THOMAS’ FAMILY OF THUE EQUATIONS OVER FUNCTION FIELDS

CLEMENS FUCHS^{∗} AND VOLKER ZIEGLER^{‡}

Abstract. We consider a function field analogue of Thomas’ family [14] of Thue equations

X^{3} − (λ − 1)X^{2}Y − (λ + 2)XY^{2} − Y^{3} = ξ, where the solutions X, Y come from the ring ℂ[T], the parameter λ ∈ ℂ[T] is some non-constant polynomial and 0 ≠ ξ ∈ ℂ. In this paper we completely solve this family.

1. Introduction and Main Result

The history of Thue equations, these are diophantine equations F (X, Y) = m where F ∈ ℤ[X, Y] is a binary irreducible form of degree at least 3 and m is a non zero integer, in particular the study of families of Thue equations dates back to Thue himself. In his paper [15] he proved that the Thue equation

(a + 1)X^{n} − aY^{n} = 1

has only the solution x = y = 1, if a is suitable large in relation to the prime n ≥ 3. In more recent time Thomas [14] considered the family

(1)  X^{3} − (λ − 1)X^{2}Y − (λ + 2)XY^{2} − Y^{3} = ±1 (λ ∈ ℤ).

This was the first time that a family of Thue equations with positive discriminant has been investigated. Thomas [14] and Mignotte [11] solved this equation completely. It has the solutions ±(1, 0), ±(0, 1), ±(1, −1) for all λ ∈ ℤ and some further solutions for λ = 0, 1, 3. Since then several authors considered many families of Thue equations up to degree 8 (cf. [6]). A survey on the subject can be found in [4].

All the families stated above were considered over the ring of rational integers. Heuberger, Pethő and Tichy solved the first time a family of relative Thue equations in [5]. In particular they studied Thomas’ family again

(2)  X^{3} − (λ − 1)X^{2}Y − (λ + 2)XY^{2} − Y^{3} = μ,

2000 Mathematics Subject Classification. 11D59,11D25,11Y50.

Key words and phrases. Thue equation, function fields.

∗The first author was supported by the Austrian Science Foundation, grants S8307-MAT and J2407-N12.

‡The second author was supported by the Austrian Science Foundation, grant S8307-MAT.

where the solutions x, y, the parameter λ and the root of unity μ now are all (algebraic) integers in the same imaginary quadratic number field k. They proved that (2) has only trivial solutions (i.e. with |x|, |y| ≤ 1), if |λ| is large enough or if the discriminant of the quadratic number field is large enough or if ℜλ = −1/2 (there are a few more solutions in this case which are explicitly listed).

Thue equations were not only considered over number fields but also over function fields starting with the paper of Gill [3]. In the next 50 years several authors as Schmidt (cf. [13]) and Mason (cf. [9], resp. [10]) considered the problem to determine effectively all solutions of a given Thue equation over some function field. In contrast to the number field case Thue equations over function fields may have infinitely many solutions. Recently, Lettl [8] could prove criteria for which a given Thue equation has only finitely many solutions. In all these investigations no families of Thue equations were considered. Recently, the authors could solve the family

X(X − Y)(X − (T + ξ)Y) + Y^{3} = 1 + ξT (1 − T)

over ℂ[T] for all ξ ∈ ℂ (cf. [2]). This was the first time that a family of Thue equations over a function field was solved.

In this paper we will go a step further and consider a family, where the parameter itself is a polynomial. In particular we revisit Thomas’ family and we completely solve the equation

(3)  X^{3} − (λ − 1)X^{2}Y − (λ + 2)XY^{2} − Y^{3} = ξ,

where λ ∈ ℂ[T] is a non-constant polynomial and ξ ∈ ℂ^{×} (which denotes as usually the unit group and in this case is just ℂ\{0}). This family can be seen as a function field analogue of the Thue equations (1) and (2), respectively. In all three cases the solutions x, y and the parameter λ come from the same commutative ring R, namely ℤ, 𝔬_{k} and ℂ[T], respectively, and the right hand side is a unit in R (we denote by 𝔬_{k} the maximal order of some number field k).

