Download Simultaneous Approximation and Algebraic Independence

THE RAMANUJAN JOURNAL 1, 379–430 (1997)
c 1997 Kluwer Academic Publishers. Manufactured in The Netherlands.
°
Simultaneous Approximation
and Algebraic Independence
DAMIEN ROY∗
[email protected]
Départment de mathématiques, Université d’Ottawa, 585 King Edward, Ottawa, Ontario K1N 6N5, Canada
MICHEL WALDSCHMIDT
[email protected]
Institut de Mathématiques de Jussieu, Case 247, Problèmes Diophantiens, 4, Place Jussieu, 75252 Paris
Cedex 05, France
Received June 24, 1996; Revised March 3, 1997; Accepted June 23, 1997
Abstract. We establish several new measures of simultaneous algebraic approximations for families of complex
numbers (θ1 , . . . , θn ) related to the classical exponential and elliptic functions. These measures are completely
explicit in terms of the degree and height of the algebraic approximations. In some instances, they imply that the
field Q(θ1 , . . . , θn ) has transcendance degree ≥2 over Q. This approach which is ultimately based on the technique
of interpolation determinants provides an alternative to Gel’fond’s transcendence criterion. We also formulate
a conjecture about simultaneous algebraic approximation which would yield higher transcendance degrees from
these measures.
Key words: simultaneous approximation, transcendental numbers, algebraic independence, approximation measures, diophantine estimates, Liouville’s inequality, Dirichlet’s box principle, Wirsing’s theorem, Gel’fond’s
criterion, interpolation determinants, absolute logarithmic height, exponential function, logarithms of algebraic
numbers, Weierstraß elliptic functions, Gamma function
1991 Mathematics Subject Classification:
Primary—11J85; Secondary—11J82, 11J89, 11J91
Introduction
For a complex number θ, Dirichlet’s box principle provides a polynomial with rational
integer coefficients whose value at the point θ is “small”. If the number θ is algebraic, this
polynomial may be a multiple of the minimal polynomial of θ over Z. On the other hand, if
θ is transcendental, one gets an algebraic approximation to θ by taking a root of this polynomial. More generally, when θ1 , . . . , θn are complex numbers in a field of transcendence
degree 1 over Q, it is possible to construct sequences (γ1(N ) , . . . , γn(N ) ), (N ≥ 1) of simultaneous algebraic approximations to θ1 , . . . , θn . Therefore if we can prove for θ1 , . . . , θn
a sharp measure of simultaneous approximation, namely if we can bound from below the
quantity max1≤i≤n |θi − γi | in terms of the heights of γi and the degree of the number field
Q(γ1 , . . . , γn ), one deduces that at least two of the numbers θi are algebraically independent.
∗ Work
of this author partially supported by NSERC and CICMA.
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ROY AND WALDSCHMIDT
This argument enables one to deduce algebraic independence results from diophantine
estimates without using Gel’fond’s transcendence criterion. We develop this point of view
by considering a few
examples.
For instance we deduce Gel’fond’s result, which states that
√
√
3
3
the two numbers 2 2 and 2 4 are algebraically independent, from the following diophantine
estimate: if γ1 and γ2 are algebraic numbers with bounded absolute logarithmic height,
and if D denotes the degree of the number field Q(γ1 , γ2 ) over Q, then
|2
√
3
2
− γ1 | + |2
√
3
4
− γ2 | > exp{−C D 2 (log D)−1/2 },
where the constant C depends only on the heights of the numbers γ1 and γ2 . The main point
in this lower bound is that the function inside the exponent is bounded by o(D 2 ). In the first
section, we produce a criterion which yields the algebraic independence of two numbers,
provided that they satisfy a suitable measure of simultaneous approximation. In Section 2 we
consider values of the usual exponential function: for numbers of the shape e xi y j , we obtain
a measure of simultaneous approximation, assuming that the complex numbers xi as well as
y j satisfy a “technical” condition (namely a measure of linear independence). This technical
hypothesis cannot be omitted, as we show with an example involving Liouville numbers.
Also in Section 2 we give a diophantine estimate related to the Lindemann-Weierstraß
theorem: for Q-linearly independent algebraic numbers β1 , . . . , βn , there exists a positive
constant C = C(β1 , . . . , βn ) such that, if γ1 , . . . , γn are algebraic numbers satisfying
[Q(γ1 , . . . , γn ) : Q] ≤ D
and
max h(γ j ) ≤ h,
1≤i≤n
then
|eβ1 − γ1 | + · · · + |eβn − γn |
ª
©
≥ exp −C D 1+(1/n) h(log h + D log D)(log h + log D)−1 .
The best known measures of simultaneous approximation for pairs of numbers like (π, eπ ),
or (e, π ), are not yet sufficient to deduce algebraic independence. The same is true for
Q-linearly independent logarithms of algebraic numbers log α1 , . . . , log αn . However we
get a sufficiently sharp estimate which implies a result of algebraic independence if we
assume that there exists a nonzero homogeneous quadratic polynomial Q which vanishes at
the point (log α1 , . . . , log αn ): under this assumption, we prove that for algebraic numbers
γ1 , . . . , γn with logarithmic height ≤ log D in a field of degree ≤D over Q, we have
n
X
| log α j − γ j | ≥ e−C D ,
2
j=1
where C depends only on log α1 , . . . , log αn and Q.
The next Section (3) deals with elliptic functions. We replace the usual exponential
function exp(z) = e z by a Weierstraß elliptic function ℘. Finally in Section 4 we propose
a measure of simultaneous approximation for the two numbers π and 0(1/4) by algebraic
numbers of bounded absolute logarithmic height:
|π − γ1 | + |0(1/4) − γ2 | > exp{−C D 3/2 log D}.
The second part of this paper (Sections 5 to 11) includes the proofs.
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
381
Here we consider only complex numbers, but the same method yields diophantine approximation estimates as well as results of algebraic independence for p-adic fields also.
1.
Simultaneous approximation of complex numbers
a)
Algebraic approximation to a complex number
Let θ be a complex number. Using Dirichlet’s box principle, we deduce that for any integer
D ≥ 1 and any real number H ≥ 1, there exists a nonzero polynomial f ∈ Z[X ], of degree
≤D and usual height (maximum absolute value of its coefficients) ≤H , such that
√
| f (θ)| ≤ 2(1 + |θ | + · · · + |θ | D )H −(D−1)/2 .
We shall keep in mind only the weaker assertion: for any θ ∈ C, there exist three positive
constants D0 , H0 and c1 such that, for any D ≥ D0 and any H ≥ H0 , there exists a
polynomial f ∈ Z[X ], f 6= 0, of degree ≤D and usual height ≤H satisfying
| f (θ )| ≤ e−c1 D log H
(admissible values are D0 = 2, H0 = 185 max{1, |θ |}20 and c1 = 1/5).
On the other hand, if the number θ is algebraic, one deduces from Liouville’s inequality
that for any nonzero polynomial f ∈ Z[X ] of degree ≤D and usual height ≤H , either
f (θ) = 0, or else
| f (θ )| ≥ e−c2 (D+log H ) ,
with c2 = d + log H (θ), where d denotes the degree of θ and H (θ ) its usual height (which
is the usual height of its minimal polynomial over Z). Therefore, provided D and H are
sufficiently large, the polynomial which arises from the pigeonhole principle vanishes at θ
(assuming θ is algebraic).
Liouville’s inequality can be phrased in terms of diophantine approximation by algebraic
numbers: if θ is an algebraic number, there exists a constant c3 > 0 such that, for any
algebraic number γ of degree ≤D and usual height ≤H with γ 6= θ, the inequality
|θ − γ | ≥ e−c3 (D+log H )
holds.
Sometimes it is more convenient to use the absolute logarithmic height h(γ ) instead of
the usual height H (γ ): if the minimal polynomial of γ over Z is
a0 X d + a1 X d−1 + · · · + ad−1 X + ad = a0
d
Y
j=1
with d = [Q(γ ) : Q] and a0 > 0, Mahler’s measure of γ is
M(γ ) = a0
d
Y
j=1
max{1, |γ j |},
(X − γ j )
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and the absolute logarithmic height of γ is
h(γ ) =
1
log M(γ ).
d
In order to get an upper bound for h(γ ), we need an upper bound for M(γ ), and a lower
bound for d. This is the reason why we use different letters to denote the exact degree d of
γ and an upper bound D for the same degree. Using the estimates
√
2−d H (γ ) ≤ M(γ ) ≤ H (γ ) d + 1,
we deduce (cf. [32], Lemma 3.11)
1
1
1
log H (γ ) − log 2 ≤ h(γ ) ≤ log H (γ ) +
log(d + 1).
d
d
2d
Liouville’s inequality (see for instance [9], Lemma 9.2, and [32], Lemma 3.14) gives
|θ − γ | ≥ exp{−δ(h(γ ) + h(θ ) + log 2)}
with δ = [Q(γ , θ) : Q]. We set d = [Q(γ ) : Q] and d0 = [Q(θ ) : Q]. Hence we have
δ ≤ d0 d and
|θ − γ | ≥ exp{−c4 d(h(γ ) + 1)},
with c4 = d0 max{1, h(θ) + log 2}. Therefore we can choose c3 = (3/2)c4 . Finally, if we
define c5 = 2c4 , we conclude:
Liouville’s inequality. If θ is an algebraic number, there exists a positive constant c5
such that, for any rational integer D ≥ 1 and any real number h ≥ 1, if γ is an algebraic
number, distinct from θ, of degree ≤D and absolute logarithmic height h(γ ) ≤ h, then
|θ − γ | ≥ exp{−c5 Dh}.
We now consider the approximation of a transcendental complex number by algebraic
numbers.
Definition. Let θ be a complex number. A function ϕ : N × R+ → R+ ∪ {∞} is an
approximation measure for θ if there exist a positive integer D0 and a real number h 0 ≥ 1
such that, for any rational integer D ≥ D0 , any real number h ≥ h 0 and any algebraic
number γ satisfying
[Q(γ ) : Q] ≤ D
and
h(γ ) ≤ h,
the following inequality holds:
|θ − γ | ≥ exp{−ϕ(D, h)}.
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
383
This definition is slightly different from the corresponding one in [29], but is more
convenient for our present purpose. We allow the value ∞ for ϕ so that algebraic numbers are
not excluded. The condition D ≥ D0 is unimportant (algebraic numbers of small degree are
not excluded). However, by assuming h ≥ h 0 , we do not take into account some refinements
which may occur when one considers the approximation of complex numbers by algebraic
numbers of small absolute height (for instance by algebraic numbers of bounded Mahler’s
measure).
Examples. Here are a few approximation measures which are known for various transcendental numbers.
• The following approximation measure for π is due to N.I. Fel’dman (Theorem 5.7 of [9],
Chap. 7, p. 120). An explicit value for the constant C is produced in [29], Theorem 3.1
and [18], Theorem 2.
There exists an absolute constant C such that the function
C D 2 (h + log D) log D
is an approximation measure for the number π.
• Let α be a nonzero algebraic number and log α a nonzero determination of its logarithm.
Again N.I. Fel’dman (Theorem 8.7 of [9], Chap. 7, p. 135) proved an approximation
measure for log α. An explicit value for C = C(log α) is given in [29], Theorem 3.6 and
[18], Theorem 5b.
There exists a constant C = C(log α) such that the function
C D 3 (h + log D)(log D)−1
is an approximation measure for log α.
• Let β be a nonzero algebraic number. The best known approximation measure for eβ is
due to G. Diaz [8], Cor. 2 (a refinement of the explicit constant C = C(β) is given in
[18], Theorem 5a):
There exists a constant C = C(β) such that the function
C D 2 h(log h + D log D)(log h + log D)−1
is an approximation measure for the number eβ .
Other approximation measures are given in [29] and in [6] (see especially [6], Theorem 2.4,
p. 41, Theorems 2.5, 2.7 and 2.8, p. 45 and Theorem 2.10, p. 47, for the numbers e, eπ
and α β ).
A natural question is to ask what is the best possible approximation measure for a
transcendental number? One expects that a function whose growth rate is slower than D 2 h
cannot be an approximation measure (compare with conjecture 1.7 below). In terms of the
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ROY AND WALDSCHMIDT
usual height, this limit corresponds to the function D log H . The first precise statement in
this direction is due to Wirsing ([34], inequality (4’)):
Let θ be a transcendental complex number. For any rational integer D ≥ 2 there exist
infinitely many algebraic numbers γ satisfying
[Q(γ ) : Q] ≤ D
and
|θ − γ | ≤ M(γ )−D/4 .
Here is a variant of Wirsing’s result which is proved in [21], Théorème 3.2.
Theorem 1.1. Let θ be a transcendental number and ϕ an approximation measure for θ .
Then, for sufficiently large h,
lim sup
D→∞
1
ϕ(D, h) ≥ 10−7 h.
D2
Here is the main idea of the proof for Theorem 1.1 as well as for Wirsing’s theorem. We
start from a polynomial produced by Dirichlet’s pigeonhole principle and we select a root
at minimal distance from θ. The arguments are similar to those which occur in the proof of
Gel’fond’s criterion ([11], Chap. III, Section 4, Lemma VII, p. 148). In fact, this criterion
of Gel’fond has been formulated by Brownawell in terms of algebraic approximation to a
complex number ([5], Theorem 2).
b)
Simultaneous approximation of several complex numbers
In the previous subsection we considered the approximation to a single complex number θ .
Now we consider simultaneous approximation to numbers θ1 , . . . , θn . We first extend the
definition of approximation measure to the case of several numbers.
Definition. Let θ1 , . . . , θn be complex numbers. A function ϕ : N × R+ → R+ ∪ {∞}
is a measure of simultaneous approximation for θ1 , . . . , θn if there exists a positive integer
D0 together with a real number h 0 ≥ 1 such that, for any integer D ≥ D0 , any real number
h ≥ h 0 and any n-tuple (γ1 , . . . , γn ) of algebraic numbers satisfying
[Q(γ1 , . . . , γn ) : Q] ≤ D
and
max h(γi ) ≤ h,
1≤i≤n
we have
max |θi − γi | ≥ exp{−ϕ(D, h)}.
1≤i≤n
There exists a finite measure of simultaneous approximation for the numbers θ1 , . . . , θn
provided they are not all algebraic.
