Algebraic independence of the values of Mahler functions satisfying implicit functional equations

XCIII.1 (2000)

Algebraic independence of the values of Mahler functions satisfying implicit functional equations

by

Bernd Greuel (K¨oln)

1. Introduction and results. In a sequence of three papers Mahler ([4]–[6]) discussed the transcendence and algebraic independence of values of functions in several variables satisfying a certain type of functional equation.

In his survey article [7], 37 years later, he stated three new problems. The third problem (for the first and second problem cf. Loxton and van der Poorten [3]) dealt with implicit functional equations of the type

(1) P (z, f (z), f (T z)) = 0

with T z = z

, d ∈ Z, d ≥ 2 and a polynomial P (z, y, u) with coefficients in Q, the algebraic closure of Q. Nishioka [8] (cf. Chapter 1.5 in [11]) solved this problem for polynomial transformations T . In [9] she extended her method to functions in several variables and suitable generalizations of the transfor- mation T z = z

.

Becker [1] generalized the result of Nishioka to algebraic transforma- tions T. T¨opfer gave in [15] a quantitative version of Becker’s result. In that article T¨opfer asked for a proof of the algebraic independence of the val- ues of several functions satisfying implicit functional equations at algebraic points.

In this paper we follow the proof of T¨opfer [15] and derive a lower bound for the transcendence degree of the values of functions f

, . . . , f

satisfy- ing a special system of implicit functional equations for the transformation T z = z

with an integer d ≥ 2. It should be easy to generalize the follow- ing result to polynomial or even rational or algebraic transformations T (cf.

Becker [1] and T¨opfer [14, 15]).

For the development of Mahler’s method in the last 15 years see the monograph of Nishioka [11] and the overview article of Waldschmidt [16] for further references.

2000 Mathematics Subject Classification: 11J82, 11J91.

[1]

Throughout the paper let K denote an algebraic number field and O

the ring of integers in K. As usual we denote by α the house of an algebraic number α, which is the maximum of the absolute values of the conjugates of α. A denominator of an algebraic number α is a positive integer D such that Dα ∈ O

. If P (z, y

, . . . , y

) =: P (z, y ) is a polynomial with complex coefficients, deg

P =: d

P denotes the partial degree of P with respect to z, deg

P =: d

P denotes the total degree in y := (y

, . . . , y

) and analogous notations in other cases. If the coefficients of P are algebraic, the height H(P ) of P is defined as the maximum of the houses of the coefficients of P , and the length L(P ) is the sum of the houses of the coefficients. In what follows let c, c

, c

, . . . and γ

, γ

, . . . denote positive constants which are independent of the parameters M, N, k, k

, k

used. For a vector µ ∈ C

we define |µ| := |µ

| + . . . + |µ

| and by N and N

we denote the positive and nonnegative integers.

Theorem 1. Let f

, . . . , f

be analytic in a neighborhood U of the ori- gin, algebraically independent over C(z) and suppose that the coefficients of their power series

f

(z) = X

f

z

(i = 1, . . . , m) belong to a fixed algebraic number field K and satisfy

f

≤ exp(c

(1 + j

)) and D

f

∈ O

for j ∈ N

and i = 1, . . . , m with suitable constants D ∈ N and L ≥ 1. Let n ∈ N

and β := n

· . . . · n

. Suppose that the functions f

, . . . , f

satisfy the functional equations

(2) a(z)f

(z

)

=

n

X

P

(z, f (z))f

(z

)

with polynomials a ∈ Q[z]\{0} and P

, . . . , P

∈ Q[z, y ] and an integer d satisfying d > max{β

, d

(P )}, where d

(P ) is defined by

d

(P ) := max{deg

(P

), . . . , deg

(P

)}.

Assume α ∈ Q

∩ U and a(α

) 6= 0 for all k ∈ N

. Let m

be the smallest integer satisfying

m

≥ m log d − L(m + 1) log β 1 +

log β + log d + L(m + 1) 1 +

+ m

(2 log β + log d

(P )) . Then

trdeg

Q(f

(α), . . . , f

(α)) ≥ m

.

As an application of this theorem we obtain easily the following

Corollary 2. Under the assumptions of Theorem 1, if α, f and the parameters d, β and d

(P ) satisfy for m > 1 the inequality

log d

(P )

log d < 1 −

2m

− m − 1 + L(m + 1) 1 +

(2m − 1)
(m − 1) L(m + 1) 1 +

+ m
,

then f

(α), . . . , f

(α) are algebraically independent.