The aim of this paper is to prove the following main theorem:

Theorem 1. If (x, y) ∈ ℂ[T] × ℂ[T] is a solution of the Thue equation

X^{3} − (λ − 1)X^{2}Y − (λ + 2)XY^{2} − Y^{3} = ξ,

with non-constant λ ∈ ℂ[T] and ξ ∈ ℂ^{×}, then

(x, y) ∈ L := {(ζ, 0), (−ζ, ζ), (0, −ζ) : ζ^{3} = ξ} .

Observe that the result contains all the (infinite) polynomial families of solutions found by Thomas [14] in his original paper and by Heuberger, Pethő and Tichy in [5] (see Table 1, page 438). So, our main theorem gives once again all the polynomial solutions in these cases and therefore we proved one part of what Thomas calls stable growth, which means that a family of Thue equations has only finitely many polynomial solutions plus finitely many sporadic ones (which means solutions for certain values of the parameter).

The rest of the paper is organized as follows. Section 2 provides a collection of useful facts on function fields and auxiliary results that we need to prove Theorem 1. In Section 3 we compute ramification indices and the genus of the function field ℂ(T)(α), where α is a root of X^{3} − (λ − 1)X^{2} − (λ+2)X −1. These computations yield an upper bound for H_{K}(x−αy) (cf. Section 4). The structure of the relevant unit group is studied in Section 5. Finally, we prove our main theorem in Section 6.

2. Preliminaries and auxiliary Results

Let us first remind the ABC-Theorem for function fields (see e.g. [12, Theorem 7.17]).

Proposition 1 (ABC-Theorem). Let K be a function field of characteristic 0 and genus g_{K}. Let u, v ∈ K^{×} satisfying u + v = 1 and put A = (u)_{0}, B = (v)_{0} and C = (u)_{∞} = (v)_{∞}, where (·)_{0} denotes the zero divisor and (·)_{∞} denotes the polar divisor. Then

deg A = deg B = deg C ≤ max (0, 2g_{K} − 2 + ∑_{P ∈Supp(A+B+C)} deg_{K} P ) .

If the constant field k is algebraically closed and of characteristic 0, Mason [10, chapter 1, Lemma 2] proved following special case.

Corollary 1. Let H(f) := − ∑_{v∈M_{K}} min(0, v(f)) denote the height of an element f ∈ K and let γ_{1}, γ_{2}, γ_{3} ∈ K with γ_{1} + γ_{2} + γ_{3} = 0. Let V be a finite set of valuations such that for all v ∉ V we have v(γ_{1}) = v(γ_{2}) = v(γ_{3}), then

H(γ_{1}/γ_{2}) ≤ max(0, 2g − 2 + |V|).

Here we denote the set of all valuations in K by M_{K}. Usually, M_{K} denotes the set of places of some field K. Since in the function field case valuations and places are one-to-one we use this notation. It is rather easy to deduce Corollary 1 from Proposition 1 (cf. [2]).

If F (X, Y) = m is a Thue equation over the integral closure O_{L} of k[T] in some function field L/k(T), then Mason [9] could prove an effective bound for the height of solutions (x, y) to F (X, Y) = m by using his fundamental inequality presented in Corollary 1. Observe that the integral closure of k[T] consists of all elements in L that may only have negative valuations above ∞. Let us remind Mason’s bound on the height of the solutions.

Proposition 2 (Mason). Let

F (X, Y) := (X − α_{1}Y) ⋯ (X − α_{d}Y) = m

be a Thue equation over the integral closure O_{L} of k[T] of some function field L/k(T) with algebraic closed constant field and characteristic 0. Then all solutions (x, y) ∈ O_{L}^{2} satisfy

H(x, y) ≤ 8H + 2g_{L} + r − 1,

where g_{L} is the genus of L, r is the number of infinite valuations and H denotes the height of the polynomial (X − α_{1}) ⋯ (X − α_{d})/m.