An alternative definition consists of replacing max1≤i≤n h(γi ) ≤ h by an upper bound
for the height of the projective point (1 : γ1 : · · · : γn ) (see for instance [32], Chap. 3,
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
385
Section 2). However, for our present purpose, this does not make a difference, since
max h(γi ) ≤ h(1 : γ1 : · · · : γn ) ≤ h(γ1 ) + · · · + h(γn ).
1≤i≤n
It makes a difference only when sharp estimates for the constants are considered, like in the
work of Schmidt.
Examples. Measures of simultaneous approximation are given in [9], [16], [22] and [23].
We shall see several other examples. To begin with, here is a result due to N.I. Fel’dman
([9], Theorem 7.7, p. 128).
Let α1 , . . . , αn be nonzero algebraic numbers. For 1 ≤ i ≤ n, let log αi be a determination of the logarithm of αi . Assume the numbers log α1 , . . . , log αn are Q-linearly
independent. Then there exists a positive constant C such that
C D 2+1/n (h + log D)(log D)−1
is a measure of simultaneous approximation for the numbers log α1 , . . . , log αn .
For n = 1 we recover Fel’dman’s above mentioned approximation measure for the number
log α.
We shall deduce from Theorem 1.1 the following corollary.
Corollary 1.2. Let θ1 , . . . , θn be complex numbers and ϕ : N × R+ → R+ a measure of
simultaneous approximation for θ1 , . . . , θn . Assume
lim inf
h→∞
1
1
lim sup
ϕ(D, h) = 0.
h D→∞ D 2
Then the field Q(θ1 , . . . , θn ) has transcendence degree ≥2 over Q.
In the next Section (Section 2 below) we shall show by an example that this sufficient
condition is not necessary: there exist fields of transcendence degree 2 which are generated by complex numbers θ1 , . . . , θn which are simultaneously very well-approximated by
algebraic numbers.
c)
Specialization lemma
We deduce Corollary 1.2 from Theorem 1.1. It is plainly sufficient to prove the following
result.
Proposition 1.3. Let n and m be positive integers with 1 ≤ n ≤ m and let θ1 , . . . , θm
be complex numbers. Assume that θn+1 , . . . , θm are algebraic over the field Q(θ1 , . . . , θn ).
Under these assumptions, there exist three positive constants c0 ∈ R, c1 ∈ N and c2 ∈ R,
which depend only on θ1 , . . . , θm , such that, if ϕ(D, h) is a simultaneous approximation
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ROY AND WALDSCHMIDT
measure for these m numbers, then c0 + ϕ(c1 D, c2 h) is a simultaneous approximation
measure for θ1 , . . . , θn .
Roughly speaking, Proposition 1.3 means that, up to constants, a simultaneous approximation measure depends only on a transcendence basis of the field Q(θ1 , . . . , θn ).
The proof of Proposition 1.3 will rest on several preliminary lemmas.
Lemma 1.4. Let
f (T ) = a0 T d + · · · + ad = a0 (T − ζ1 ) · · · (T − ζd )
be a polynomial with complex coefficients and degree d ≥ 1. Assume f is separable (i.e.,
has no multiple root). There exist two positive constants c = c( f ) and η = η( f ) such that,
if ã0 , . . . , ãd are complex numbers satisfying
max |ai − ãi | < η,
0≤i≤d
then the polynomial
f˜(T ) = ã0 T d + · · · + ãd
can be written
f˜(T ) = ã0 (T − ζ̃1 ) · · · (T − ζ̃d )
with
max |ζ j − ζ̃ j | ≤ c max |ai − ãi |.
1≤ j≤d
0≤i≤d
Proof: (Compare with [10], Chap. I, Lemma 8.7 and Corollary 8.8). We shall obtain
explicit values for η and c. Set
r=
1
min |ζi − ζ j |,
2 i6= j
r0 = max{|ζ1 |, . . . , |ζd |},
R = max{1, r + r0 }
and define
η=
|a0 |r d
,
(d + 1)R d
c = r/η.
For |z| ≤ R, we have
| f (z) − f˜(z)| ≤ max |ai − ãi |(1 + R + · · · + R d ) < |a0 |r d .
0≤i≤d
For |z − ζi | = r , we have
| f (z)| = |a0 |
d
Y
j=1
|z − ζ j | ≥ |a0 |r d .
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
387
According to Rouché’s theorem, since the estimate | f (z) − f˜(z)| < | f (z)| holds for z
on the circumference of the disk |z − ζi | ≤ r , the two functions f and f˜ have the same
number of zeroes in this open disk. Hence f˜ has a single zero ζ̃i in the same disk. For
i 6= j, according to the definition of r , we also have |ζ̃ j − ζi | ≥ r . Therefore
| f (ζ̃ j )| = |a0 |
d
Y
|ζ̃ j − ζi | ≥ |a0 |r d−1 |ζ̃ j − ζ j |.
i=1
On the other hand
| f (ζ̃ j )| = | f (ζ̃ j ) − f˜(ζ̃ j )| ≤ max |ai − ãi |(d + 1)R d .
0≤i≤d
2
This completes the proof of Lemma 1.4.
Lemma 1.5. Let P ∈ Z[Y1 , . . . , Ym ] be a polynomial in m variables of degree D j in Y j ,
(1 ≤ j ≤ m). Let γ1 , . . . , γm be algebraic numbers. Then
h(P(γ1 , . . . , γm )) ≤ log L(P) +
m
X
D j h(γ j ).
j=1
2
Proof: See for instance [32], Lemma 3.6.
Lemma 1.6. Let F ∈ Z[X 1 , . . . , X n , T ] be a polynomial in n + 1 variables of degree
D j in X j , (1 ≤ j ≤ n), and let α1 , . . . , αn , β be algebraic numbers which satisfy
F(α1 , . . . , αn , β) = 0. Assume that F(α1 , . . . , αn , T ) ∈ Q(α1 , . . . , αn )[T ] is not the
zero polynomial. Then
h(β) ≤ 2 log L(F) + 2
n
X
D j h(α j ).
j=1
Proof: (We are thankful to Guy Diaz who kindly provided us with the following proof).
Let t be the degree of F in the variable T . Write α for (α1 , . . . , αn ), X for (X 1 , . . . , X n ),
and write
F( X, T ) = T t Q t ( X ) + T t−1 Q t−1 ( X ) + · · · + Q 0 ( X ).
Since F(α, T ) ∈ Q(α)[T ] is not the zero polynomial and since F(α, β) vanishes, at least
one of the numbers Q t (α ), . . . , Q 1 (α) does not vanish. Denote by m the largest index j,
(1 ≤ j ≤ m) such that Q j (α) 6= 0. From F(α, β) = 0 we deduce
−β m Q m (α) = β m−1 Q m−1 (α ) + · · · + β Q 1 (α ) + Q 0 (α ).
Define
Q̃( X, T ) = T m−1 Q m−1 ( X ) + · · · + T Q 1 ( X ) + Q 0 ( X ),
so that
−β m Q m (α ) = Q̃(α, β).
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An upper bound for h(β m ) is
h(β m ) = h(β m Q m (α)Q m (α)−1 ) ≤ h(β m Q m (α )) + h(Q m (α )) = h( Q̃(α, β)) + h(Q m (α )).
Using Lemma 1.5, we bound h( Q̃(α, β)) and h(Q m (α )) from above:
h(Q m (α)) ≤ log L(Q m ) +
n
X
¡
¢
deg X j Q m h(α j )
j=1
and
h( Q̃(α, β)) ≤ log L( Q̃) +
n
X
¡
¢
deg X j Q̃ h(α j ) + (degT Q̃)h(β).
j=1
The degrees deg X j Q m and deg X j Q̃ are bounded by deg X j F = D j . Also we have degT Q̃ ≤
m − 1. Coming back to h(β m ), we deduce
mh(β) = h(β m ) ≤ log L(Q m ) + log L( Q̃) + 2
n
X
D j h(α j ) + (m − 1)h(β).
j=1
Hence
h(β) ≤ log L(Q m ) + log L( Q̃) + 2
n
X
D j h(α j ).
j=1
Since L(Q m ) + L( Q̃) ≤ L(F), we see that log L(Q m ) + log L( Q̃) is bounded by 2 log L(F),
which yields the desired estimate.
2
Remark. The proof of Lemma 5 in [2] (which deals with the case n = 1) yields, under
the assumptions of Lemma 1.6, the following upper bound for Mahler’s measure of β:
M(β) ≤ L(F)d
n
Y
M(α j ) D j ,
j=1
where d = [Q(α1 , . . . , αn ) : Q]. The advantage of Lemma 1.6 is to produce an upper
bound for h(β) which does not depend on d.
Proof of Proposition 1.3: By induction, it is sufficient to deal with the case m = n + 1.
Write θ for (θ1 , . . . , θn ). Let F ∈ Z[X 1 , . . . , X n , T ] be a nonzero polynomial of degree d
in T such that f (T ) = F(θ, T ) ∈ C[T ] is separable, of degree d, and has θn+1 as a root.
Denote by η and c the constants related to f by Lemma 1.4. Write
F(X 1 , . . . , X n , T ) =
d
X
j=0
a j (X 1 , . . . , X n )T d− j .
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
389
There exists a positive constant c3 such that, for any θ 0 = (θ10 , . . . , θn0 ) ∈ Cn satisfying
max1≤i≤n |θi − θi0 | ≤ 1, we have
max |a j (θ ) − a j (θ 0 )| ≤ c3 max |θi − θi0 |.
0≤ j≤d
1≤i≤n
Since a0 (θ ) 6= 0, there exists a constant c4 > 0 such that a0 (θ 0 ) 6= 0 for max1≤i≤n |θi −θi0 | ≤
c4 .
Let γ1 , . . . , γn be algebraic numbers, D a positive integer and h ≥ 1 a real number such
that
max h(γi ) ≤ h,
1≤i≤n
[Q(γ1 , . . . , γn ) : Q] ≤ D
and
max |θi − γi | < min{1, η/c3 , c4 }.
1≤i≤n
Since a0 (γ1 , . . . , γn ) 6= 0, we have F(γ1 , . . . , γn , T ) 6= 0. Using Lemma 1.4 we see that
the polynomial F(γ1 , . . . , γn , T ) has a root γn+1 which satisfies
|θn+1 − γn+1 | ≤ c5 max |θi − γi |
1≤i≤n
with c5 = max{1, cc3 }. Since
[Q(γ1 , . . . , γn , γn+1 ) : Q(γ1 , . . . , γn )] ≤ d,
we have [Q(γ1 , . . . , γn , γn+1 ) : Q] ≤ d D. Using Lemma 1.6 we bound h(γn+1 ) by c2 h
with
c2 = 2 log L(F) + 2
n
X
deg X j F.
j=1
By assumption, for sufficiently large D and h, we have
max |θi − γi | ≥ exp{−ϕ(d D, c2 h)}.
1≤i≤n+1
Hence
max |θi − γi | ≥ c5−1 exp{−ϕ(d D, c2 h)},
1≤i≤n
which gives the desired estimate with c1 = d and c0 = log c5 .
d)
2
Large transcendence degrees
In the present paper we consider only “small transcendence degrees”. However we are
tempted to propose the following conjecture.
Conjecture 1.7. Let a ≥ 1, b ≥ 1 be real numbers and θ1 , . . . , θn be complex numbers.
Denote by t the transcendence degree over Q of the field Q(θ1 , . . . , θn ). Let ϕ be a simultaneous approximation measure for these n numbers. Let (Dν )ν≥1 be a non-decreasing
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ROY AND WALDSCHMIDT
sequence of positive integers and (h ν )ν≥1 be a non-decreasing sequence of positive real
numbers with Dν + h ν → ∞. Assume
Dν+1 ≤ a Dν
and
h ν+1 ≤ bh ν ,
(ν ≥ 1).
Then
lim sup
ν→∞
1
1+(1/t)
Dν
hν
ϕ(Dν , h ν ) > 0.
According to Proposition 1.3, it would be sufficient to establish this result under the
assumption that θ1 , . . . , θn are algebraically independent (that is n = t).
2.
Usual exponential function in a single variable
a)
The numbers a β
j
In 1949 Gel’fond (see [11], Chap. III, Section 4) established the algebraic independence of
2
the two numbers α β and α β when α is a nonzero algebraic number (with a determination
log α 6= 0 of its logarithm, giving rise to α β = exp(β log α)), and β is a cubic irrational
algebraic number. Here, we deduce this algebraic independence result from a simultaneous
2
approximation estimate for the two numbers α β and α β . Like Gel’fond, we consider the
more general situation where β is algebraic of degree d ≥ 2.
Theorem 2.1. Let a be a nonzero complex number and β an algebraic number of degree
d ≥ 2. Choose a nonzero determination log a for the logarithm of a. There exists a positive
constant C such that
C D (d+1)/(d−1) h d/(d−1) (log D + log h)−1/(d−1)
is a simultaneous approximation measure for a, a β , . . . , a β
d−1
.
Notice that for each h ≥ h 0 , there exists a positive constant C(h) such that this approximation measure is bounded by C(h)D (d+1)/(d−1) (log D)−1/(d−1) . If d ≥ 3, this function is
o(D 2 ). From Corollary 1.2 we deduce:
Corollary 2.2. Let a be a nonzero complex number, log a a nonzero determination of its
logarithm and β an algebraic number of degree d ≥ 3. Then at least two of the d numbers
2
d−1
a, a β , a β , . . . , a β are algebraically independent. In particular, if a is algebraic and β
2
is cubic irrational, then the two numbers a β and a β are algebraically independent.
For d = 2 and a = α algebraic, Theorem 2.1 gives the following approximation measure
for α β when β is a quadratic irrational number:
C D 3 h 2 (log D + log h)−1 .
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
b)
391
An effective version of the six exponentials theorem
The six exponentials theorem, due to Lang and Ramachandra (see for instance [1], Theorem 12.3, or [28], Cor. 2.2.3) states that
if x 1 , . . . , xd are Q-linearly independent and y1 , . . . , y` are also Q-linearly independent,
with d` > d + `, then at least one of the d` numbers exp(xi y j ), (1 ≤ i ≤ d, 1 ≤ j ≤ `)
is transcendental.