Remarks. (i) Nishioka [8] proved the transcendence of f (α) under the condition d

> n

max{d, deg

(P )}, where f satisfies the functional equation (1) and n = deg

(P ).

Under the hypotheses of Theorem 1 we get the transcendence of f (α) only under the stronger condition d > max{n

, deg

(P )}. The reason for this is that we have to construct a sequence of polynomials (Q

)

, where the difference k

− k

has to be relatively large (cf. Lemma 8). In the simpler case m = 1 it suffices to find just one integer k to obtain a contradic- tion. By an improvement of the method of proof we get the transcendence of f (α) under the condition d > max{n

, deg

(P )}, which coincides with the condition of Nishioka in the case d > deg

(P ). Note that we have to assume d > d

(P ) only for technical reasons (cf. formula (24)).

(ii) T¨opfer proved in [15] a transcendence measure for f (α) under the condition d > n max{n, deg

(P )}.

(iii) For m ≥ 1 and β = 1 we get the result of Nishioka [10]. In [10]

one can also find a lot of applications. For other examples in this case, but d

(P ) = 1, see Chirski˘ı [2] and T¨opfer [14].

Our next example deals with infinite products of the form f

(z) :=

Y

(1 − z

)

, where d and n are positive integers with d ≥ 2.

Let 1 ≤ n

< . . . < n

(m ≥ 2). Then the functions f

are analytic for

|z| < 1 and satisfy the functional equations

f

(z) = (1 − z)f

(z

)

(i = 1, . . . , m).

Hence we have the following:

Corollary 3. Let 1 ≤ n

< . . . < n

be integers and β := n

· . . . · n

. If α is algebraic with 0 < |α| < 1 and d is an integer with

log d > (2m

− 1 + p

4m

− 2m

+ m) log β,

then the values

Y

(1 − α

)

, . . . , Y

(1 − α

)

are algebraically independent over Q. Under the corresponding conditions on α, d and n we get the algebraic independence of

Y

(1 − α

), Y

(1 − α

)

, . . . , Y

(1 − α

)

.

Remark. Nishioka proved (Theorem 3.4.13 in [11]) the algebraic inde- pendence of

Y

(1 − α

) (d = 2, 3, . . .) for any algebraic number α with 0 < |α| < 1.

P r o o f (of Corollary 3). The algebraic independence of the functions f

, . . . , f

over C(z) will be shown in the last section.

By the remark after Lemma 4, f

, . . . , f

satisfy the conditions for the houses and denominators of the coefficients in Theorem 1 for any L > 1.

Then the assumption of Corollary 3 follows immediately from Theorem 1 and Corollary 2.

2. Preliminaries and auxiliary results. For µ ∈ N

, µ ∈ N

and f

(z) := P

j=0

f

z

(i = 1, . . . , m) we define f

(z)

:=

X

f

z

, f

:= X

ν1,...,νµ∈N0

ν1+...+νµ=j

f

· . . . · f

, (3)

f (z)

:= f

(z)

· . . . · f

(z)

= X

f

z

,

f

:= X

ν₁,...,ν_m∈N₀ ν1+...+νm=j

f

· . . . · f

. (4)

Lemma 4. If f

≤ exp(c

(1 + j

)) and D

f

∈ O

for i = 1, . . . , m and all j ∈ N

with L ≥ 1 and D ∈ N, then for all µ ∈ N

and µ ∈ N

the following assertions hold:

(i) f

≤ exp(c

(µ + j

)), D

f

∈ O

,

(ii) f

≤ exp(c

(|µ| + j

)), D

f

∈ O

.

P r o o f. Assertions (i) and (ii) are consequences of the identities (3) and (4) using the fact that the number of ν ∈ N

with ν

+ . . . + ν

= j is bounded by

≤ 2

.

Remark. If the functions f

, . . . , f

satisfy functional equations of type P

(z, f

(z), f

(z

)) = 0 (i = 1, . . . , m)

with polynomials P

∈ Q[z, y, u]\{0} and deg

(P

) ≥ 1, we see that there exist an algebraic number field K, an explicit computable constant c > 0 and a positive integer D ∈ N such that for j ∈ N

and all ε > 0:

(i) f

∈ K,

(ii) f

≤ exp(c(1 + j

)), (iii) D

f

∈ O

hold, i.e. the conditions of Lemma 4 are fulfilled for all L > 1. For a proof of this remark see Lemma 1.5.3 of Nishioka [11] and Proposition 1 of Becker [1] for a more general result.