In order to compute the genus of some function field we will use the Riemann-Hurwitz Theorem (see e.g. [12, Theorem 7.16])

Proposition 3 (Riemann-Hurwitz). Let L/K be an extension of function fields of characteristic 0 and let g_{K} and g_{L} be the genus of K and L, respectively, then

(4)  2g_{L} − 2 = [L : K](2g_{K} − 2) + ∑_{w∈M_{L}} e_{w} − 1,

where M_{L} is the set of valuations of L and e_{w} denotes the ramification index of w in the extension L/K.

Let us fix for the rest of the paper some notations.

(5)  F_{λ}(X, Y) :=X^{3} − (λ − 1)X^{2}Y − (λ + 2)XY^{2} − Y^{3},
(6)  f_{λ}(X) := F_{λ}(X, 1) :=X^{3} − (λ − 1)X^{2} − (λ + 2)X − 1,

where λ ∈ ℂ[T] is some non-constant polynomial. We denote the degree of λ by 𝔞 := deg λ > 0. Let α := α_{1} be a root of f_{λ} and let α_{2} and α_{3} be the conjugates of α. Then K := ℂ(T)(α) is a finite extension of the function field ℂ(T). Observe that we have the identities

F_{λ}(Y, −(X + Y)) = F_{λ}(−(X + Y), X) = F_{λ}(X, Y) and

f_{λ} (−1 − \frac{1}{X} ) = \frac{f_{λ}(X)}{X^{3}}.

Therefore,

α_{2} = −1 − \frac{1}{α} and α_{3} = −1 − \frac{1}{α_{2}} = −\frac{1}{α + 1}.

Observe that f_{λ} is irreducible over ℂ(T), since otherwise one of the roots would be in ℂ[T] (because all the root are integral over ℂ[T]), which contradicts the fact that the constant term of f_{λ} is −1, unless this root is in ℂ\{0}. But this implies by the above formulas that all roots are in ℂ, which is clearly not possible. Consequently, K is the splitting field of f_{λ} and hence a Galois extension of ℂ(T). Furthermore, the Galois group is the cyclic group with three elements. We collect these facts in the following

Lemma 1. The extension K/ℂ(T) is a Galois extension. In particular the Galois group G of K/ℂ(T) is isomorphic to the cyclic group C_{3} with three elements.

Further, we denote by (x, y) ∈ ℂ[T]×ℂ[T] some solution to F_{λ}(X, Y) = ξ with ξ ∈ ℂ^{×} and we define

α_{i,j} := α_{i} − α_{j},  β_{i} := x − α_{i}y,  γ_{i,j,k} := β_{i}α_{j,k},

and we write β := β_{1} = x − αy. The polynomial F_{λ} may be expressed as F_{λ}(X, Y) = N_{K/ℂ(T)}(X − αY), where N_{K/ℂ(T)} denotes the norm from K to ℂ(T). From this norm notation we deduce β_{i} ∈ ℂ[T][α]^{×}.

Note that since ℂ is algebraically closed all discrete valuation rings with quotient field ℂ(T) are isomorphic to O_{a} := {f(T)/g(T) : f(T), g(T) ∈ ℂ[T], g(a) ≠ 0} for some a ∈ ℂ or O_{∞} := {f(T)/g(T) : f(T), g(T) ∈ ℂ[T], deg f ≤ deg g}. Valuations corresponding to O_{a} are denoted by v_{a} (finite valuations) and the valuation corresponding to O_{∞} is denoted by v_{∞} (infinite valuation). To simplify notations we will use the following definition.

Definition 1. Let L/ℂ(T) be an extension of degree d, f ∈ L and for each a ∈ ℂ ∪ {∞} let us fix a d-tuple (w_{1}, …, w_{d}) of valuations in L with w_{i}|v_{a} for i ∈ {1, …, d}, where the valuations are used with multiplicity, i.e. a valuation with ramification index e is written down e times. Let us define

(·)_{a} : L → ℤ^{d}, f ↦ (f)_{a} := (w_{1}(f), …, w_{d}(f)).