We give a simultaneous approximation measure for these d` numbers, assuming an extra
“technical hypothesis”. Next we show that such a hypothesis cannot be omitted.
Definition. Let n be a positive integer, ν a positive real number and x1 , . . . , xn complex
numbers. We say that x1 , . . . , xn satisfy a measure of linear independence with exponent
ν if there exists a positive integer T0 satisfying the following property: for any n-tuple
(t1 , . . . , tn ) ∈ Zn and any real number T ≥ T0 with
0 < max{|t1 |, . . . , |tn |} ≤ T,
we have
|t1 x1 + · · · + tn xn | ≥ exp(−T ν ).
According to this definition, if the numbers x1 , . . . , xn satisfy a measure of linear independence, then they are linearly independent over Q.
Theorem 2.3. Let d and ` be positive integers satisfying d` > d + `. Set
κ=
d`
.
d` − d − `
Let x 1 , . . . , xd be complex numbers which satisfy a measure of linear independence with
exponent 1/(3d), and also let y1 , . . . , y` be complex numbers which satisfy a measure of
linear independence with exponent 1/(3`). Then, there exists a positive constant C such
that
C(Dh)κ (log D + log h)1−κ
is a simultaneous approximation measure for the d` numbers e xi y j , (1 ≤ i ≤ d, 1 ≤ j ≤ `).
Remark. Earlier estimates of this type were known ([16, 22, 23]), but with a weaker
dependence on D. Because of that, they are not sufficient to prove results of algebraic
independence.
Assuming d` ≥ 2(d + `), for fixed h the simultaneous approximation measure is bounded
by o(D 2 ). Therefore, in this case, at least two of the d` numbers exp(xi y j ), (1 ≤ i ≤ d,
1 ≤ j ≤ `) are algebraically independent. We remark that this algebraic independence
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ROY AND WALDSCHMIDT
result is known, given only that x1 , . . . , xd on one side, y1 , . . . , y` on the other, are linearly
independent over Q. Here we need to add technical hypotheses (measures of linear independence). In fact, these conditions are similar to those which occur in the known results for
large transcendence degrees [6, 31] (see also the comment after the proof of Theorem 2.4).
However, for small transcendence degrees, these extra conditions (which Gel’fond needed
in his original work—cf. [11], Chap. III, Section 4, Theorems I, II and III), have been shown
to be unnecessary by Tijdeman [24] (see also [28], Chap. 7). The underlying method of the
present work also enables us to avoid these assumptions for small transcendence degrees
[21]. But in order to do so, we need to bypass Theorem 2.3.
The proof of Theorem 2.3 relies on the following statement which does not require a
diophantine hypothesis.
Denote by Im(z) the imaginary part of a complex number z.
Theorem 2.4. Let d, ` and κ be as in Theorem 2.3. There exists a positive constant C
which satisfies the following property. Let λi j , (1 ≤ i ≤ d, 1 ≤ j ≤ `) be complex numbers
whose exponentials γi j = eλi j are algebraic. Assume that the d numbers λ11 , . . . , λd1
are linearly independent over Q, and also that the ` numbers λ11 , . . . , λ1` are linearly
independent over Q. Let D be the degree of the number field generated over Q by the d`
numbers γi j , (1 ≤ i ≤ d, 1 ≤ j ≤ `), and let h ≥ 3, E ≥ e, F ≥ 1 be real numbers
satisfying
max h(γi j ) ≤ h,
1≤i≤d
1≤ j≤`
max |λi j | ≤ Dh/E
1≤i≤d
1≤ j≤`
and
F = 1 + max |Im(λi j )|.
1≤i≤d
1≤ j≤`
Then, we have
max |λi j λ11 − λi1 λ1 j | ≥ exp{−C(Dh)κ (log E)1−κ F κ/m },
1≤i≤d
1≤ j≤`
where m = max{d, `} − 1.
The conclusion is a lower bound for at least one of the 2 × 2 minors of the matrix
(λi j )1≤i≤d,1≤ j≤` . Extensions of this result to minors of larger size can also be produced—
the corresponding algebraic independence statements are given in [21].
Remark. One can prove variants of Theorems 2.3 and 2.4 which contain Gel’fond’s algebraic independence results concerning the numbers xi , e xi y j , (resp. xi , y j , e xi y j )—see [11],
Chap. III, Section 4, Theorems I, II and III. More generally, one can prove simultaneous
approximation measures which contain the results of algebraic independence obtained by
Chudnovski using Baker’s method in Chapter 3 of [6]. By the way, it is necessary to add
the assumption r2 > 0 in Theorem 3.1, p. 136 of [6]: it was not known at that time that
the two numbers π and eπ are algebraically independent; and it is still unknown that, for
β a quadratic irrational number and λ a nonzero logarithm of an algebraic number, the two
numbers λ and eβλ are algebraically independent.
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
c)
393
Liouville numbers
We show that the conclusion of Theorem 2.3 may fail if we omit the hypotheses concerning
measures of linear independence.
Let φ be a strictly increasing function N → N and let (u n )n≥0 be a bounded sequence of
rational integers: |u n | ≤ c1 for all n ≥ 0. Assume u n 6= 0 for an infinite set of n (hence
c1 ≥ 1). Consider the number
ξ=
X un
.
2φ(n)
n≥0
For N ≥ 0, define
q N = 2φ(N ) ,
pN =
N
X
u n 2φ(N )−φ(n) .
n=0
Then ( p N , q N ) ∈ Z2 , q N > 0 and
¯
¯ ¯¯ X
¯
¯
X
¯ ¯
¯
2c1
p
N
¯
−φ(n)
¯=¯
¯ξ −
u
2
2−φ(N +1)−n+N +1 ≤ φ(N +1) .
¯ ≤ c1
n
¯
¯
¯
¯n>N
qN
2
n≥N +1
From the upper bound
|ea − eb | ≤ |a − b| max{ea , eb }
for real numbers a and b,
we deduce
|2ξ − 2 p N /q N | <
c2
φ(N
2 +1)
with a constant c2 = 21+2c1 c1 independent of N . For a/b ∈ Q, we have h(2a/b ) =
(a/b) log 2, hence the absolute logarithmic height of the algebraic number 2 p N /q N is bounded
independently of N by
h(2 p N /q N ) ≤ 2c1 log 2,
while its degree is ≤q N .
(d)
)n≥0 , (1 ≤ i ≤ d) by
Let d be a positive integer. Define d sequences (u in
½
if n ≡ i (mod d),
otherwise,
(1 ≤ i ≤ d)
X u (d)
X
1
in
=
,
φ(n)
φ(qd+i)
2
2
n≥1
q≥0
(1 ≤ i ≤ d).
(d)
=
u in
1
0
and set
ξid =
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ROY AND WALDSCHMIDT
Define, for N ≥ 1,
pi(d)
N =
N
X
(d) φ(N )−φ(n)
u in
2
.
n=1
We get
¯
¯
¯
p (d) ¯
2
max ¯¯ξid − i N ¯¯ < φ(N +1) .
1≤i≤d
qN
2
Moreover, assume φ(n + 1) − φ(n) → ∞ as n → ∞. Then, for any tuple (t1 , . . . , td ) of
rational integers, not all of which are zero, the number t1 ξ1d + · · · + td ξdd has a lacunary
2-adic expansion. Hence it is irrational, and therefore does not vanish (in fact if the function φ grows sufficiently fast, this number is transcendental). It follows that the numbers
ξ1d , . . . , ξdd are linearly independent over Q.
Define x1 , . . . , xd by xi = ξid , (1 ≤ i ≤ d) and define y1 , . . . , y` by y j = ξ j` log 2 for
1 ≤ j ≤ `. The transcendence degree t over Q of the field generated by the d` numbers
e xi y j satisfies t ≥ 1 provided d` > d + ` and satisfies t ≥ 2 provided d` ≥ 2(d + `) (cf.
(`)
ai N b j N /q N2
. We get
[4] and [28], Chap. 7). Define further ai N = pi(d)
N , b j N = p j N and γi j = 2
¯
¯
¯
¯
bjN
2 log 2
¯
max y j −
log 2¯¯ ≤ φ(N +1)
1≤ j≤` ¯
qN
2
¯
¯
¯
ai N ¯¯
2
¯
,
≤
max xi −
1≤i≤d ¯
q N ¯ 2φ(N +1)
and
max |e xi y j − γi j | ≤
1≤i≤d
1≤ j≤`
8 log 2
.
2φ(N +1)
The absolute logarithmic height of the numbers γi j is bounded independently of N :
h(γi j ) ≤ log 2.
The field generated over Q by the d` numbers γi j has degree ≤q N2d` = 4d`φ(N ) over Q. If
ϕ(D, h) is a simultaneous approximation measure for the d` numbers e xi y j , then for any
h ≥ max{h 0 , log 2} and any D ≥ max{D0 , 4d`φ(N ) }, we have
ϕ(D, h) ≥ φ(N + 1) log 2 − 2.
We can choose for φ a function such that
lim sup
N →∞
φ(N + 1)
= ∞.
24d`φ(N )
In this case lim sup N →∞ D −2 ϕ(D, h) = ∞, hence the hypothesis of Corollary 1.2 is not
satisfied. Therefore we cannot deduce from Corollary 1.2 the algebraic independence of at
least two of these numbers.
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
395
The argument used for lifting the obstructing subgroup in [21] shows that the underlying
method to the present work yields not only simultaneous approximation measures, but also
results of algebraic independence without technical hypothesis.
Remark.
Consider the number
θ=
X 1
,
2φ(n)
n≥0
where φ : N → N is a strictly increasing function, such that
φ(n + 1) ≥ n 2 φ(n)
for all n ≥ 0. Define two sequences (Dν )ν≥1 and (h ν )ν≥1 by
Dν = ν
and
h ν = ν −1/2 φ(ν) (ν ≥ 1).
Then Dν + h ν → ∞ for ν → ∞. However, for any positive real number C and for any
sufficiently large ν, there is no algebraic number γ of degree ≤Dν and height ≤h ν which
satisfies
ª
©
|θ − γ | ≤ exp −C Dν2 h ν .
We prove this claim by contradiction. Assume such a γ exists for some value of ν with
ν ≥ max{16, 2C −2 }. We compare γ to the rational number
α=
ν
X
1
.
φ(n)
2
n=1
Since h(α) = φ(ν) log 2 > h ν , we have γ 6= α. From Liouville’s inequality we deduce
log |γ − α| ≥ −Dν (log 2 + φ(ν) log 2 + h ν ) ≥ −νφ(ν).
This is not possible, according to the following computation:
log |γ − α| ≤ log 2 + log max{|θ − γ |, |θ − α|}
≤ log 2 + max{−Cν 3/2 φ(ν), −φ(ν + 1) log 2 + log 2}
< −νφ(ν).
This example shows that the condition h ν+1 ≤ bh ν in Conjecture 1.7 cannot be omitted.
d)
Schanuel’s conjecture
Let x1 , . . . , xn be Q-linearly independent complex numbers. Schanuel’s conjecture (see
[13], Chap. 3, Historical Note) states that the transcendence degree over Q of the field
Q(x1 , . . . , xn , e x1 , . . . , e xn ) is ≥ n. Here we produce a simultaneous approximation measure for the 2n numbers x1 , . . . , xn , e x1 , . . . , e xn . If we select xi = ξin , (1 ≤ i ≤ n), with
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ROY AND WALDSCHMIDT
the previously defined numbers ξin , we see that the hypothesis of linear independence for
the numbers xi is not sufficient to get any estimate at all. This is why we assume a measure
of linear independence.
Theorem 2.5. Let x1 , . . . , xn be complex numbers which satisfy a measure of linear
independence with exponent 2n + 1. There exists a positive constant C = C(n) such that
the function
C D 2+1/n h(h + log D)(log h + log D)−1
is a simultaneous approximation measure for the 2n numbers x1 , . . . , xn , e x1 , . . . , e xn .
We shall also prove a variant of this statement, where no technical hypothesis is needed:
we produce a lower bound for
n
X
|βi − log αi |
i=1
when log α1 , . . . , log αn are logarithms of algebraic numbers, while β1 , . . . , βn are
Q-linearly independent algebraic numbers (see Theorem 8.1 below).
e)
Lindemann-Weierstraß theorem
Chudnovsky [7] has shown how to prove the Lindemann-Weierstraß theorem on the algebraic independence of the numbers eβ1 , . . . , eβn by means of Gel’fond’s method. Here is a
simultaneous approximation measure for these numbers.
Theorem 2.6. Let β1 , . . . , βn be Q-linearly independent algebraic numbers. There exists
a positive constant C = C(β1 , . . . , βn ) such that the function
C D 1+(1/n) h(log h + D log D)(log h + log D)−1
is a simultaneous approximation measure for the numbers eβ1 , . . . , eβn .
The estimate is not sharp enough to apply Corollary 1.2 to the numbers θi = eβi , (1 ≤ i≤
n). However the function
ϕ(D, h) = C D 1+(1/n) h(log h + D log D)(log h + log D)−1
satisfies, for n ≥ 2,
lim sup
D→∞
1
1
lim sup ϕ(D, h) = 0.
D 1+1/(n−1) h→∞ h
Therefore Conjecture 1.7 would enable us to deduce the Lindemann-Weierstrass theorem
from Theorem 2.6.
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
f)
397
Other examples
Theorem 2.5 contains many results of simultaneous approximation. In certain cases the
estimate can be refined. Here is a first example.
Theorem 2.7. Let β be an irrational quadratic number and let λ be a nonzero logarithm
of an algebraic number. There exists a positive constant C = C(β, λ) such that
C D 2 h(h + log D)1/2 (log h + log D)−1/2
is a simultaneous approximation measure for the two numbers λ and eβλ .
If we choose λ = 2iπ and β = i, we deduce the existence of a positive absolute constant
C such that
C D 2 h(h + log D)1/2 (log h + log D)−1/2
is a simultaneous approximation measure for π and eπ . This is so far the best known
estimate, but it is not sharp enough to yield the algebraic independence of the two numbers
π and eπ . We come back to this question in Section 4.