Lemma 5. For N ∈ N there exists a polynomial R ∈ O

[z, y ]\{0} with the following properties:

(i) deg

R ≤ N , deg

R ≤ N , (ii) log H(R) ≤ c

N

, (iii) ν := ord

R(z, f (z)) ≥ c

N

for suitable constants c

, c

∈ R

.

P r o o f. Put

R(z, y ) :=

X

X

|µ|≤N

r

z

y

with (N + 1)

unknown coefficients r

. Then R(z, f (z)) :=

X

X

|µ|≤N

r

z

f (z)

= X

β

z

(say) with (cf. the identity (4))

(5) β

=

min{h,N }

X

λ=0

X

|µ|≤N

r

f

.

Assertion (iii) is equivalent to the condition β

= 0 for 0 ≤ h < c

N

, and this yields at most [c

N

] + 1 equations in the

(N + 1)

 N + m m



≥ 1

m! N

> 2c

N

+ 1

unknowns r

for a suitable constant c

. After multiplication with a suit- able denominator D

according to Lemma 4 the coefficients f

are algebraic integers and their houses are bounded by exp(c

(N

)).

Siegel’s lemma (cf. Hilfssatz 31 in Schneider [12]) yields the assertion.

Lemma 6. Let ν be as in Lemma 5 and β

denote the Taylor coefficients of R(z, f (z)) as in the proof. Then

(i) |β

| ≤ exp(c

(h + N

)) ≤ exp(c

(h + ν

)).

(ii) |β

| ≥ exp(−c

ν

).

(iii) Suppose that k ∈ N satisfies d

≥ c

ν

with ν, N, L as above and a suitable constant c

∈ R

depending only on f and α. Then there exist constants c

, c

∈ R

depending only on f and α such that

−c

νd

≤ log |R(T

(α), f (T

(α)))| ≤ −c

νd

, where T

(α) denotes the kth iterate of T at the point α.

P r o o f. From (5) we get β

=

min{h,N }

X

λ=0

X

|µ|≤N

r

f

.

This representation together with Lemma 5 and the inequality |f

| ≤ exp(γ

(j + 1)) (notice that the functions f

, . . . , f

are analytic in a neigh- borhood of 0), hence |f

| ≤ exp(γ

(|µ| + h)) with γ

, γ

∈ R

, implies the first estimate of Lemma 6.

For D, L, c

as above and ν as in Lemma 5 we get (recall ν ≥ c

N

) D

β

∈ O

and

β

≤ exp(γ

(N

+ ν

+ N )) ≤ exp(γ

ν

).

By a Liouville estimate we obtain the second part.

We now come to the last part of Lemma 6. By Lemma 5 we write R(T

(α), f (T

(α))) = β

(T

(α))

 1 +

X

β

β

(T

(α))



and by the assumption on k and the first two parts of Lemma 6 we get  

  X

β

β

(T

(α))

    ≤

X

exp(c

(ν

+ h) + c

ν

− γ

hd

)

≤ X

exp(γ

ν

− γ

hd

) < 1

2 .

Now the assertion follows from |T

(α)|

= exp(−γ

νd

) and exp(−c

ν

) ≤

|β

| ≤ exp(2c

ν

).

Lemma 7. Let S, U

, . . . , U

∈ C satisfy S

+ U

S

+ . . . + U

= 0 and

−X

≤ log |S| ≤ −X

, log |U

| ≤ Y (1 ≤ i ≤ d) for X

, X

, Y ∈ R

. Then there exists j ∈ {1, . . . , d} such that

−dX

− Y − log d ≤ log |U

| ≤ −X

+ Y + log d.

P r o o f. This is Lemma 4.2.3 of Wass [17].

Remark. The examples S

+ U

= 0 and S

+ U

S

= 0 show that the bounds for |U

| cannot be improved.

The proof of Theorem 1 depends on the following result from elimination theory, which can be found in T¨opfer [13, Theorem 1] with slight modifica- tions.