In the next lemma we calculate all valuations of the quantities α_{1}, α_{2}, α_{3}, respectively, in the function field K/ℂ(T).

Lemma 2. We have

(α_{1})_{∞} = (𝔞, 0, −𝔞),  (α_{2})_{∞} = (0, −𝔞, 𝔞),  (α_{3})_{∞} = (−𝔞, 𝔞, 0)

and (α_{i})_{a} = (0, 0, 0) for a ∈ ℂ and i = 1, 2, 3, where 𝔞 = deg λ.

Proof: Let us first consider the infinite valuations. We compute α_{i} as formal Laurent series in λ and obtain

α_{1} =λ + 1 + \frac{1}{λ} − \frac{3}{λ^{2}} + ⋯ = λ_{𝔞}T^{𝔞} + ⋯,
α_{2} = − 1 − \frac{1}{λ} + \frac{1}{λ^{2}} + ⋯ = −1 − ⋯,
α_{3} = − \frac{1}{λ} + \frac{2}{λ^{2}} + ⋯ = −\frac{1}{λ_{𝔞}}T^{−𝔞} + ⋯,

where λ_{𝔞} denotes the leading term of λ. From the expansion above we obtain the first part of the lemma by using a suitable triple of the infinite valuations. The second part is immediate, since α_{1}, α_{2} and α_{3} are integer units in K. ■

3. Ramification

In this section we compute, which places are ramified in the extension K/ℂ(T). Since K/ℂ(T) is Galois, the ramification indices e_{w} may only take the value 1 (unramified) or 3 (ramified). Hence,it suffices to know how often ramification occurs in order to compute the genus g_{K} of K using Proposition 3. Let us first prove

Lemma 3. If w ∈ M_{K} is ramified then w|v_{a}, where a ∈ ℂ such that a is a root of δ := λ^{2} + λ + 7 ∈ ℂ[T].

Proof: Assume K is not ramified at a valuation that lies over v_{a} with a ≠ ∞. By Puiseux’s Theorem [1, 7] there exists a formal Power series

α(T) := ∑_{n=0}^{∞} a_{n}(T − a)^{n}

such that f_{λ}(α(T)) = 0. On the other hand it is well known that given an equation f(X, T) = 0 with f holomorphic and \frac{∂f}{∂X} |_{T =a} ≠ 0, then there exists

a holomorphic function X(T) = ∑_{n=0}^{∞} a_{n}(T − a)^{n} in an open neighborhood U ⊂ ℂ of a, such that f(X(T), T) = 0. We conclude that ramification over v_{a} may only occur if

\frac{∂f_{λ}}{∂X} |_{T =a} = 3X^{2} − 2X(λ(a) − 1) − (λ(a) + 2) = 0 and
f_{λ}(X)|_{T =a} = X^{3} − X^{2}(λ(a) − 1) + X(λ(a) + 2) − 1 = 0

or equivalently f_{λ}|_{T =a} has a multiple zero. This is δ(a) := δ|_{T =a} = λ(a)^{2} + λ(a) + 7 = 0.

We are left to prove that v_{∞} is not ramified. Since the infinite valuations of K have different values α by Lemma 2, these are different and thus unramified over ℂ(T). ■

Corollary 2. If r is the number of ramified valuations in K/ℂ(T) then g_{K} = r − 2 ≤ 2𝔞 − 2.

Proof: For the proof we utilize Proposition 3. We already know that if w ∈ M_{K} is ramified then e_{w} = 3. Moreover, the genus of ℂ(T) is zero. With this data we obtain

2g_{K} − 2 = 3(0 − 2) + (3 − 1)r = 2r − 6,

hence, g_{K} = r − 2. By Lemma 3 we therefore have g_{K} = r − 2 ≤ 2𝔞 − 2. ■

4. Upper bound for the height of β

In order to solve equation (3) we have to compute all values of β = x−αy that solve the norm equation

(7)  N_{K/ℂ(T)}(β) = ξ (ξ ∈ ℂ^{×}).