As far as the two numbers e and π are concerned, we do not know anything better than
the above mentioned individual approximation measures for e and for π, due to Fel’dman—
also, we do not know anything better than Theorem 2.7 concerning a simultaneous approximation measure for the three numbers e, π and eπ .
The measure of simultaneous approximation for the numbers log α1 , . . . , log αn due to
Fel’dman (cf. Section 1) is not sufficient to settle the open problem of the existence of
two algebraically independent logarithms of algebraic numbers. However we can improve
Fel’dman’s measure for large D by adding a hypothesis.
Theorem 2.8. Let n ≥ 2 be an integer and λ1 , . . . , λn be Q-linearly independent logarithms of algebraic numbers. Assume that there exists a nonzero homogeneous polynomial
Q ∈ Q[X 1 , . . . , X n ] of degree 2, such that Q(λ1 , . . . , λn ) = 0. Then there is a positive
constant C such that the function
C D 2 (h + log D)2 (log D)−2
is a simultaneous approximation measure for the numbers λ1 , . . . , λn .
Therefore, under the assumptions of this Theorem 2.8, at least two of the numbers
λ1 , . . . , λn are algebraically independent (cf. [20], Theorem 2, and [21]).
3.
Elliptic functions
Elliptic analogues of all results in the preceding section can be proved. The estimates are
sharper in the CM case. Here is the analogue of Theorem 2.1 for elliptic functions.
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ROY AND WALDSCHMIDT
Theorem 3.1. Let ℘ be a Weierstraß elliptic function with algebraic invariants g2 and
g3 , let u be a complex number and let β be an algebraic number. Assume that none of
the numbers u, βu, . . . , β d−1 u is a pole of ℘. There exists a positive constant C with the
following property:
a) if β is of degree d ≥ 3 over Q, the function
C D (d+1)/(d−2) h d/(d−2) (log D + log h)−2/(d−2)
is a simultaneous approximation measure for the d numbers ℘ (β i u), (0 ≤ i ≤ d − 1).
b) If the elliptic curve E associated with ℘ has complex multiplication, and if β is of degree
d ≥ 2 over the field of endomorphisms of E, the function
C D (d+1)/(d−1) h d/(d−1) (log D + log h)−1/(d−1)
is a simultaneous approximation measure for the d numbers ℘ (u), ℘ (βu), . . . ,
℘ (β d−1 u).
The assumption that none of the numbers u, βu, . . . , β d−1 u is a pole of ℘ is not a serious
restriction: if for instance u is pole of ℘, we deduce a simultaneous approximation measure
for the remaining d − 1 numbers ℘ (βu), . . . , ℘ (β d−1 u) by applying the theorem with u
replaced by u/n, where n is a sufficiently large integer.
In the CM case, the estimate is the same as in the exponential case. In fact we shall
give a single proof for the two statements (Theorems 2.1 and 3.1). From Theorem 3.1 we
deduce the algebraic independence of at least two of the numbers ℘ (β i u), (0 ≤ i ≤ d − 1),
provided β is of degree ≥5 over Q in the general case, and provided β is of degree ≥3 over
the field of endomorphisms in the CM case. In particular, in the CM case, we deduce the
algebraic independence of the two numbers ℘ (βu) and ℘ (β 2 u) when ℘ (u) is an algebraic
number and β a cubic irrational number. These algebraic independence results are due to
Masser and Wüstholz [14].
In the same way we can produce other diophantine approximation estimates which yield
algebraic independence results. Elliptic analogues of Theorems 2.3 and 2.4 can be proved,
as well as simultaneous approximation measures related to results by Chudnovsky [6, 7]
and Tubbs [25, 26].
4.
Gamma function
Our last theorem deals with periods of elliptic integrals of first or second kind.
Theorem 4.1. Let ℘ be a Weierstraß elliptic function with algebraic invariants g2 and g3 .
Denote by (ω1 , ω2 ) a fundamental pair of periods of ℘, and by η1 , η2 the corresponding
quasi-periods of the Weierstraß zeta function associated with ℘:
ζ 0 = −℘,
ζ (z + ωi ) = ζ (z) + ηi ,
(i = 1, 2).
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
399
There exists a positive constant C such that
C D 3/2 (h + log D)3/2
is a simultaneous approximation measure for the four numbers ω1 , ω2 , η1 , η2 .
We deduce a well known result due to Chudnovsky [6] (Chap. 1, Section 2, Theorem 1):
the field Q(ω1 , ω2 , η1 , η2 ) has transcendence degree ≥2 over Q. In the CM case, the
transcendence degree is 2.
Theorem 4.1 shows that there exists an absolute positive constant C satisfying the following property: let γ1 and γ2 be algebraic numbers. Define
h = max{3, h(γ1 ), h(γ2 )}
and
D = [Q(γ1 , γ2 ) : Q].
Then
©
ª
|0(1/4) − γ1 | + |π − γ2 | ≥ exp −C D 3/2 (h + log D)3/2 .
It follows that the two numbers 0(1/4) and π are algebraically independent (cf. [6],
Chap. 7, Section 1, Cor. 1.7).
Remarks.
1. One can prove a more general statement than Theorem 4.1 which deals with the universal
g
extension of an abelian variety of dimension g by Ga . The corresponding result of
algebraic independence is stated by Chudnovsky in [6], Introduction, Theorem 9, p. 9.
2. Very recently, dramatic progress has been achieved on this topic. The story started with
the solution by Barré-Sirieix et al. [2] of a conjecture of Mahler (in the complex case)
and Manin (in the p-adic case) on the transcendence of the values of the modular function
J (q) for algebraic q with 0 < |q| < 1. This work gave the impetus to Nesterenko’s
proof of the algebraic independence of the three numbers π , eπ and 0(1/4) (see [17]).
At the same time, Nesterenko produces diophantine approximation results, including a
measure of algebraic independence. A slightly different approach is used by Philippon
[19] who also produces diophantine estimates, including a simultaneous approximation
measure for these three numbers: for any ² > 0 there exists a positive constant C(²)
such that the function
C(²)(Dh)(4/3)+²
is a simultaneous approximation measure for the three numbers π, eπ and 0(1/4).
Conjecture 1.7 suggests that such a measure for three numbers with an exponent <3/2 can
take place only when these numbers are algebraically independent. On the other hand this
estimate does not include the simultaneous approximation measure for π and eπ which we
have deduced from Theorem 2.7, nor the simultaneous approximation measure for π and
0(1/4) which follows from Theorem 4.1.
400
5.
ROY AND WALDSCHMIDT
A special case of the explicit algebraic subgroup theorem
The main tool in the proof of each of the above results is Theorem 2.1 of [33]. This theorem
provides an explicit version of the algebraic subgroup theorem, in the context of an arbitrary
commutative algebraic group. For the convenience of the reader, we prove here a special
case of this theorem that is sufficient for the proof of all our results, with the exception of
Theorem 4.1. The proof of the latter, given in Section 11, relies directly on Theorem 2.1 of
[33]. We try to conform, as much as possible, with the notations of [33].
Let d0 , d1 and d2 be integers ≥0 whose sum d is positive, let K ⊂ C be a number field,
let D be its degree over Q, and let E ⊂ P2 be an elliptic curve defined by a Weierstraß
equation
E : X 0 X 22 − 4X 13 + g2 X 02 X 1 + g3 X 03 = 0
(5.1)
with g2 , g3 ∈ K . Put G 0 = Gda 0 and n = d1 + d2 . We embed G 0 in Pd0 and Gm in P1 in
such a way that their exponential maps be given respectively by
¡
¢ ¡
¢
and expGm (z) = (1 : e z ).
expG 0 z 1 , . . . , z d0 = 1 : z 1 : · · · : z d0
We also denote by ℘ the Weierstraß ℘-function attached to E, so that the exponential map
of E, from C to E(C), is given, away from the poles of ℘, by
expE (z) = (1 : ℘ (z) : ℘ 0 (z)).
The product
G = G 0 × Gdm1 × Ed2
therefore embeds into Pd0 × (P1 )d1 × (P2 )d2 . Its exponential map
expG : TG (C) = Cd → G(C)
is given by the product of the exponential maps of its factors. For simplicity however,
we often identify G 0 (C) with the additive group Cd0 and Gm (C) with the multiplicative
group C× .
Given an algebraic subset V of G, we denote by H(V ; T0 , T1 , . . . , Tn ) the product by
(dim V )! of the homogeneous part of degree dim V of its Hilbert-Samuel polynomial (with
respect to the above embedding of G). Given a finitely generated subgroup 0 of G(K ),
a finite set of generators γ1 , . . . , γ` of 0, and positive integers S1 , . . . , S` , we denote by
0(S1 , . . . , S` ) the set of all elements of 0 of the form s1 γ1 + · · · + s` γ` with integral
coefficients s j satisfying 0 ≤ s j < S j for j = 1, . . . , `. When S1 = · · · = S` = S, we
simply denote this set by 0(S). Similarly, if Y is a subgroup of Cd generated by a given
finite set of points η1 , . . . , η` , we define Y (S1 , . . . , S` ) as the set of all linear combinations
of these points with integral coefficients in the same range. Finally, we denote by |z| the
supremum norm of a point z of Cd :
|(z 1 , . . . , z d )| = max |z i |.
1≤i≤d
This definition applies in particular to elements of Zd .
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
401
For w = (α1 , . . . , αd ) ∈ K d , we denote by h(1 : w) the absolute logarithmic height of
the projective point (1 : α1 : · · · : αd ).
Theorem 5.1. There exist positive constants C1 = C1 (d) ≥ e and C2 = C2 (d, `, E) with
the following property. Let `0 and `1 be integers ≥ 0, let w10 , . . . , w`0 0 and η10 , . . . , η`0 1 be
elements of Cd , let w1 , . . . , w`0 be elements of K d , and let η1 , . . . , η`1 be elements of Cd
whose images γ1 , . . . , γ`1 under expG all belong to G(K ). Denote by r the dimension of
the subspace of Cd generated by w10 , . . . , w`0 0 and η10 , . . . , η`0 1 . Denote by W the subspace
of K d generated by w1 , . . . , w`0 , and denote by 0 the subgroup of G(K ) generated by
γ1 , . . . , γ`1 . Assume that η10 , . . . , η`0 1 do not all belong to Cd0 × {0}n . Suppose that we are
given real numbers A1 , . . . , An ≥ e2 , B1 , B2 ≥ 2d and E ≥ e, integers S0 , S1 , . . . , S`1 ≥ 1
and T0 , T1 , . . . , Tn ≥ 1, and real numbers U, V > 0. Suppose B2 ≥ C2 if d2 > 0. Write
¡
¢
η j = β1,`0 + j , . . . , βd0 ,`0 + j , u 1 j , . . . , u n j
for j = 1, . . . , `1 , and suppose
Ã
!
`1
`1
X
X
¡
¢
log
Sj +
h βi,`0 + j ≤ log B1 ,
j=1
(1 ≤ i ≤ d0 ),
j=1
h(1 : w j ) ≤ log B2 ,
)
`1
`1
X
E X
ui j
S j h(e ),
S j |u i j | ≤ log Ai ,
max
D j=1
j=1
½
¾
E2 2
S j |u i j |2 ≤ log Ai ,
C2 max S 2j (h(℘ (u i j )) + 1),
1≤ j≤`1
D
(
(1 ≤ j ≤ `0 ),
(1 ≤ i ≤ d1 ),
(d1 < i ≤ n).
Suppose also
|w 0j − w j | ≤ e−2V
(1 ≤ j ≤ `0 ) and
|η0j − η j | ≤ e−2V
(1 ≤ j ≤ `1 ),
and that the parameters satisfy
D log B1 ≥ log E,
DT0 log B1 ≤ U,
D log B2 ≥ log E,
DS0 log B2 ≤ U,
B2 ≥ d S0 + T0 + T1 + · · · + Tn , (5.2)
n
X
DTi log Ai ≤ U, C1 U ≤ V
(5.3)
i=1
and the main condition
¶r
µ ¶ µ
µ
¶
1
T0 + d0
U
V
>
exp
.
(T1 + 1) · · · (Tn + 1) ≥ 4
8 · 3d2
2D
d0
log E
(5.4)
Then, there exists a connected algebraic subgroup G ∗ of G, distinct from G, incompletely
defined in G by polynomials of multidegree
¢
¡
≤ T0 , T1 , . . . , Td1 , 2Td1 +1 , . . . , 2Tn ,
402
ROY AND WALDSCHMIDT
such that, if we put
`∗0 = dim((W + TG ∗ (K ))/TG ∗ (K )),
K
we obtain
`∗
S0 0
µ
0(S1 , . . . , S`1 ) + G ∗ (K )
Card
G ∗ (K )
¶
· H(G ∗ ; T0 , T1 , . . . , Tn ) ≤
6d2 d! d0
T T1 · · · Tn .
d0 ! 0
Proof: Let M = S1 · · · S`1 , and let S be the set of all points s = (s1 , . . . , s`1 ) ∈ Z`1 with
0 ≤ s j < S j , (1 ≤ j ≤ `1 ). Choose an enumeration s1 , . . . , s M of S in such a way that, if
sk = (s1k , . . . , s`1 k ), then, for k = 1, . . . , `1 , we have
ηk0 =
`1
X
s jk η0j
and
j=1
ηk =
`1
X
s jk η j .
j=1
Use the same formulas, to define ηk0 and ηk for k = `1 + 1, . . . , M, and put γk = expG (ηk )
for k = 1, . . . , M. Then the set 6 = {γ1 , . . . , γ M } is simply 0(S1 , . . . , S`1 ). We now show
that these data and the present choice of parameters satisfy all constraints in Theorem 2.1
of [33].
We first verify the height constraints in Section 2e) of [33]. ForPi = 1, . . . , d0 , the
1
s j βi,`0 + j with
point of P M (K ) whose coordinates are 1 and the linear combinations `j=1
coefficients (s1 , . . . , s`1 ) ∈ S has height
!
Ã
`1
`1
X
X
¡
¢
Sj +
h βi,`0 + j ≤ log B1 .
≤ log
j=1
j=1
For any i = 1, . . . , d1 and any (s1 , . . . , s`1 ) ∈ S, we have
!!
