, we will use that (7.2) is the complete factorization of the product e\overline{e}.

Proof of Theorem 7.4.

Let S = {a, b, c} with a < b < c be polynomial D(n)-triple with linear n ∈ ℤ[x] and let deg a = deg b = deg c = 1. Then by (7.1) we get deg \overline{e} = 3. Now (7.2) implies that

deg e ≤ 1.

From ab + n = r^{2} it follows that at most one of the elements in S is divisible by n. Indeed, assume that a and b are divisible by n. Then n|r and n^{2}|n, which contradicts the assumption that n is linear polynomial. Thus we may assume that a, b, c are not divisible by n. We have

e + \overline{e} = 2n(a + b + c) + 4abc.  (7.4)
\overline{e} − e = 4rst.  (7.5)

If n|e, then (7.2) implies that n|\overline{e} and therefore, by (7.4), we get n|abc, a contradiction. Therefore, e = δ · (c − a − b ± 2r), δ ∈ ℚ. Assume that δ ≠ 0. We have

\overline{e} = n^{2}(c − a − b ∓ 2r)\frac{1}{δ}.

This implies

\frac{e}{δ} − \frac{\overline{e}δ}{n^{2}} = ±4r

or

\frac{n^{2}e}{δ} − \overline{e}δ = ±4n^{2}r

or

\frac{n^{2}e}{δ} − eδ = 4r(δst ± n^{2}).

This can be written as

\frac{e}{δ}(n^{2} − δ^{2}) = 4r(δst ± n^{2}).

Hence, there are two possibilities: r|e or r|n ± δ.

If r|e, then by (7.5) we have r|\overline{e} which yields

r^{2}|n^{2}(c − a − b − 2r)(c − a − b + 2r).

If r|n, then from ab + n = r^{2} we conclude r|a or r|b. Both cases lead to a contradiction since this would imply n|a or n|b. Observe that r and n only differ by a constant factor since they are both linear. Thus, r|c − a − b, say c = a + b + r · ρ, ρ ∈ ℚ. But from this we get

ac + n = a^{2} + ab + arρ + n = a^{2} + r^{2} + ρar = s^{2}

or

(2r + ρa)^{2} − (ρ^{2} − 4)a^{2} = (2s)^{2}.

Now, if ρ = ±2 then c = a + b ± 2r. Observe that by considering leading coefficients it is clear that a + b − 2r < b so this case is impossible and it remains the possible case c = a + b + 2r. Otherwise if ρ ≠ ±2, we have

(ρ^{2} − 4)a^{2} = (2r + ρa − 2s)(2r + ρa + 2s)

which implies a|r and a|s, and moreover, using ab + n = r^{2}, we get a|n or equivalently n|a, a contradiction.

Therefore, it remains the case r|n ± δ. It means that δ is unique. It is defined by n ≡ ∓δ (mod r). Let n ≡ δ_{0} (mod r). We have the following possibilities for c, namely c = a + b + 2r and c(e), where e = (c − a − b + 2r)(−δ_{0}) or e = (c − a − b − 2r) · δ_{0}. Each of this two e’s induce at most two c’s as we have seen at the end of the proof of Theorem 7.3. Therefore, we have at most seven linear polynomials in S which are not divisible by n. We get

L_{1} ≤ 8,

and so the proof is finished. □

Now we are ready to prove our bound for L.

Proof of Theorem 7.5.

Let S be a polynomial D(ax + b)-m-tuple with some integers a ≠ 0 and b. From the fact that the product of each two elements from S plus ax + b is a square of a polynomial with integer coefficients it follows that the set S contains a polynomial with degree ≥ 2, then it contains either polynomials with even or polynomials with odd degree only. Together with the upper bound for the number of polynomials in S with degree ≥ 4, this implies that we have

|S| ≤ 11 + 16 = 27.