Note that β is a unit of ℂ[T][α]. A first step is to bound the height of β, which will be done in this section.

Lemma 4. We have

H_{K} (\frac{γ_{1,2,3}}{γ_{2,3,1}} ) ≤ 6𝔞 − 3.

Proof: It is well known that

γ_{1,2,3} + γ_{2,3,1} + γ_{3,1,2} = 0

(Siegel’s Identity), hence by Corollary 1 we have

H_{K} (\frac{γ_{1,2,3}}{γ_{2,3,1}} ) ≤ max(0, 2g_{K} − 2 + |V|).

Let r be the number of ramified places, then g_{K} = r − 2 (cf. Corollary 2). Since γ_{i,j,k} with pairwise distinct i, j, k ∈ {1, 2, 3} may only have non zero valuations at places w with w|v_{a}, where a ∈ {k ∈ ℂ : δ(k) = 0} ∪ {∞}, we have |V| ≤ r + 3(2𝔞 − r) + 3. This yields

H_{K} (\frac{γ_{1,2,3}}{γ_{2,3,1}} ) ≤ 2r − 4 − 2 + r + 3(2𝔞 − r) + 3 = 6𝔞 − 3.

■

Next, we want to compute an upper bound for H_{K}(β_{1}/β_{2}). Let us denote by

H_{a}(α) := − ∑_{w|v_{a}} min(0, w(α)), a ∈ ℂ ∪ {∞}

the local height. Obviously, we have

(8)  H_{K}(α) = ∑_{a∈ℂ∪{∞}}H_{a}(α).

From equation (8) one computes

\begin{matrix}H_{K} (\frac{γ_{1,2,3}}{γ_{2,3,1}} ) = & ∑_{δ(a)=0} H_{a} (\frac{α_{2,3}}{α_{3,1}} ) + H_{∞} (\frac{γ_{1,2,3}}{γ_{2,3,1}} ) \\ ≥ & H_{∞} (\frac{β_{1}}{β_{2}} · \frac{α_{2,3}}{α_{3,1}} ) \\ ≥ & H_{∞} (\frac{β_{1}}{β_{2}} ) − H_{∞} (\frac{α_{3,1}}{α_{2,3}} ) . \end{matrix} ; (9)

By Lemma 2 we obtain (\frac{α_{3,1}}{α_{2,3}} )_{∞} = (0, 𝔞, −𝔞), hence

H_{K} (\frac{β_{1}}{β_{2}} ) = H_{∞} (\frac{β_{1}}{β_{2}} ) ≤ H_{K} (\frac{γ_{1,2,3}}{γ_{2,3,1}} ) + H_{∞} (\frac{α_{3,1}}{α_{2,3}} ) ≤ 7𝔞 − 3.

Let us now compute H_{K}(β). Since β_{i} ∈ ℂ[T][α]^{×}, with i = 1, 2, 3, we only have to consider infinite valuations. After a suitable permutation of valuations we may assume

(β_{1})_{∞} = (−b_{1}, −b_{2}, −b_{3})

where b_{1} ≥ b_{2} ≥ b_{3}. Therefore, the indices of the conjugates of β_{1} are fixed. By Lemma 1 we either have (case 1)

(β_{2})_{∞} = (−b_{3}, −b_{1}, −b_{2})
(β_{3})_{∞} = (−b_{2}, −b_{3}, −b_{1})

or (case 2)

(β_{2})_{∞} = (−b_{2}, −b_{3}, −b_{1})
(β_{3})_{∞} = (−b_{3}, −b_{1}, −b_{2})

which implies that

(β_{1}/β_{2})_{∞} = \{ \begin{matrix}(b_{3} − b_{1}, b_{1} − b_{2}, b_{2} − b_{3}) & (case 1), \\ (b_{2} − b_{1}, b_{3} − b_{2}, b_{1} − b_{3}) & (case 2). \end{matrix}

Consequently, we get

H_{K}(β_{1}/β_{2}) = \{ \begin{matrix}b_{1} − b_{3} \\ b_{1} − b_{2} − b_{3} + b_{2} = b_{1} − b_{3} \end{matrix} \} = b_{1} − b_{3}.