à Ã
`1
`1
X
X
s j ui j
S j h(eu i j ) ≤ log Ai ,
≤
h exp
j=1
j=1
¯
¯
`1
`1
¯
E ¯¯ X
E X
¯
s j ui j ¯ ≤
S j |u i j | ≤ log Ai ,
¯
¯
D ¯ j=1
D j=1
and 2/D ≤ 2 ≤ log Ai . Similarly, Section 4c) of [33] shows that there is a positive constant
C3 = C3 (E) such that for any i = d1 + 1, . . . , n and any (s1 , . . . , s`1 ) ∈ S, we have
Ã
!!
Ã
`1
X
ª
©
s j ui j
≤ `21 C3 max S 2j (h(expE (u i j )) + 1)
h expE
j=1
1≤ j≤`1
ª
©
≤ `21 C32 max S 2j (h(℘ (u i j )) + 1)
1≤ j≤`1
≤ log Ai ,
403
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
¯
!
à ¯
`1
¯
¯X
©
ª
1 +
(d`1 + 1)2
¯
¯
s j u i j ¯ + 2 ≤ C3
H Ed ¯
max max 1, E 2 S 2j |u i j |2 ≤ log Ai ,
1≤ j≤`1
¯
¯ j=1
D
D
and
¯!
à ¯
`1
¯
1 − ¯¯ X
¯
s j u i j ¯ ≤ C3 ≤ log Ai ,
H d¯
¯
¯
D
j=1
provided C2 ≥ max{`21 C32 , (d`1 + 1)2 C3 }.
The conditions in Section 2f) of [33] are fulfilled because of (5.2), (5.3) and the hypothesis
B2 ≥ C2 if d2 > 0. Moreover, the assumption that not all of η10 , . . . , η0M belong to Cd0 ×{0}n
implies that the integer r3 defined in Section 2c) is positive. Therefore, since the HilbertSamuel function of G verifies
¶ n
µ
¶ n
µ
T0 + d0 Y
T0 + d0 Y
(Ti + 1) ≤ H (G ; T0 , . . . , Tn ) ≤ 3d2
(Ti + 1),
d0
d0
i=1
i=1
the main constraint in Section 2g) is satisfied if
¶ µ
¶r3
µ
µ
¶ n
¶µ
¶µ
T0 + d0 Y
U
T0 + r1 d S0 + r2
V
1
>
exp
(Ti + 1) ≥ 4
.
8 · 3d2
2D
d0
r1
r2
log E
i=1
This last condition follows from (5.4) because we have r1 + r2 + r3 = r ,
µ
µ
¶
¶r1 µ
¶r1
T0 + r1
2U
V
r1
≤
,
≤ (2T0 ) ≤
r1
D log B1
log E
and
µ
d S0 + r2
r2
µ
¶
≤ ((d + 1)S0 ) ≤
r2
(d + 1)U
D log B2
µ
¶r2
≤
V
log E
¶r2
,
provided C1 ≥ d + 1. Finally, for any (s1 , . . . , s`1 ) ∈ S, we have
¯
¯ Ã
!
`1
`1
`1
¯X
¯
X
X
¯
¯
0
s η −
sjηj¯ ≤
S j e−2V ≤ B1 e−2V ≤ e−V ,
¯
¯ j=1 j j
¯
j=1
j=1
since B1 ≤ eU . Thus, all the hypotheses of Theorem 2.1 of [33] are satisfied. The conclusion
follows.
2
6.
Proofs of Theorems 2.1 and 3.1
We prove simultaneously both theorems. Let D be a positive integer, let c0 , h be positive
real numbers with h ≥ c0 ≥ 3, and let γ0 , . . . , γd−1 be algebraic numbers with
max{h(γ0 ), . . . , h(γd−1 )} ≤ h
and
[Q(γ0 , . . . , γd−1 ) : Q] ≤ D.
404
ROY AND WALDSCHMIDT
In the case of Theorem 2.1, we assume
¯ i
¯
max ¯a β − γi ¯ ≤ e−4V
0≤i≤d−1
where
V d−1 = c09d−1 D d+1 h d (log D + log h)−1 .
(6.1)
In the case of Theorem 3.1, we denote by E the elliptic curve associated to the function ℘.
For part a), we assume
max |℘ (β i u) − γi | ≤ e−4V
0≤i≤d−1
(6.2)
where
V d−2 = c09d−2 D d+1 h d (log D + log h)−2 .
For part b), we assume that (6.2) holds with V given by (6.1). We will derive a contradiction
assuming that c0 is sufficiently large (depending only on β and log a for Theorem 2.1, on
β, u and E for Theorem 3.1). Since V is an increasing function of D, we may assume
D = [Q(γ0 , . . . , γd−1 ) : Q].
√
In the situation of Theorem 2.1, we put A = Z, k = Q, Ä = 2π −1 Z, % = 1 and
ν = 1, and we apply Theorem 5.1 with the group
G = Ga × G 1 ,
where
G 1 = Gdm .
Thus, we have d0 = 1, d1 = d, d2 = 0 and, in the notations of Theorem 5.1, d is replaced
by d + 1. We put u = log a, and, for 0 ≤ i ≤ d − 1, we denote by u i the determination of
the logarithm of γi which is closest to β i u. Then, we have
max |β i u − u i | ≤ e−3V ,
0≤i≤d−1
(0 ≤ i ≤ d − 1).
(6.3)
In the situation of Theorem 3.1, we denote by Ä the group of periods of E. For part a), we
put A = Z, k = Q, % = 2 and ν = 1. For part b), we choose a non trivial endomorphism of
E corresponding to some imaginary quadratic number τ , and we put A = Z[τ ], k = Q(τ ),
% = ν = 2. For both, we apply Theorem 5.1 with the group
G = Ga × G 2 ,
where
G 2 = Ed .
Thus, we have d0 = 1, d1 = 0, d2 = d and again, in the notations of Theorem 5.1, d is
replaced by d + 1. For i = 0, . . . , d − 1, we denote by u i the complex number which is
closest to β i u and satisfies ℘ (u i ) = γi . Then (6.3) holds in this case as well.
In all cases, we have ν = [k : Q], % = rankZ Ä and d > %/ν. Moreover, β is algebraic
over k of degree d. So, we can write
β d = b0 + b1 β + · · · + bd−1 β d−1 ,
(6.4)
405
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
with b0 , b1 , . . . , bd−1 ∈ k. We define recursively complex numbers u d , . . . , u 2d−2 by the
conditions
u i+d = b0 u i + b1 u i+1 + · · · + bd−1 u i+d−1 ,
(0 ≤ i ≤ d − 2).
We put `0 = 0, `1 = νd, and
η0j = (β j−1 ; β j−1 u, β j u, . . . , β j+d−2 u),
η j = (β
j−1
; u j−1 , u j , . . . , u j+d−2 ),
(1 ≤ j ≤ d),
(1 ≤ j ≤ d).
If ν = 2, we also define
0
0
ηd+
j = τηj
and
ηd+ j = τ η j ,
(1 ≤ j ≤ d).
Thanks to (6.3) and (6.4), these points satisfy
|η0j − η j | ≤ e−2V ,
(1 ≤ j ≤ `1 ).
Moreover, since η10 , . . . , η`0 1 generate a subspace of Cd+1 of dimension 1, we have r = 1.
Let 0 be the subgroup of G(C) generated by the points expG (η j ), (1 ≤ j ≤ `1 ), and let
K be the smallest extension of k such that G is defined over K and 0 ⊂ G(K ). Then, the
degree of K over Q satisfies
D ≤ [K : Q] ≤ c0 D.
The parameter V defined above is given in all cases by
9d−%/ν
V d−%/ν = c0
D d+1 h d (log D + log h)−%/ν .
Define U = c0−1 V and the remaining parameters by
¸
·
£¡
¢1/d ¤
U
, T1 = · · · = Td = T = c05 D
,
T0 = 3
c0 D(log D + log h)
¶1/% ¸
·µ
U
,
S0 = 1, S1 = · · · = S`1 = S =
c03 DT h
A1 = · · · = Ad = exp{c0 S % h},
B1 = S c0 ,
B2 = exp{U/(c0 D)},
E = (Dh)1/% .
Then, S0 , S, T0 and T are positive integers and S satisfies
(Dh)1/(dν) ≤ S ≤ (Dh)c0 .
Moreover, one verifies that all the conditions of Theorem 5.1 are fulfilled, and that we have
S dν > 6d2 (d + 1)!T0 T d .
(6.5)
406
ROY AND WALDSCHMIDT
This last condition ensures that the conclusion of Theorem 5.1 is non trivial as we will now
see.
Theorem 5.1 shows that there exists an algebraic subgroup G ∗ = G ∗0 × G ∗% of G =
Ga × G % , which is defined over K , which is also incompletely defined by equations of
multidegrees ≤ (T0 , %T, . . . , %T ), with G ∗ 6= G, such that
µ
Card
0(S) + G ∗ (K )
G ∗ (K )
¶
d∗
∗
≤ 6d2 (d + 1)!T0 0 × T d% ,
where d0∗ is the codimension of G ∗0 in Ga , while d%∗ is the codimension of G ∗% in G % . Note
that, because of (6.5), the right hand side of this inequality is <S `1 . Moreover, since
1, β, . . . , β d−1 are linearly independent over k, the projection of 0(S) on the factor Ga of
G has cardinality S `1 . We deduce G ∗0 = Ga and d0∗ = 0. Let 0% be the projection of 0
on the factor G % . For j = 1, . . . , `1 , we denote by y j the element of Cd formed by the
last d coordinates of η j . Then, the points expG % (y j ), (1 ≤ j ≤ `1 ), constitute a system of
generators of 0% , and we have
¶
µ
¶
0% (S) + G ∗% (K )
0(S) + G ∗ (K )
= Card
≥ S `1 Card(E)−1
Card
G ∗ (K )
G ∗% (K )
µ
where E denotes the set of points s = (s1 , . . . , s`1 ) ∈ Z`1 with |s| < S and
Ã
expG %
`1
X
!
∈ G ∗% (K )
sjηj
(6.6)
j=1
(compare with Lemma 10.3 in [32]). This implies
S `1 Card(E)−1 ≤ (d + 1)!6d2 T d
and so, by (6.5), we get Card(E) ≥ T0 . Since T0 ≥ 4S % > (2S − 1)% , this means that E
contains at least % + 1 elements which are linearly independent over Q. We will show that
this is impossible.
Since the subgroup G ∗ is distinct from G and incompletely defined in G by polynomials
of multidegrees ≤(T0 , %T, . . . , %T ), there exists a point t = (t1 , . . . , td ) in Ad with
|t| ≤ c0 T %/ν ≤ c0 T 2
such that G ∗% is contained in the connected algebraic subgroup of G % of codimension 1
whose Lie algebra is the kernel of the linear map:
g:
→
Cd
(z 1 , . . . , z d ) 7→
C
t1 z 1 + · · · + td z d
(see [32], Chap. 8 for multiplicative groups, [15], Theorem III, p. 425 for elliptic curves).
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
407
For any point (s1 , . . . , s`1 ) ∈ Z`1 satisfying (6.6), we have
!
Ã
`1
X
s j y j ∈ Ä.
g
j=1
Define a mapping f : Z`1 → C by
¢
¡
f s1 , . . . , s`1 = g
Ã
`1
X
!
sj yj ,
j=1
and fix ρ + 1 linearly independent points of E. Since the images under f of these points
are elements of Ä of norm ≤c02 ST 2 , the pigeonhole principle shows that the kernel of f
contains a linear combination of these points with integral coefficients which are bounded
above in absolute value by
¡
¢%
c0 c02 ST 2 ≤ c05 S 2 T 4 ,
and not all zero. Let s = (s1 , . . . , s`1 ) ∈ Z`1 be such a point. The relation f (s) = 0 gives
d
d X
X
½
a j ti u j+i−2 = 0
j=1 i=1
where a j =
sj
s j + sd+ j τ
if ν = 1
if ν = 2.
Since the numbers |β i u − u i | are bounded from above by e−2V for i = 0, . . . , 2d − 2, we
deduce
¯
¯Ã
!Ã
!¯ ¯
d
d
d
d X
¯ ¯X
¯
¯ X
X
¯
¯
j−1
i−1 ¯
j+i−2 ¯
ajβ
ti β
a j ti β
¯=¯
¯ ≤ e−V .
¯
¯
¯
¯
¯ j=1
i=1
j=1 i=1
This is impossible because, since β is of degree d ≥ 2 over k, the above product does not
vanish, and Liouville’s inequality shows that it is bounded below by (c07 S 3 T 6 )1−dν .
2
Remark. A variant of this proof can be given with the group G = G % , using d0 = 0, `0 = 1,
`1 = dν,
w1 = (1, β, . . . , β d−1 )
and by deleting the first coordinate of the points η10 , . . . , η`0 1 and η1 , . . . , η`1 .
7.
Proof of Theorems 2.3 and 2.4
Proof of Theorem 2.4: Fix a constant c0 ≥ 1, and define
V = c010κ+1 (Dh)κ (log E)1−κ F κ/m .
408
ROY AND WALDSCHMIDT
We will assume
max |λi j λ11 − λi1 λ1 j | < e−3V ,
1≤i≤d
1≤ j≤`
and show that, if c0 is sufficiently large in terms of d and `, this hypothesis leads to a
contradiction. This will prove the theorem with C = 3c010κ+1 . Each computation below
assumes that c0 is sufficiently large as a function of d and `.
By replacing the matrix (λi j ) by its transpose if necessary, we may assume that ` ≥ d.
Since d` > d + `, this implies d ≥ 2 and ` ≥ 3, and we have m = ` − 1. By permuting
between themselves the columns of this matrix, we may also assume that γ11 6= 1. Then,
Liouville’s inequality applied to |γ11 − 1| gives
|λ11 | ≥
1
min{|γ11 − 1|, 1} ≥ exp{−D(log 2) − Dh} ≥ exp{−2Dh}.
2
Since Dh/E ≥ |λ11 |, this implies log E ≤ 3Dh. So, we get V ≥ c010κ Dh, and thus
¯¡
¯
¢
−1 −3V
¯
max ¯ λ−1
≤ e−2V .