Lemma 8. Suppose ω ∈ C

and K is an algebraic number field. Then there exists a constant c

= c

(ω, K) ∈ R

with the following prop- erty: If there exist increasing functions ψ

, ψ

, Λ : N → R

, real numbers Φ

≥ Φ

≥ c

, positive integers k

< k

, m

∈ {0, . . . , m} and polynomials (Q

)

∈ O

[ y ] such that the following assumptions are satisfied:

(i) 1 ≤ ψ

(k + 1)/ψ

(k) ≤ Λ(k) and ψ

(k) ≥ c

(log H(Q

) + deg

Q

) for k ∈ {k

, . . . , k

},

(ii) the polynomials (Q

)

satisfy, for k ∈ {k

, . . . , k

}, (a) deg

Q

≤ Φ

,

(b) log H(Q

) ≤ Φ

,

(c) −ψ

(k) ≤ log |Q

(ω)| ≤ −ψ

(k),

(iii) ψ

(k

) ≥ c

Λ(k

)

Φ

max{ψ

(k

), Φ

}, then trdeg

Q(ω) ≥ m

.

3. Construction of an auxiliary function. Since the case β = 1 (i.e.

n

= . . . = n

= 1) was treated by Nishioka [10] we can assume β > 1.

The proof is rather long, so we give a short sketch of the main steps.

In the first step we show how the powers of f (α) can be reduced by using

the functional equations. In the second step we consider R(T

(α), f (T

(α)))

for a polynomial R and construct by induction a polynomial R

, with de-

grees and height depending only on the degrees and height of R and on

d, β, d

(P ) and k, such that |R

(α, f (α))| has almost the same analytic

bounds as |R(T

(α), f (T

(α)))|. In the last step we use this polynomial R

to construct a suitable sequence of polynomials Q

∈ O

[ y ] satisfying the

assumptions of Lemma 8 and prove Theorem 1 by Lemma 8.

For a real number a we define a

:= max{a, 0} =

(a + |a|).

Let K be an algebraic number field containing α, the coefficients of f

, . . . , f

(cf. the assumption of Theorem 1 and Lemma 4) and the co- efficients of the polynomials a, P

, . . . , P

. Without loss of generality we can assume a ∈ O

[z] and P

, . . . , P

∈ O

[z, y ].

In what follows let k ∈ N be fixed. Under the conditions of Theorem 1 on α, d and f we put for abbreviation

τ

:= α

, ϕ

:= f

(α

) and ϕ

:= (f

(α

), . . . , f

(α

)).

For j = 1, . . . , m let P

:= a and we define the following notations:

d

(P ) := max{deg

(P

), . . . , deg

(P

)}, d

(P ) := max{deg

(P

), . . . , deg

(P

)},

L(P ) := max{L(P

), . . . , L(P

)}.

Lemma 9. Suppose that k ∈ N and λ ∈ N

. Then for all j = 1, . . . , m we have

(a(τ

)f

(τ

))

=

n

X

P

(τ

, ϕ

)(a(τ

)f

(τ

))

with polynomials P

∈ O

[z, y ] satisfying

d

(P

) ≤ (λ − i)

d

(P ), d

(P

) ≤ (λ − i)

d

(P ),

L(P

) ≤ 2

L(P )

.

P r o o f. For λ ∈ {0, . . . , n

− 1} we choose P

= δ

, where δ

is the Kronecker symbol, and the assertions are obvious.

Let now λ = n

+ l for l ∈ N

. We show the assertion by induction on l.

This is obvious for l = 0 because of (2) and (a(τ

)f

(τ

))

=

n

X

P

(τ

, ϕ

)a(τ

)

(a(τ

)f

(τ

))

, with P

j,j

(z, y ) := P

(z, y )a(z)

.

In the induction step the assertion follows from

(a(τ

)f

(τ

))

= (a(τ

)f

(τ

))

(a(τ

)f

(τ

))

=

n

X

P

j+l,j

(τ

, ϕ

)(a(τ

)f

(τ

))

=

n

X

P

(τ

, ϕ

)(a(τ

)f

(τ

))

+ P

j−1,nj+l,j

(τ

, ϕ

)(a(τ

)f

(τ

))

=

n

X

P

(τ

, ϕ

)(a(τ

)f

(τ

))

+ P

(τ

, ϕ

)

×

n

X

P

(τ

, ϕ

)(a(τ

)f

(τ

))

=

n

X

P

(τ

, ϕ

)(a(τ

)f

(τ

))

. So we get

P

(z, y ) := P

(z, y ) + P

(z, y )P

(z, y ), where P

j+l,j

(z, y ) := 0.