Since H_{K}(β) = H_{K}(β^{−1}) we may also assume b_{1} ≥ b_{2} ≥ 0. From b_{1} + b_{2} + b_{3} = 0 we deduce −b_{3} = b_{1} + b_{2} ≤ 2b_{1}, hence \frac{−b_{3}}{2} ≤ b_{1}. This yields

\frac{3}{2}H_{K}(β) = −\frac{3}{2}b_{3} ≤ b_{1} − b_{3} = H_{K}(β_{1}/β_{2}) ≤ 7𝔞 − 3,

hence following Proposition:

Proposition 4. We have H_{K}(β) ≤ \frac{14}{3}𝔞 − 2.

We remark that by using Proposition 2 directly we get the weaker result that if (x, y) is a solution to (3) then

max(deg x, deg y) ≤ \frac{28𝔞 − 2}{3}.

This can be obtained just by computing H = H_{K}(f_{λ}), since K is the splitting field of f_{λ}:

H = max (H_{K}(1), H_{K}(λ − 1), H_{K}(λ + 2), H_{K}(−1)) = 3𝔞,

where H_{K} denotes the height with respect to the function field K. Now r = 3 by Lemma 3 and g_{K} ≤ 2𝔞 − 2 by Corollary 2, hence,

max(H_{K}(x), H_{K}(y)) ≤ 24𝔞 + 4𝔞 − 4 + 3 − 1 = 28𝔞 − 2.

Since x, y ∈ ℂ[T] we have 3 deg x = H_{K}(x), 3 deg y = H_{K}(y), respectively, and therefore the claimed bound.

5. Unit structure

Since β ∈ ℂ[T][α]^{×} we have to investigate this unit group. The aim of this section is to prove the following precise description of this group

Proposition 5. The units α_{1} and α_{2} form a fundamental system, so ℂ[T][α]^{×} = ℂ⟨α_{1}, α_{2}⟩, i.e. for every ε ∈ ℂ[T][α]^{×} we have

ε = ηα_{1}^{x}α_{2}^{y},

where η ∈ ℂ and x, y are integers.

In order to prove Proposition 5 we consider first the following Lemma.

Lemma 5. Let ε ∈ ℂ[T][α]^{×} then either H_{K}(ε) ≥ 𝔞 or ε ∈ ℂ^{×}.

Proof: Let ε_{1}, ε_{2}, ε_{3} elements of K with (ε_{1})_{∞} = (a_{1}, b_{1}, c_{1}), (ε_{2})_{∞} = (a_{2}, b_{2}, c_{2}) and (ε_{3})_{∞} = (a_{3}, b_{3}, c_{3}). We will write by abuse of language (a_{1}, b_{1}, c_{1}) ∘ (a_{2}, b_{2}, c_{2}) = (a_{3}, b_{3}, c_{3}), if ε_{1} ∘ ε_{2} = ε_{3}, where ∘ = +, −, ·. A computation (if ∘ = +, − and e.g. a_{1} = a_{2}) may not be unique in this vector notation. In this case we place a ? at that component.

Let ε ∈ ℂ[T][α]^{×} and (ε)_{∞} = (e_{1}, e_{2}, e_{3}). Since H_{K}(ε) = H_{K}(ε_{i}) = H_{K}(ε_{i}^{−1}) with i = 1, 2, 3 and where ε_{i} are the conjugates of ε, we may assume e_{1}, e_{3} ≥ 0 and e_{2} < 0. We may exclude e_{2} = 0, since otherwise ε would be constant. We have