11 λ1 j λi1 − λi j ≤ |λ11 | e
1≤i≤d
1≤ j≤`
Denote by K the field generated over Q by the d` numbers γi j . Put U = c0−1 V , and define
T and S to be the largest integers satisfying respectively
T d ≤ c02
U
log E
S ` ≤ c08
and
U
F `/(`−1) .
log E
We apply Theorem 5.1 to the algebraic group G = Gdm with the parameters d0 = `0 = 0,
d1 = d, `1 = `,
log A1 = · · · = log Ad = c0 Sh,
T0 = S0 = 1,
B1 = B2 = eU/D ,
T1 = · · · = Td = T,
S1 = · · · = S` = S,
and the points
η0j = λ−1
11 λ1 j (λ11 , . . . , λd1 )
and
η j = (λ1 j , . . . , λd j ),
(1 ≤ j ≤ `).
We have G(K ) = (K × )d , and 0 is the subgroup of G(K ) generated by the points
γ j = expG (η j ) = (γ1 j , . . . , γd j ),
(1 ≤ j ≤ `).
Since η10 , . . . , η`0 span a one-dimensional subspace of Cd , one verifies that all the conditions
of Theorem 5.1 are satisfied with r = 1. Thus, there exists a connected algebraic subgroup
G ∗ of G, different from G, which is defined by equations of partial degrees ≤T , such that
µ
¶
0(S) + G ∗ (K )
Card
H(G ∗ ; T, . . . , T ) ≤ d!T d .
G ∗ (K )
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
409
Let d ∗ > 0 be the codimension of G ∗ in G, and let 8 be the subgroup of Zd consisting of
all points (t1 , . . . , td ) ∈ Zd for which the monomial Y1t1 · · · Ydtd ∈ Z[Y1±1 , . . . , Yd±1 ] induces
the trivial character on G ∗ . Then, 8 has rank d ∗ , and we obtain
H(G ∗ ; T, . . . , T ) ≥ c0−1 det(8)T d−d
∗
where det (8) denotes the determinant of 8 (see Section 3(b) of [3]). Put
(
)
X̀
E = (s1 , . . . , s` ) ∈ Z` ;
s j γ j ∈ G ∗ (K ) and |s j | < S for j = 1, . . . , ` .
j=1
Then, we also have
µ
Card
0(S) + G ∗ (K )
G ∗ (K )
¶
Card(E) ≥ S `
(see for example the remark after Lemma 10.3 in [32]). Combining the last three inequalities,
we obtain
∗
S ` det(8) ≤ d!c0 T d Card(E).
First, assume d ∗ < d, and let ϕ be a nonzero element of 8 of smallest supremum norm.
By Minkowski’s first convex body theorem, and Vaaler’s cube slicing inequality [27], we
get
∗
|ϕ| ≤ det(8)1/d ≤ det(8),
since 8 has rank d ∗ . Write ϕ = (t1 , . . . , td ) and consider the function f : Z` → C given by
X
ti s j λi j .
f (s1 , . . . , s` ) =
1≤i≤d
1≤ j≤`
√
By construction, it maps elements of E to elements of 2π −1Z of absolute value ≤c0 S F|ϕ|.
On the other hand, since ` ≥ d and ` ≥ 3, we have S `−1 ≥ c03 T d−1 F, and so,
Card(E) ≥
S ` det(8)
> c0 S F|ϕ|.
d!c0 T d−1
Thus, f is not injective on E and so, there exists a nonzero element (s1 , . . . , s` ) of Z` with
|s j | < 2S, (0 ≤ j ≤ `), such that f (s1 , . . . , s` ) = 0. We deduce
¯
¯
¯ ¯
d
d X̀
¯X
¯
¯ ¯X
X̀
¯
¯
¯ ¯
ti λi1
s j λ1 j ¯ = ¯
ti s j (λi1 λ1 j − λ11 λi j )¯ ≤ e−V .
(7.1)
¯
¯ i=1
¯
¯
¯
j=1
i=1 j=1
Q and since
Since λ11 , . . . , λd1 are linearly independent
P λ11 , . . . , λ1` are also linearly
Pover
d
ti λi1 and `j=1 s j λ1 j vanishes. Moreover,
independent over Q, none of the numbers i=1
410
ROY AND WALDSCHMIDT
we have
|ϕ| = max |ti | ≤ T,
1≤i≤d
because G ∗ is defined by polynomials of partial degrees ≤T . Thus, Liouville’s inequality
yields lower bounds:
¯
¯
¯)
( ¯
d
d
¯X
¯
¯ 1
¯
Y
¯
¯
¯
ti ¯
ti λi1 ¯ ≥ min 1, ¯1 −
γi1 ¯ ≥ exp{−c0 T Dh}
¯
¯ i=1
¯
¯ 2
¯
i=1
and similarly
¯
¯
¯
¯ X̀
¯
¯
s j λ1 j ¯ ≥ exp{−c0 S Dh}.
¯
¯
¯ j=1
Since V > c0 T S Dh, this is a contradiction.
Finally, assume d ∗ = d. In this case, the subgroup G ∗ is reduced to the neutral element
of G, and we have 8 = Zd . Since S ` ≥ c02 T d , we get Card(E) > 1. Let s = (s1 , . . . , s` ) be
a nonzero element of E. For each (t1 , . . . , td ) ∈ Zd with 0 ≤ ti < T , (1 ≤ i ≤ d), the sum
d X̀
X
ti s j λi j
(7.2)
i=1 j=1
√
is an element of 2π −1 Z of absolute value ≤c0 T S Dh/E. Since c0 T S Dh/E < T d ,
Dirichlet’s box principle yields a point t = (t1 , . . . , td ) ∈ Zd with |ti | < T , (1 ≤ i ≤ d),
for which the sum (7.2) vanishes. For these choices of s and t, inequality (7.1) again holds,
and we get a contradiction as above.
2
A direct proof of Theorem 2.3 can be given along the same lines. It shows that the
technical hypothesis can be replaced by the following weaker one: the numbers x1 , . . . , xd
satisfy a measure of linear independence with exponent d, and the numbers y1 , . . . , y`
satisfy a measure of linear independence of exponent `.
Here, we prefer to derive Theorem 2.3 from Theorem 2.4, in order to show the link
between the two results. To this end, we will need the following lemma:
Lemma 7.1. Let z 1 , . . . , z n be complex numbers satisfying a measure of linear independence with exponent ν > 0, and let θ > 3nν be a real number. There exists a positive
constant c satisfying the following property. Let λ1 , . . . , λn be logarithms of algebraic
numbers, let D be a positive integer, and let h ≥ c be a real number. Define γ j = eλ j ,
(1 ≤ j ≤ n), and assume
[Q(γ1 , . . . , γn ) : Q] ≤ D,
max{h(γ j ), |λ j |} ≤ h,
and
max |z j − λ j | ≤ exp{−(Dh)θ }.
1≤ j≤n
Then the numbers λ1 , . . . , λn are Q-linearly independent.
(1 ≤ j ≤ n),
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
411
Proof: By assumption there exists a real number T0 such that, for T ≥ T0 and for
(t1 , . . . , tn ) ∈ Zn satisfying
0 < max{|t1 |, . . . , |tn |} ≤ T,
the inequality
|t1 z 1 + · · · + tn z n | ≥ exp{−T ν }
holds. Since ν < θ/(3n), we may assume
T ν + log(nT ) < T θ/(3n)
for T ≥ T0 . Suppose h ≥ c where
c1 (n) = (103 n)n−1
and
©
1/(3n) ª
c = max c1 (n)1/(2n) , T0
.
If the numbers λ1 , . . . , λn were linearly dependent over Q, we could use Lemma 7.2 of [32]
and write
t1 λ1 + · · · + tn λn = 0,
with a n-tuple (t1 , . . . , tn ) ∈ Zn such that
0 < max{|t1 |, . . . , |tn |} ≤ c1 (n)(D 3 h)n−1 .
Put T = (Dh)3n . Then, we get T ≥ T0 , max{|t1 |, . . . , |tn |} ≤ T , and
©
ª
|t1 z 1 + · · · + tn z n | ≤ nT max |x j − λ j | ≤ nT exp − T θ/(3n) < exp{−T ν },
1≤ j≤n
which contradicts the hypothesis on the measure of linear independence of z 1 , . . . , z n . 2
Proof of Theorem 2.3: Let D be a positive integer, let c2 and h be real numbers with
h ≥ c22 ≥ 3, and let γ11 , . . . , γd` be algebraic numbers which generate over Q a field K of
degree ≤D, and satisfy h(γi j ) ≤ h for i = 1, . . . , d and j = 1, . . . , `. Suppose
|γi j − e xi y j | ≤ e−3V
where
V = c2κ+1 (Dh)κ (log D + log h)1−κ .
We will derive a contradiction, assuming that c2 is sufficiently large as a function of
x1 , . . . , xd and y1 , . . . , y` .
For this purpose, we may assume, without loss of generality, that K has degree exactly
D over Q. For each i = 1, . . . , d and j = 1, . . . , `, choose a determination λi j of the
logarithm of γi j so that
|λi j − xi y j | ≤ e−2V .
We get
max |λi j λ11 − λi1 λ1 j | ≤ e−V .
1≤i≤d
1≤ j≤`
Moreover, since x1 y1 , . . . , xd y1 satisfy a measure of linear independence with exponent
<κ/(3d), Lemma 7.1 shows that λ11 , . . . , λd1 are linearly independent over Q. Similarly,
412
ROY AND WALDSCHMIDT
since x1 y1 , . . . , x1 y` satisfy a measure of linear independence with exponent <κ/(3`),
Lemma 7.1 shows that λ11 , . . . , λ1` are linearly independent over Q. The contradiction
follows by applying Theorem 2.4 with E = Dh/c2 . Note that E ≥ (Dh)1/2 since
2
h ≥ c22 .
8.
Proof of Theorems 2.5 and 2.6
In this section we give a lower bound for
and 2.6.
P
|β j − log α j | and we deduce Theorems 2.5
Theorem 8.1. Let n be a positive integer. There exist two positive constants c1 and c2
with the following property. Let α1 , . . . , αn and β1 , . . . , βn be algebraic numbers, let D be
the degree of the number field they generate, and let A, B, B 0 , E be real numbers which
are ≥e and satisfy
max h(α j ) ≤ log A
1≤ j≤n
max h(β j ) ≤ log B.
and
1≤ j≤n
For 1 ≤ j ≤ n, assume that the number α j does not vanish, and choose a determination
λ j of its logarithm. Assume
max |λ j | ≤
1≤ j≤n
log E ≤ D log A,
D
log A,
E
log E ≤ D log B,
log E ≤ D log B 0 ,
0
D(log A)(log B) ≥ (log B )(log E),
B ≥ D(log B 0 )(log E)−1
and
B 0 ≥ D(log A)(log B).
Furthermore, assume
s1 β1 + · · · + sn βn 6= 0
for any (s1 , . . . , sn ) ∈ Zn with
µ
0 < max |s j | ≤ c1
1≤ j≤n
D log B 0
log E
¶1/n
.
Then, we have
n
X
|β j − λ j | ≥ exp{−c2 D 2+1/n (log A)(log B)(log B 0 )1/n (log E)−1−1/n }.
j=1
Proof: Let c0 ≥ 1 be a real number. Set
5+4/n
V = c0
D 2+1/n (log A)(log B)(log B 0 )1/n (log E)−1−1/n .
(8.1)
(8.2)
(8.3)
413
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
5/n
Assume c1 ≥ c0
and
n
X
|β j − λ j | < e−2V .
j=1
We will show that, if c0 is sufficiently large as a function of n, then the last inequality leads
5+4/n
.
to a contradiction. This will prove the theorem with c2 = 2c0
To this end, we apply Theorem 5.1 with the group G = Ga × Gm , so that d0 = d1 = 1
and d = 2. For the number field K , we take Q(α1 , . . . , αn , β1 , . . . , βn ). Its degree over Q
is D. We choose the points
w10 = w1 = (1, 1),
η0j = (β j , β j ),
η j = (β j , λ j ),
(1 ≤ j ≤ n),
so that `0 = 1 and `1 = n and r = 1. Note that n has a different meaning here than in the
statement of Theorem 5.1. Let 0 be the subgroup of G(K ) = K × K × generated by
γ j = expG (η j ) = (β j , α j ),
(1 ≤ j ≤ n).
We use the parameters
B1 = B c0 ,
·
T1 =
c02
·
¸
¸
U
U
, S0 =
,
D log B1
D log B2
¶1/n ¸
·µ
3 D log B2
S1 = · · · = Sn = S = c0
and A1 = Ac0 S .
log E
B2 = (B 0 )c0 ,
¸
D log B1
,
log E
U = c0−1 V,
·
T0 =
By (8.1), the integers T0 , T1 and S are positive. The inequality (8.2) combined with (8.1)
gives as well
·
µ
0 ¶1/n ¸
3+4/n D log B
S0 ≥ c0
≥ 1.
log E
1/2
Assuming c0 sufficiently large, the left inequality in (8.3) gives S ≤ (c04 B)1/n ≤ B1 , thus
log(nS) +
n
X
j=1
h(β j ) ≤
1
log B1 + log n + n log B ≤ log B1 .
2
Using the right inequality in (8.3), we also find U ≤ B2 . So, we get
2S0 + T0 + T1 ≤ U ≤ B2 .
The remaining conditions of Theorem 5.1 are plainly satisfied.
Therefore, there is a connected algebraic subgroup G ∗ of G, distinct from G, such that
¶
µ
0(S) + G ∗ (K )
H(G ∗ ; T0 , T1 ) ≤ 2T0 T1 .
S0 Card
G ∗ (K )
414
ROY AND WALDSCHMIDT
Here, the exponent `∗0 of S0 is 1 because the only possibilities for G ∗ are {0}×{1}, {0}×Gm ,
or Ga × {1}, and, in all cases, w1 does not belong to the tangent space of G ∗ at the identity.
Since
2T0 T1 ≤ 2c02
U
< S0 S n ,
log E
this inequality implies
¶
0(S) + G ∗ (K )
H(G ∗ ; T0 , T1 ) < S n .
Card
G ∗ (K )
µ
(8.4)
Explicitly, we have
ª
¢
©¡
0(S) = s1 β1 + · · · + sn βn , α1s1 · · · αnsn ; 0 ≤ s j < S, (1 ≤ j ≤ n) .