This proves our theorem. □

Curriculum vitae

Personal Data

Date of Birth: 10.11.1976

Place of Birth: Lienz (Austria)

Nationality: Austrian

Parents: Josef Fuchs (Unterassling 47, 9911 Thal-Assling)

Notburga Fuchs, maiden name: Salcher (deceased in June 1998)

Brothers and Sisters: Thomas, Michael, Daniela, Ursula, Anna

Address: Karl-Morre Str. 46/15, 8020 Graz (Austria)

Phone: +43-316-573513

Education

1983–1987 Primary School in Assling, Tyrol

1987–1991 Secondary School in Lienz, Tyrol

1991–1995 High School in Lienz (with special emphasis on natural science) School Leaving Examination on June 14, 1995 with honours

1995–2000 Study of Technical Mathematics at the Technical University of Vienna, Branch: Mathematical Computer Science First Diploma Examination on March 23, 1997 with first honours Second Diploma Examination on October 23, 2000 with first honours Diploma Thesis: “Algebraisch-geometrische Codes” Advisor: Ao.Univ-Prof. Dr. M. Drmota Graduation to “Diplom-Ingenieur” (corresponding to Master of Science)

Since 2000 PhD-Studies in Mathematics at the Technical University of Graz Working Title of the PhD-Thesis: “Quantitative Finiteness Results for Diophantine Equations” Advisor: O.Univ-Prof. Dr. R. F. Tichy

Career History

WS 1998/99 Teaching assistent at the Department of Applied and Numerical Mathematics at the Technical University of Vienna

WS 1999/00 Teaching assistent at the Department of Applied and Numerical Mathematics at the Technical University of Vienna

WS 2000/01 Teaching assistent at the Department of Mathematics at the Technical University of Graz

SS 2001 Teaching assistent at the Department of Mathematics and Applied Geometry at the Montanistic University of Leoben and Teaching assistent at the Department of Mathematics at the Technical Univeristy of Graz

WS 2001/02 Teaching assistent at the Department of Mathematics and Applied Geometry at the Montanistic University of Leoben and Teaching assistent at the Department of Mathematics at the Technical Univeristy of Graz

June 2000 Collaborator in the research project “Algorithmic Diophantine Problems” (S-8307 MAT) via working contract

July-September 2000 Collaborator in the research project “Algorithmic Diophantine Problems” (S-8307 MAT) via a “Forschungsbeihilfe”

Since November 2000 Employment in the research project “Algorithmic Diophantine Problems” (S-8307 MAT)

Research Interests

Algebraic Number Theory, Diophantine equations, Algebraic Geometry, Algebraic Coding Theory.

Publications

1. A. Dujella and C. Fuchs (2001)

A polynomial variant of a problem of Diophantus and Euler, to appear in Rocky Mount. J. Math.

2. C. Fuchs and R. F. Tichy (2001)

Perfect Powers in Linear Recurring Sequences, to appear in Acta Arith.

3. C. Fuchs, A. Pethő and R. F. Tichy (2001)

On the Diophantine equation G_{n}(x) = G_{m}(P (x)), submitted for publication.

4. C. Fuchs (2001)

Exponential-polynomial Equations and Linear Recurring Sequences, submitted for publication.

5. C. Fuchs (2001)

An upper bound for the G.C.D. of two linear recurring sequences, manuscript.

6. C. Fuchs, A. Pethő and R. F. Tichy (2001)

On the Diophantine equation G_{n}(x) = G_{m}(P (x)) for third order linear recurrrences, manuscript.

7. A. Dujella, C. Fuchs and R. F. Tichy (2001)

Diophantine m-tuples for linear polynomials, manuscript.

Posters and Oral Contributions

1. C. Fuchs: Algebraisch-geometrische Codes, Zahlentheoretisches Kolloquium, Graz, Austria, 17.11.2000.

2. C. Fuchs: On the equation G_{n}(x) = G_{m}(P (x)), AMS-Sectional Meeting, Hoboken, New Jersey, USA, 28.04.2001.

3. A. Dujella and \underline{C. Fuchs}: A Polynomial Variant of a Problem of Diophantus and Euler, 15. ÖMG-Kongress, Vienna, Austria, 20.09.2001.

4. C. Fuchs: Exponential-polynomial Equations and Linear Recurring Sequences, Workshop on “Effective methods for Diophantine Equations”, Debrecen, Hungary, 26.10.2001.

5. C. Fuchs: Exponential-polynomial Equations and Linear Recurring Sequences, Zahlentheoretisches Kolloquium, Graz, Austria, 30.11.2001.

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