(10)  ε_{i} = h_{0} + h_{1}α_{i} + h_{2}α_{i}^{2} (1 ≤ i ≤ 3),

with h_{0}, h_{1}, h_{2} ∈ ℂ[T]. Solving this linear system by Cramer’s rule one obtains

\begin{aligned}h_{0} & = \frac{ε_{1}α_{2}α_{3}(α_{3} − α_{2}) + ε_{2}α_{3}α_{1}(α_{1} − α_{3}) + ε_{3}α_{1}α_{2}(α_{2} − α_{1})}{δ}, \\ h_{1} & = \frac{ε_{1}(α_{2} + α_{3})(α_{2} − α_{3}) + ε_{2}(α_{3} + α_{1})(α_{3} − α_{1}) + ε_{3}(α_{1} + α_{2})(α_{1} − α_{2})}{δ}, \\ h_{2} & = \frac{ε_{1}(α_{3} − α_{2}) + ε_{2}(α_{1} − α_{3}) + ε_{3}(α_{2} − α_{1})}{δ}, \end{aligned} (11)

where

δ = det(α_{j}^{i−1})_{1≤i,j≤3} = (α_{1} − α_{2})(α_{2} − α_{3})(α_{3} − α_{1}) = λ^{2} + λ + 7 ∈ ℂ[T]

is the discriminant of f_{λ}.

Let us assume h_{0} ≠ 0. We want to compute deg h_{0}. Using the vector notation we obtain

(α_{2}α_{3}(α_{3}−α_{2}))_{∞} = (0, −𝔞, 𝔞)(−𝔞, 𝔞, 0)((−𝔞, 𝔞, 0)−(0, −𝔞, 𝔞)) = (−2𝔞, −𝔞, 𝔞)

and

(δ)_{∞} = (−2𝔞, −2𝔞, −2𝔞),

hence

(h_{0})_{∞} = (e_{1}, 𝔞 + e_{2}, 3𝔞 + e_{3}) + ⋯,

where the other summands are the conjugates of the first summand. We have 0 ≥ − deg h_{0} ≥ min(e_{1}, e_{2} +𝔞). But min(e_{1}, e_{2} +𝔞) ≤ 0, if either e_{1} = 0 and H_{K}(ε) = −e_{2} < 𝔞 or H_{K}(ε) = −e_{2} ≥ 𝔞. So either the Lemma is true or h_{0} ∈ ℂ.

If h_{0} ∈ ℂ one has H_{K}(ε) = H_{K}(h_{1}α + h_{2}α^{2}). Therefore, without loss of generality we may assume that h_{0} = 0. Let us assume now h_{1} ≠ 0, h_{2} ≠ 0 and furthermore, deg h_{1} = 𝔥_{1} and deg h_{2} = 𝔥_{2}. Since ε = α(h_{1} + h_{2}α), we have

(ε)_{∞} = (𝔞, 0, −𝔞) ((−𝔥_{1}, −𝔥_{1}, −𝔥_{1}) + (𝔞 − 𝔥_{2}, −𝔥_{2}, −𝔞 − 𝔥_{2})) .

We distinguish between the two cases 𝔥_{1} ≠ 𝔞 + 𝔥_{2} and 𝔥_{1} = 𝔞 + 𝔥_{2}. In the first case we have (ε)_{∞} = (?,?, min(−𝔥_{1}, −𝔞 − 𝔥_{2}) − 𝔞), hence H_{K}(ε) ≥ max(𝔥_{1}, 𝔥_{2} + 𝔞) + 𝔞 ≥ 𝔞. In the second case we obtain (ε)_{∞} = (?, −𝔥_{1},?), hence H_{K}(ε) ≥ 𝔥_{1} = 𝔥_{2} + 𝔞 ≥ 𝔞.