By induction it follows that P

j+l+1,j

∈ O

[z, y ] and d

(P

j+l+1,j

) ≤ (n

+ l + 1 − i)d

(P ), d

(P

j+l+1,j

) ≤ (n

+ l + 1 − i)d

(P ), L(P

j+l+1,j

) ≤ 2

L(P )

.

In the reduction step we replace R(τ

, ϕ

) =: R

(τ

, ϕ

) for an arbitrary polynomial R ∈ O

[z, y ] inductively by R

(τ

, ϕ

) and finally get a polynomial R

with almost the same bounds for |R

(α, f (α))|, the degrees and the height of R

as R

.

Lemma 10. Suppose k ∈ N and R ∈ O

[z, y ]. Then there exists a poly- nomial

R

(z, u, y ) := X

µ∈M

R

(z, u)y

∈ O

[z, u, y ] with M := {0, 1, . . . , n

− 1} × . . . × {0, 1, . . . , n

− 1} and

d

(R

) ≤ n

− 1 (j = 1, . . . , m), d

(R

) ≤ dd

(R) + d

(P )d

(R), d

(R

) ≤ d

(P )d

(R),

L(R

) ≤ L(R)L(P )

2

such that

a(τ

)

R(τ

, ϕ

) = R

(τ

, ϕ

, a(τ

)ϕ

).

P r o o f. From the representation R(z, y ) :=

d

X

X

|j|≤dy(R)

R

z

y

we get, by Lemma 9, a(τ

)

R(τ

, ϕ

) =

d

X

X

|j|≤dy(R)

R

τ

a(τ

)

(a(τ

)ϕ

)

= X

µ∈M

R

(τ

, ϕ

)(a(τ

)ϕ

)

, where

R

(z, u) :=

d

X

X

|j|≤dy(R)

R

z

a(z)

P

1,j₁,1

(z, u) · . . . · P

m,j_m,m

(z, u).

Now the bounds for the partial degrees d

are obvious. From Lemma 9 we get

d

(R

) ≤ dd

(R) + d

(P )d

(R) + d

(P ) max

n X

i=1

(j

− µ

)

− j

: |j| ≤ d

(R) o

≤ dd

(R) + d

(P )d

(R)

and similarly we derive the upper bound for d

. The length can be bounded in an analogous way by

L(R

) ≤ L(R)2

L(P )

≤ L(R)L(P )

2

.

Lemma 11. Suppose that R

∈ O

[z, u, y ] is the polynomial in Lemma 10. Then there exist polynomials U

, . . . , U

∈ O

[z, u] such that

R

+ U

R

+ . . . + U

= 0 at the point (z

, u

, y

) := (τ

, ϕ

, a(τ

)ϕ

) and

d

(U

) ≤ βdd

(R) + βd

(P )(d

(R) + |n|), d

(U

) ≤ βd

(P )(d

(R) + |n|),

L(U

) ≤ exp(c

(d

(R) + d

(R)))H(R)

.

P r o o f. With R

(z, u, y ) := P

µ∈M

R

(z, u)y

as in Lemma 10 we put for ν ∈ M ,

R

(τ

, ϕ

, a(τ

)ϕ

)(a(τ

)ϕ

)

= X

µ∈M

R

(τ

, ϕ

)(a(τ

)ϕ

)

= X

λ∈M

R

(τ

, ϕ

)(a(τ

)ϕ

)

, with (cf. Lemma 9)

R

(z, u) := X

µ∈M

R

(z, u)P

1,µ₁+ν₁,1

(z, u) · . . . · P

m,µ_m+ν_m,m

(z, u).

The degrees and length of R

can be bounded by Lemmas 9 and 10:

d

(R

) ≤ max

µ∈M

n

d

(R

) + X

d

(P

j,µj+νj,j

) o

≤ dd

(R) + d

(P )d

(R) + d

(P ) max

µ∈M

n X

j=1

(µ

+ ν

− λ

)

o

≤ dd

(R) + d

(P )(d

(R) + |n| + |ν| − |λ|).

Similarly

d

(R

) ≤ d

(P )(d

(R) + |n| + |ν| − |λ|),

L(R

) ≤ L(R)L(P )

2

≤ γ

L(R)γ

, where the constants γ

, γ

∈ R

depend only on P and n.