5/n
Since c1 ≥ c0 , the hypothesis tells us that, for any nonzero element s = (s1 , . . . , sn ) of
Z with |s| < 2S, we have
n
s1 β1 + · · · + sn βn 6= 0.
Thus, the set
{s1 β1 + · · · + sn βn ; 0 ≤ s j < S, 1 ≤ j ≤ n}
has cardinality S n . Since this set is the projection of 0(S) on the factor Ga of G, the
condition (8.4) is not satisfied if G ∗ is equal to {0} × {1} or {0} × Gm . It remains the case
where G ∗ = Ga × {1}. In this case, the condition (8.4) becomes
©
ª
Sn
Card α1s1 · · · αnsn ; 0 ≤ s j < S, 1 ≤ j ≤ n <
.
T0
Let E be the set of all points (s1 , . . . , sn ) ∈ Zn with supremum norm <S such that
α1s1 · · · αnsn = 1. Since
ª
©
Card α1s1 · · · αnsn ; 0 ≤ s j < S, 1 ≤ j ≤ n ≥
Sn
,
Card(E)
we get Card(E) > T0 . On the other hand, for each (s1 , . . . , sn ) ∈ E, we have
√
s1 λ1 + · · · + sn λn = 2π −1 s0 ,
for an integer s0 with
|s0 | ≤
nS
nS D log A
c0 S D log A
max |λ j | ≤
≤
≤ c0−1 T0 .
1≤
j≤n
2π
2π E
log E
By Dirichlet box principle, this means that there are at least two different elements of E with
the same value of s0 . Taking their difference, we get a nonzero element s = (s1 , . . . , sn ) of
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
415
Zn with |s| < 2S such that
s1 λ1 + · · · + sn λn = 0.
This implies
|s1 β1 + · · · + sn βn | ≤ 2nSe−2V ≤ e−V .
Since s1 β1 + · · · + sn βn is a nonzero algebraic number with absolute logarithmic height
≤ log(2nS) + n log B ≤ log(2B1 ), Liouville’s inequality gives
|s1 β1 + · · · + sn βn | ≥ (2B1 )−D .
2
This is impossible since V ≥ c0 D log B1 . This contradiction ends the proof.
Remark. Here, we applied Theorem 2.1 of [21] with the algebraic group Ga × Gm ,
the direction (1, 1) and the n points (β1 , λ1 ), . . . , (βn , λn ). A similar argument can be
given with the algebraic group Ga × Gnm , the direction (1, β1 , . . . , βn ) and a single point
(1, λ1 , . . . , λn ). The choice of parameters is similar but the conditions which arise concerning B and B 0 are not exactly the same. The transition from one proof to the other is merely
the transposition of the interpolation matrix. Compare with the remark (a) in Section 7
of [33].
Proof of Theorem 2.5: Let D ≥ 1 be an integer, and let h ≥ e be a real number. Put
ϕ(D, h) = 16c2 D 2+1/n h(h + log D)(log h + log D)−1 ,
where c2 is as in Theorem 8.1. Assume that α1 , . . . , αn , β1 , . . . , βn are algebraic numbers
in a number field of degree ≤D, which satisfy
max h(α j ) ≤ h
1≤ j≤n
and
max h(β j ) ≤ h.
1≤ j≤n
We will show that, if h is sufficiently large, with a lower bound depending only on x1 , . . . , xn ,
then we have
max max{|x j − β j |, |e x j − α j |} ≥ exp{−ϕ(D, h)}.
1≤ j≤n
(8.5)
We argue by contradiction, assuming that (8.5) does not hold. Since ϕ(D, h) is an
increasing function of D, we may assume that D is the degree of the extension of Q
generated by α1 , . . . , αn and β1 , . . . , βn . For j = 1, . . . , n, denote by λ j the determination
of the logarithm of α j which is closest to x j . If h is sufficiently large, we obtain
n
X
|β j − λ j | < exp{−(1/2)ϕ(D, h)}.
j=1
Define
A = eh ,
B = Deh ,
B 0 = (Dh)2
and
E = (Dh)1/2 .
416
ROY AND WALDSCHMIDT
Then, if h is sufficiently large, all the hypotheses of Theorem 8.1 are satisfied, except
maybe the condition about linear combinations of β1 , . . . , βn . Moreover, the conclusion of
the theorem fails. Therefore, there are integers s1 , . . . , sn with
n
X
sjβj = 0
and
j=1
This implies
0 < max |s j | ≤ c1 (4D)1/n .
1≤ j≤n
¯
¯
n
¯X
¯
¯
¯
s j x j ¯ ≤ 4nc1 D 1/n exp{−ϕ(D, h)}
¯
¯ j=1
¯
(8.6)
which contradicts the hypothesis that x1 , . . . , xn satisfy a measure of linear independence
with exponent 2n + 1.
2
Remark. Note that, for fixed D, we get only finitely many possibilities for (s1 , . . . , sn ). For
each of them, (8.6) cannot hold if h is sufficiently large. This does not require that x1 , . . . , xn
satisfy a measure of linear independence but simply that they be linearly independent over
Q. Thus, (8.5) holds with this weaker condition for h ≥ h 0 (D, x1 , . . . , xn ).
Proof of Theorem 2.6: Let D ∈ N and h ∈ R satisfy D ≥ 1 and h ≥ e. Assume that
there exist algebraic numbers α1 , . . . , αn such that
[Q(α1 , . . . , αn , β1 , . . . , βn ) : Q] ≤ D,
max h(α j ) ≤ h
1≤ j≤n
and
©
ª
max |eβ j − α j | < exp −8c2 D 1+(1/n) h(log h + D log D)(log h + log D)−1
1≤ j≤n
with c2 as in Theorem 8.1. It remains to show that this is impossible if h is larger than
some constant h 0 depending only on β1 , . . . , βn . The argument is similar to the proof of
Theorem 2.5. We may again assume that D is the degree of the number field generated by
α1 , . . . , αn and β1 , . . . , βn . Set
A = eh ,
B = Dh 1/D ,
B 0 = (Dh)2 ,
E = (Dh)1/2 .
Since β1 , . . . , βn are linearly independent over Q, Theorem 8.1 yields the desired contradiction.
2
Remark. It is possible to refine Theorem 8.1 in such a way as to include Fel’dman’s simultaneous approximation measure for logarithms of algebraic numbers. A slight modification of
the proof in [33] is needed (cf. [33], Section 7, subsection e), where the basis 1, z, . . . , z n
of the space of polynomials of degree ≤ n is replaced by z(z − 1) · · · (z − j + 1)/j!,
(0 ≤ j ≤ n). See [9], Lemma 19.7, p. 127.
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
9.
417
Proof of Theorem 2.7
Let c0 be a real number ≥1, and let γ1 and γ2 be algebraic numbers. Put α = eλ , and define
K = Q(α, β, γ1 , γ2 ),
D = [K : Q]
and
h = max{3, h(γ1 ), h(γ2 )}.
Suppose
|λ − γ1 | + |eβλ − γ2 | ≤ e−3V
where
V = c06 D 2 h(h + log D)1/2 (log h + log D)−1/2 .
We will derive a contradiction assuming, at each step, that c0 is sufficiently large as a
function of β and λ alone.
Let m X 2 − bX − a be the irreducible polynomial of β over Z, and denote by log γ2 the
determination of the logarithm of γ2 which satisfies
|βλ − log γ2 | ≤ c0 e−3V .
We apply Theorem 5.1 to the group G = Ga × G2m with the points
w10 = (1, 1, β),
η10
η20
w1 = (1, 1, β),
= (λ, λ, βλ),
η1 = (γ1 , λ, log γ2 ),
= (mβλ, mβλ, mβ λ),
2
η2 = (mβγ1 , m log γ2 , aλ + b log γ2 ).
So, we have d0 = `0 = 1, d1 = `1 = 2, d = 3 and r = 1. The group G(K ) is K × (K × )2 ,
and its subgroup 0 is generated by
¡
¢
expG (η1 ) = (γ1 , α, γ2 ) and expG (η2 ) = mβγ1 , γ2m , α a γ2b .
Put
£ √ ¤
S = c02 D ,
U = c0−1 V,
and define the remaining parameters by
A1 = A2 = ec0 Sh , B1 = (Deh )c0 , B2 = (Dh)c0 , E = Dh,
·
·
¸
¸
U
U
, T1 = T2 = T = 7/2
T0 =
,
c0 D(h + log D)
c0 D 3/2 h
·
¸
U
and S1 = S2 = S.
S0 =
c0 D(log h + log D)
Then, all conditions of Theorem 5.1 are satisfied. In particular, the main condition (5.4)
holds and we have
S0 S 2 > 6T0 T 2
and
S0 S > 6T 2 .
418
ROY AND WALDSCHMIDT
Moreover, 0(S) is the set of points
¢
¡
(s1 + ms2 β)γ1 , α s1 γ2ms2 , α as2 γ2s1 +bs2 ,
(0 ≤ s1 , s2 < S).
By Theorem 5.1, there is a connected algebraic subgroup G ∗ of G, of dimension ≤2, such
that
¶
µ
0(S) + G ∗ (K )
H(G ∗ ; T0 , T1 , T2 ) ≤ 6T0 T 2 .
S0 Card
G ∗ (K )
The exponent of S0 in the left hand side is 1 because, since β is irrational, there is no
algebraic subgroup of G, apart from G itself, whose tangent space at the origin contains
w1 = (1, 1, β). Write G ∗ = G ∗0 × G ∗1 where G ∗0 is an algebraic subgroup of Ga and G ∗1 is
an algebraic subgroup of G2m . The factor G ∗0 is either {0} or Ga . If we had G ∗0 = {0}, then,
since β 6∈ Q and γ1 6= 0, we would get
¶
µ
0(S) + G ∗ (K )
≥ S 2 > 6T0 T 2 /S0 ,
Card
G ∗ (K )
which is not possible. Hence, we have G ∗ = Ga × G ∗1 and G ∗1 is an algebraic subgroup of
G2m of dimension ≤1. Denote by 01 (S) the projection of 0(S) on the factor G2m of G. This
set consists of the points
¡ s1 ms2 s1 ¡ a b ¢s2 ¢
,
(0 ≤ s1 , s2 < S).
α γ2 , γ2 α γ2
Then, we find
µ
01 (S) + G ∗1 (K )
Card
G ∗1 (K )
¶
≤ 6T 2 /S0 < S.
This last inequality shows that the images of (α, γ2 ) and (γ2m , α a γ2b ) in the quotient
(K × )2 /G ∗1 (K ) are torsion points (of order <S). Thus, there exist integers k1 , k2 , not
both zero, such that
¡
¢k
α k1 γ2k2 = 1 and γ2mk1 α a γ2b 2 = 1.
We deduce
(γ2 )mk1 +bk1 k2 −ak2 = 1.
2
2
The polynomial m X 2 − bX Y − aY 2 being irreducible over Q, this exponent is nonzero,
and so, γ2 is a root of unity. It follows that α is also a root of unity. Since γ2 belongs to K
and since K has degree D over Q, the order of γ2 must be ≤2D 2 . Write (log γ2 )/λ = p/q
with relatively prime integers p and q with q > 0. Since λ is fixed, we get q < c0 D 2 , and,
because β is quadratic irrational, Liouville’s inequality yields
|βλ − log γ2 | ≥
in contradiction with the choice of log γ2 .
1
1
≥ 3 4,
c0 q 2
c0 D
2
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
10.
419
Proof of Theorem 2.8
Denote by L the Q-vector space of logarithms of algebraic numbers:
L = exp−1 (Q̄× ).
The main result of this section is the following:
Theorem 10.1. Let d1 and `1 be positive integers and let M = (λi j ) be a d1 × `1 matrix
with coefficients in L. Set κ = (1/d1 ) + (1/`1 ). Denote by X the Q-vector subspace of
Cd1 which is spanned by the columns of M. Let r be the rank of M. Assume that, for any
∗
surjective linear map g : Cd1 → Cd1 which is defined over Q, the dimension `∗1 of the
Q-vector space g(X ) satisfies
`∗1 /`1 ≥ d1∗ /d1 .
Then, there exists a positive constant C such that the function
C Dr κ+1 (h + log D)r κ+1 (log D)−r κ−1
is a measure of simultaneous approximation for the d1 `1 numbers λi j , (1 ≤ i ≤ d1 ,
1 ≤ j ≤ `1 ).
Note that under the hypotheses of this theorem, we have r κ ≥ 1 by virtue of Corollary 7.2
of [30].
Proof: Let D be a positive integer, let c0 and h be positive real numbers with h ≥ c0 ≥ 3,
and let γi j , (1 ≤ i ≤ d1 , 1 ≤ j ≤ `1 ), be algebraic numbers. Assume that the number field
K generated over Q by the 2d1 `1 numbers
αi j := eλi j ,
γi j ,
(1 ≤ i ≤ d1 , 1 ≤ j ≤ `1 )
has degree ≤D. Suppose also
max h(γi j ) ≤ h
1≤i≤d1
1≤ j≤`1
and
where
V =
c03+4r κ
µ
max |λi j − γi j | ≤ exp(−2V )
1≤i≤d1
1≤ j≤`1
D(h + log D)
log(eD)
¶r κ+1
.
Note that, since r κ ≥ 1, we have V ≥ c03+4r κ D(h + log D). We will derive a contradiction
assuming, at each step, that c0 is a sufficiently large constant, independent of h, D and the
algebraic numbers γi j . This will prove the theorem.
420
ROY AND WALDSCHMIDT
Without loss of generality, we may assume D = [K : Q]. By permuting the rows and
columns of M, we may also assume that the principal minor of order r of M is nonzero:
det(λi j )1≤i, j≤r 6= 0.
We claim that the matrix M 0 = (γi j ) also has rank r , and that its principal minor of order r is
nonzero. To justify this, let I be a subset of {1, . . . , d1 } and let J be a subset of {1, . . . , `1 },
both with the same cardinality. We find
|det(λi j )i∈I, j∈J − det(γi j )i∈I, j∈J | ≤ c0 exp(−2V ) < exp(−V ).