We are left to the case h_{1} = 0 or h_{2} = 0 and obtain

(ε)_{∞} =(2𝔞, 0, −2𝔞)(−𝔥_{2}, −𝔥_{2}, −𝔥_{2}) = (?,?, −2𝔞 − 𝔥_{2}) or
(ε)_{∞} =(𝔞, 0, −𝔞)(−𝔥_{1}, −𝔥_{1}, −𝔥_{1}) = (?,?, −𝔞 − 𝔥_{1}),

i.e. in both cases H_{K}(ε) ≥ 𝔞. ■

Corollary 3. There does not exist a rational integer k with |k| > 1 and a unit ε ∈ ℂ[T][α]^{×} such that α = ε^{k}. Furthermore, there exists a unit ε ∈ ℂ[T][α]^{×} such that α and ε form a fundamental system, i.e. ℂ[T][α]^{×} = ℂ⟨α, ε⟩.

Proof: The first part is obvious, since if such k and ε would exist, then 0 < H_{K}(ε) < 𝔞, a contradiction to Lemma 5.

Suppose now (ε_{1}, ε_{2}) is a fundamental system such that k > 0 is minimal with ξε_{1}^{k}ε_{2}^{l} = α and ξ ∈ ℂ^{×}. Now k = lq+r with 0 ≤ r < l and (ε_{1}, ε = ε_{1}^{q}ε_{2}) is again a fundamental system, such that ξε_{1}^{r}ε^{l} = α. Since we assume k is minimal, we have r = 0 and from the first part of the Corollary we deduce l = 1, hence (α, ε) is a fundamental system. ■

Now, Proposition 5 follows easily from Lemma 5 or from Corollary 3.

Proof of Proposition 5: Suppose ε is a unit with (ε)_{∞} = (e_{1}, e_{2}, e_{3}). Multiplying with appropriate powers of α_{1} and α_{2} one obtains a unit ε′ with (ε′) = (e′_{1}, e′_{2}, e′_{3}) and 0 ≤ e′_{1} < 𝔞 and −𝔞 < e′_{2} ≤ 0. But now

H_{K}(ε′) = − min{0, e′_{1}} − min{0, e′_{2}} − min{0, −(e′_{1} + e′_{2})}
= − e′_{2} + max{0, e′_{1} + e′_{2}} = max{e′_{1}, −e′_{2}} < 𝔞

implies by Lemma 5 that ε′ must be a constant. From this Proposition 5 follows immediately. ■

Corollary 4. Let ε ∈ ℂ[T][α]^{×}, then H_{K}(ε) = n𝔞 for some n ∈ ℤ.

Proof: From Proposition 5 we now may deduce that all components of (ε)_{∞} are divisible by 𝔞, hence the Corollary follows. ■

6. End of the proof

By Proposition 4 every solution β ∈ ℂ[T][α]^{×} to (7) satisfies H_{K}(β) ≤ 4.67𝔞 − 2. Furthermore, we know from Proposition 5 that

β = ζ (α_{1})^{a_{1}} (α_{2})^{a_{2}} ,

where ζ ∈ ℂ^{×} and a_{1}, a_{2} ∈ ℤ. These two propositions together yield |a_{1}|, |a_{2}| ≤ 4 and |a_{1} − a_{2}| ≤ 4. By computing all possibilities (there are exactly 61 such pairs (a_{1}, a_{2})), we see that only for (a_{1}, a_{2}) ∈ E, where

E = {(1, 1), (0, 0), (1, 0)},

we get a β which has the form x − αy with x, y ∈ ℂ[T], This yields β ∈ L′ := {ζ(−1 − α), ζ, ζα : ζ ∈ ℂ^{×}}. Consequently, we have

N_{K/ℂ[T]}(β) = ζ^{3} = ξ (β ∈ L′)

and therefore finally Theorem 1. ■

7. Acknowledgement

The authors are most grateful to the anonymous referee for several useful comments which enabled us to shorten some of the proofs and to improve the presentation of the paper.

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Institut für Mathematik A, Technische Universität Graz, Steyrergasse 30, A-8010 Graz, Austria Current address: Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, P.O. Box 9512, 2300 RA Leiden, The Netherlands

   E-mail address: clemens.fuchs@tugraz.at

Institut für Mathematik A, Technische Universität Graz, Steyrergasse 30, A-8010 Graz, Austria

   E-mail address: ziegler@finanz.math.tugraz.at