Thus the system of β linear equations with β unknowns, X

λ∈M

{R

(τ

, ϕ

) − δ

R

(τ

, ϕ

, a(τ

)ϕ

)}ω

= 0, where

δ

:=

n 1 if λ = ν, 0 else

is the generalized Kronecker symbol, has for ω := (ω

)

a nontrivial solution

ω

:= (a(τ

)ϕ

)

.

Hence the determinant of the matrix of coefficients must vanish at the point

(z

, u

, y

) := (τ

, ϕ

, a(τ

)ϕ

), and the expansion of the determinant

with respect to the powers of R

(τ

, ϕ

, a(τ

)ϕ

) implies

0 = det(R

− δ

R

)

= ±(R

+ U

R

+ . . . + U

) with polynomials U

∈ O

[z, u ].

Since the polynomials U

are sums of products of the form R

· . . . · R

,

where λ

, . . . , λ

∈ M are pairwise distinct and σ := (σ

, . . . , σ

) is a per- mutation of {0, . . . , n

− 1} × . . . × {0, . . . , n

− 1}, for l ∈ {1, . . . , β} we get

d

(U

) ≤ max

σ

n X

λ∈M

d

(R

) o

≤ βd

(P )(d

(R) + |n|)

because X

λ∈M

(|λ| − |σ(λ)|) = 0.

By analogy we obtain d

(U

) ≤ max

σ

n X

λ∈M

d

(R

) o

≤ βdd

(R) + βd

(P )(d

(R) + |n|).

The length of U

can be bounded by

L(U

) ≤ β! max{L(R

) : λ, ν ∈ M }

, with

L(R

) ≤ exp(c

(d

(R) + d

(R)))H(R).

Lemma 11 is proved.

Now the necessary tools for the reduction step from R

to R

are com- plete, and we prove for j = 0, . . . , k the existence of polynomials R

∈ O

[z, y ] such that for j = 0,

(6) d

(R

) := d

, d

(R

) := d

, log H(R

) := H

, exp(−ψ

(0)) ≤ |R

(τ

, ϕ

)| ≤ exp(−ψ

(0)), and for j ≥ 1:

d

(R

) =: d

≤ βd

(P )(d

+ |n|), (7)

d

(R

) =: d

≤ βdd

+ βd

(P )(d

+ |n|), (8)

log H(R

) =: H

≤ βH

+ c

(d

+ d

).

(9)

Here the constant c

> 0 depends only on f and α and (10) exp(−ψ

(j)) ≤ |R

(τ

, ϕ

)| ≤ exp(−ψ

(j)).

The functions ψ

, ψ

satisfy for j ≥ 1 the following recurrence equalities:

ψ

(j) := βψ

(j − 1) + βH

+ c

(d

+ d

d

) + log β, (11)

ψ

(j) := ψ

(j − 1) − βH

− c

(d

+ d

) − log β

(12)

provided that

(13) ψ

(0) ≥ c

β

(H

+ d

(d

+ d

)),

where c

, c

, c

∈ R

are suitable constants depending only on f and α.

The existence of the polynomials will be proved in the next section. First we will derive upper bounds for d

, d

, H

and ψ

(j) and a lower bound for ψ

(j).

Obviously (7) implies

d

≤ γ

(βd

(P ))

(d

+ |n|) ≤ c

(βd

(P ))

d

,

and for d

we get inductively (note that d > d

(P ) by the condition of Theorem 1)

d

≤ (βd)

d

+ βd

(P )

j−1

X

i=0

(βd)

(d

+ |n|) ≤ c

(βd)

(d

+ d

).

For H

, the logarithm of the height of R

, we get in a similar way H

≤ β

H

+ γ

j−1

X

i=0

β

(d

+ d

) ≤ β

H

+ c

(d

+ d

)(βd)

. Now we can easily deduce from (11) and the above estimates that

ψ

(k) = β

ψ

(0) (14)

+

k−1

X

i=0

β

{βH

+ c

(d

+ d

d

) + log β}

≤ β

ψ

(0) + kβ

H

+ c

(βd)

(d

+ d

).

In a similar way (cf. (13)) we can derive a lower bound for ψ

(k):

ψ

(k) = ψ

(0) −

k−1

X

i=0

{βH

+ c

(d

+ d

) + log β}

(15)

≥ ψ

(0) − c

β

(H

+ d

(d

+ d

)).

Now we prove by induction on j = 0, . . . , k the existence of a sequence of polynomials R

∈ O

[z, y ] satisfying the conditions (6)–(10). For j = 0, this is a consequence of Lemmas 5 and 6 with R

:= R and

(16) d

, d

≤ N, H

≤ c

N

, ψ

(0) := c

νd

, ψ

(0) := c

νd

provided that d

≥ c

ν

for a suitable constant c

> 0. Now suppose that

the assertions are true for j − 1 (j ∈ {1, . . . , k}). We apply Lemmas 10

and 11 with R replaced by R

. This yields the existence of polynomials

U

, . . . , U

∈ O

[z, u] with

d

(U

) ≤ βdd

+ βd

(P )(d

+ |n|), d

(U

) ≤ βd

(P )(d

+ |n|),

log H(U

) ≤ γ

(d

+ d

) + βH

for l = 1, . . . , β such that

R

+ U

R

+ . . . + U

= 0

for (z

, u

, y

) := (τ

, ϕ

, a(τ

)ϕ

). Here R

∈ O

[z, u, y ] is defined analogously to Lemma 10 by

a(τ

)

R

(τ

, ϕ

)

= R

(τ

, ϕ

, a(τ

)ϕ

).

The induction hypothesis together with the fact that −γ

d

≤ log |a(τ

)| ≤ γ

for k ∈ N

, implies

−ψ

(j − 1) − γ

d

d

≤ log |R

(τ

, ϕ

, a(τ

)ϕ

)|

≤ − ψ

(j − 1) + γ

d

.

For l = 1, . . . , β we obtain by a standard estimate together with Lemma 11,

|U

(τ

, ϕ

)| ≤ L(U

) max{1, |τ

|, |ϕ

|, . . . , |ϕ

|}

≤ exp(βH

+ γ

(d

+ d

)), where the constant γ

∈ R

depends only on f and α.

By (13) and (16) we see that

ψ

(j − 1) − (βH

+ γ

(d

+ d

) + log β) > 0 and by Lemma 7 we get the existence of l

∈ {1, . . . , β} such that

log |U

(τ

, ϕ

)| ≤ − ψ

(j − 1) + γ

d

+ βH

+ γ

(d

+ d

) + log β

≤ − ψ

(j − 1) + βH

+ c

(d

+ d

) + log β

= − ψ

(j) and

log |U

(τ

, ϕ

)|

≥ − βψ

(j − 1) − γ

βd

d

− βH

− γ

(d

+ d

) − log β

≥ − βψ

(j − 1) − βH

− c

(d

+ βd

d

) − log β

= − ψ

(j).

Thus we put R

(z, y ) := U

(z, y ) ∈ O

[z, y ] and see that (6)–(10) are proved for the polynomial R

.

4. Proof of Theorem 1. Now the necessary tools for the proof of Theorem 1 are complete. From the preceding section with j = k we know that for k, N ∈ N sufficiently large with

d

≥ c

ν

, (17)

νd

≥ c

β

(N

+ d

N ) (18)

for sufficiently large constants c

, c

> 0, there exist polynomials R

∈ O

[z, y ] with

d

(R

) ≤ c

(βd)

N, (19)

d

(R

) ≤ c

(βd

(P ))

N, (20)

log H(R

) ≤ c

(βd)

N, (21)

−c

(βd)

ν ≤ log |R(α, f (α))| ≤ −c

d

ν.

(22)

The estimates for the degrees (19) and (20) are obvious from (16) and the above estimates. The upper bound for the height (21) of R

and a lower bound for the right-hand side of (22) could be derived from (18) and (15).

With (14) and (16) it follows from (18) that

ψ

(k) ≤ γ

β

d

ν + γ

kβ

(N

+ d

N ) ≤ γ

β

d

ν + γ

kd

ν and this gives the left-hand inequality of (22); note that β ≥ 2.

In order to use Lemma 8 we define the polynomials (Q

)

∈ O

[ y ] by

Q

(y ) := D

R

(α, y ), where D ∈ N is a denominator of α.

Because of (18) and (19) and the condition d

(P ) < d we obtain, for k ∈ N,

d

(Q

) ≤ c

(βd

(P ))

N, log H(Q

) ≤ c

(βd)

N,

log |Q

( f (α))| ≤ − c

d

ν + c

(βd)

N ≤ −c

νd

, log |Q

( f (α))| ≥ − c

ν(βd)

.

Now for N ∈ N we define a number M ≥ N by ν := c

M

and for

positive integers k

≤ k ≤ k

, where k

< k

will be specified later, we