Since there are only finitely many possibilities for I and J , we deduce det(γi j )i∈I, j∈J 6= 0
if det(λi j )i∈I, j∈J 6= 0. Conversely, if det(γi j )i∈I, j∈J 6= 0, then Liouville’s inequality gives
|det(γi j )i∈I, j∈J | ≥ exp(−c0 Dh).
Since V > c0 Dh, this implies det(λi j )i∈I, j∈J 6= 0. So, a minor of M vanishes if and only
if the corresponding minor of M 0 vanishes.
We apply Theorem 5.1 to the group G = Gra × Gdm1 . So, we have d0 = r and d = r + d1 .
We also choose `0 = `1 and define four families of `1 points by
¡
¢
w 0j = η0j = λ1 j , . . . , λr j ; λ1 j , . . . , λd1 j ∈ TG (C) = Cd ,
¡
¢
(1 ≤ j ≤ `1 ).
w j = γ1 j , . . . , γr j ; γ1 j , . . . , γd1 j ∈ TG (K ) = K d ,
¡
¢
d
η j = γ1 j , . . . , γr j ; λ1 j , . . . , λd1 j ∈ TG (C) = C ,
By construction, the subspace of Cd generated by w10 , . . . , w`0 1 and η10 , . . . , η`0 1 has dimension r . So, there is no conflict of notations concerning this parameter r . Consider the
matrices
¶
µ
¶
µ 0
M M
M M0
.
and L =
L0 =
M0 M
M M
The reader may also think of w10 , . . . , w`0 1 (resp. w1 , . . . , w`1 ) as the first `1 column vectors
of L0 (resp. L), and of η10 , . . . , η`0 1 (resp. η1 , . . . , η`1 ) as their last `1 column vectors. We
have G(K ) = K r × (K × )d1 , and the subgroup 0 of G(K ) is generated by the `1 points
¡
¢
γ j = expG (η j ) = γ1 j , . . . , γr j ; α1 j , . . . , αd1 j ,
(1 ≤ j ≤ `1 ).
Finally, W is the subspace of Cd spanned by w1 , . . . , w`1 . Define
B1 = B2 = B = (De ) ,
h c0
E = eD,
U=
c0−1 V,
·
T0 = S0 =
¸
U
.
D log B
The last two parameters satisfy T0 = S0 ≥ c04r κ . Let T and S be the integers given by
·µ
T =
c02
D log B1
log E
¶r/d1 ¸
,
S=
¶ ¸
·µ
D log B2 r/`1
c03
.
log E
421
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
Since log E ≤ D, these integers as well are positive. For the remaining parameters, we
choose
log A1 = · · · = log Ad1 = c0 S,
T1 = · · · = Td1 = T,
S1 = · · · = S`1 = S.
Then, all the hypotheses of Theorem 5.1 are satisfied. Therefore, there exists a connected
algebraic subgroup G ∗ of G, defined over K , incompletely defined in G by polynomials of
degree ≤T in each of the last `1 variables, and distinct from G, such that
`∗
S0 0
µ
¶
0(S) + G ∗ (K )
d!
Card
H(G ∗ ; T0 , T, . . . , T ) ≤ T0r T d1 ,
G ∗ (K )
r!
(10.1)
where
`∗0 = dim((W + TG ∗ (K ))/TG ∗ (K )).
K
Write G ∗ = G ∗0 × G ∗1 where G ∗0 and G ∗1 are algebraic subgroups of Gra and Gdm1 respectively.
Put
d0∗ = r − dim G ∗0
and
d1∗ = d1 − dim G ∗1 ,
and denote by λ∗ the largest integer for which there exist λ∗ elements γ1∗ , . . . , γλ∗∗ among
γ1 , . . . , γ`1 such that
λ∗
X
s j γ j∗ ∈
/ G ∗ (K )
j=1
∗
for any (s1 , . . . , sλ∗ ) ∈ Zλ with 0 < max1≤ j≤λ∗ |s j | < S. Then, (0(S) + G ∗ (K ))/G ∗ (K )
∗
has cardinality ≥S λ , and (10.1) implies
`∗
∗
∗
∗
d∗
∗
S0 0 S λ ≤ d! T0 0 T d1 .
Since S0 = T0 ≥ S, this gives
S `0 +λ
−d0∗
∗
≤ d! T d1 .
Finally, since S `1 ≥ (1/2)c0r T d1 , we deduce that either we have
d∗
`∗0 + λ∗ − d0∗
< 1,
`1
d1
(10.2)
or `∗0 + λ∗ − d0∗ = d1∗ = 0.
td
Let 8 be the set of points ϕ = (t1 , . . . , td1 ) ∈ Zd1 such that the monomial Y1t1 · · · Yd11
induces the trivial character on G ∗1 . Since G ∗1 is incompletely defined in Gdm1 by polynomials
of degree ≤T in each variable, Lemma 4.8 of [21] shows that 8 admits a basis ϕ1 , . . . , ϕd1∗
422
ROY AND WALDSCHMIDT
with |ϕk | ≤ c0 T for k = 1, . . . , d1∗ . Write ϕk = (tk1 , . . . , tkd1 ) for k = 1, . . . , d1∗ , and
∗
denote by g1 : Cd1 → Cd1 the linear map given by
!
Ã
d1
d1
X
X
¡
¢
t1i u i , . . . ,
td1∗ i u i
g1 u 1 , . . . , u d1 =
i=1
i=1
for any point (u 1 , . . . , u d1 ) ∈ Cd1 . By construction, this map is surjective, defined over Q,
and its kernel is TG ∗1 (C). Moreover, a point ξ = (u 1 , . . . , u d1 ) ∈ Cd1 satisfies
expGdm1 (ξ ) = (eu 1 , . . . , eu d1 ) ∈ G ∗1 (K )
√
∗
if and only if g1 (ξ ) ∈ (2π −1 Z)d1 .
Denote by ξ1 , . . . , ξ`1 the columns of the matrix M, and let X be the Q-vector subspace
of Cd1 that they generate. Denote by `∗1 the dimension of g1 (X ) over Q. Then, if (10.2)
holds, the hypothesis of the theorem gives
`∗1 > `∗0 − d0∗ + λ∗ .
(10.3)
We will show that, nor can this inequality hold, nor can we have `∗0 + λ∗ − d0∗ = d1∗ = 0.
The proof will then be complete.
∗
To show this, choose a linear map g0 : Cr → Cd0 defined over K whose kernel is TG ∗0 (C),
∗
∗
and define g : Cr × Cd1 → Cd0 × Cd1 to be the product map g = (g0 , g1 ). Since TG ∗ (C) is
the kernel of g, we have
`∗0 = dim g(W ).
√
∗
∗
∗
∗
Define Ä∗ = {0}d0 × (2π −1 Z)d1 ⊂ Cd0 × Cd1 . By construction, we also have, for any
(s1 , . . . , s`1 ) ∈ Z`1 ,
`1
X
s j γ j ∈ G ∗ (K ) ⇔
j=1
`1
X
s j g(η j ) ∈ Ä∗ .
j=1
Thus, λ∗ is also the largest integer for which there exist λ∗ elements η1∗ , . . . , ηλ∗ among
η1 , . . . , η`1 such that
λ∗
X
s j g(η∗j ) ∈
/ Ä∗
j=1
λ∗
for any (s1 , . . . , sλ∗ ) ∈ Z with 0 < max1≤ j≤λ∗ |s j | < S.
Denote by
π0 : Cd → Cr
(resp. π1 : Cd → Cd1 )
the projection on the first r coordinates, (resp. on the last d1 coordinates). Similarly, denote
by
¡
∗
∗
∗
∗
∗
∗¢
π0∗ : Cd0 × Cd1 → Cd0
resp. π1∗ : Cd0 × Cd1 → Cd1
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
423
the projection on the first d0∗ coordinates, (resp. on the last d1∗ coordinates). By construction,
∗
we have π0 (W ) = Cr . Since g0 ◦ π0 = π0∗ ◦ g, this implies π0∗ (g(W )) = Cd0 , and thus
`∗0 = d0∗ + dim(g(W ) ∩ ker(π0∗ )).
(10.4)
If we had `∗0 + λ∗ − d0∗ = d1∗ = 0, this would imply λ∗ = 0 and Ä∗ = {0}, and so, the
kernel TG ∗ (C) of g would contain η1 , . . . , η`1 . Since d1∗ = 0, this kernel would also contain
{0} × Cd1 . So, it would be all of Cd , in contradiction with the hypothesis G ∗ 6= G. It remains
to show that (10.3) cannot hold.
For simplicity, assume that g1 (ξ1 ), . . . , g1 (ξ`∗1 ) are linearly independent over Q. Since
ξ j = π1 (η j ) for j = 1, . . . , `1 , this implies that g(η1 ), . . . , g(η`∗1 ) also are linearly independent over Q. By virtue of (10.4), we have `∗0 − d0∗ + λ∗ ≥ λ∗ . So, (10.3) does not
hold if λ∗ ≥ `∗1 . Assume λ∗ < `∗1 , and define m = `∗1 − λ∗ . Then, there exist m linearly
∗
independent elements (s11 , . . . , s1`∗1 ), . . . , (sm1 , . . . , sm`∗1 ) of Z`1 with supremum norm <S
such that
∗
`1
X
sk j g(η j ) ∈ Ä∗ ,
(1 ≤ k ≤ m).
(10.5)
j=1
Let A be the (d0∗ + d1∗ ) × m matrix whose column vectors are given by (10.5). Since these
vectors are linearly independent over Q and belong to Ä∗ , the first r rows of A are zero, and
of
the rank of A is equal to m. Moreover, let A0 be a non-singular square matrix consisting
√
0
rows of A of indices, say, j1 , . . . , jm . Then, since
the
coefficients
of
A
belong
to
2π
−1
Z,
√
its determinant is an integral multiple of (2π −1)m . Now, consider the matrix B of the
same size as A, whose column vectors are
∗
`1
X
sk j g(w j ),
(1 ≤ k ≤ m).
(10.6)
j=1
Then, B also has rank m. To prove this, denote by B 0 the square matrix formed by the rows
of B of indices j1 , . . . , jm . The coefficients of A0 being bounded above, in absolute value,
by c02 T S, and the distance between A and B being ≤c02 T S exp(−2V ), we get
|det A0 − det B 0 | ≤ c02m+1 (T S)m exp(−2V ) < 1.
Since |det A0 | ≥ (2π)m , this implies det B 0 6= 0. Thus, B has rank m. Moreover, since η j
and w j have the same first r coordinates for any j, the first d0∗ rows of A and B are the same,
that is, equal to zero. Therefore, the vectors in (10.6) are m elements of W which are linearly
independent over C and belong to ker π0∗ . By virtue of (10.4), this gives `∗0 ≥ d0∗ + m and
2
thus `∗0 − d0∗ + λ∗ ≥ λ∗ + m = `∗1 . This shows that (10.3) does not hold.
Remark. The proof simplifies notably when all the logarithms λi j are real. In this case,
the condition expG (η) ∈ G ∗ is equivalent to g(η) = 0, and thus, we have λ∗ ≥ `∗1 . This
inequality combined with `∗0 ≥ d0∗ show that (10.3) is impossible. Moreover, this does not
424
ROY AND WALDSCHMIDT
require S0 ≥ S. This allows one to use a smaller value for V . The reader may check that,
in this case, one can use
¶
µ
D(h + log D) r κ
.
V = c03+4r κ D
log(eD)
Thus, there exists a constant C > 0 such that
C Dr κ+1 (h + log D)r κ (log D)−r κ
is a measure of approximation of the d1 `1 numbers λi j .
Corollary 10.2. Let W ⊂ C2n be the set of zeroes of the polynomial X 1 Y1 + · · · + X n Yn .
Let w = (x1 , . . . , xn , y1 , . . . , yn ) be a point in L2n ∩ W which does not belong to any vector
subspace of C2n defined over Q and contained in W . Then there exists a positive constant C
such that
C D 2 (h + log D)2 (log D)−2
is a measure of simultaneous approximation for the 2n numbers x1 , . . . , xn , y1 , . . . , yn .
Put N = 2n . The proof of Corollary 10.2 rests on the next lemma which involves the
map
θn : C2n → Mat N ×N (C)
defined in [21] (see the remark at the end of Section 7). This map has the form
¶
µ
0
ψn (v)
,
θn (v) =
0
ϕn (v)
where ϕn and ψn are C-linear maps from C2n to Mat2n−1 ×2n−1 (C). The latter maps are
defined by induction on n. For n = 1, they are given by
ϕ1 (x1 , y1 ) = (x1 ),
ψ1 (x1 , y1 ) = (y1 ).
For n ≥ 1 and v 0 = (x1 , . . . , xn+1 , y1 , . . . , yn+1 ) ∈ C2n+2 , we put
µ
¶
xn+1 I2n−1
ψn (v)
0
ϕn+1 (v ) =
,
−ϕn (v) yn+1 I2n−1
µ
¶
yn+1 I2n−1 −ψn (v)
,
ψn+1 (v 0 ) =
xn+1 I2n−1
ϕn (v)
where v = (x1 , . . . , xn , y1 , . . . , yn ), and where, for a positive integer m, Im denotes the
m × m identity matrix. The main properties of the map θn are that, for any v ∈ C2n written
as above, we have
θn (v)2 = (x1 y1 + · · · + xn yn )2 I2n
and that the matrix θn (v) has rank ≤N /2 when v ∈ W .
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
425
We will need the following lemma:
Lemma 10.3. Let W be as in Corollary 10.2, and let w = (x1 , . . . , xn , y1 , . . . , yn ) be a
point of this hypersurface. Set N = 2n and denote by X the Q-vector subspace of C N which
is spanned over Q by the column vectors of the matrix θn (w). Assume that there exists a
vector subspace U of C N , defined over Q, such that
dimQ (X/(X ∩ U )) < dimC (C N /U ).
Then w belongs to a vector subspace of C2n , defined over Q and contained in W .
Proof: Define
λ = dimQ (X/(X ∩ U )) and
δ = dimC (C N /U ).
There exist two matrices P and Q in GL N (Q) such that Pθn (w)Q can be written in block
form
µ
¶
A 0
,
B C
where the matrix A has size δ × λ. Let E be the vector subspace of C2n over C which
consists of the points v ∈ C2n such that Pθn (v)Q can be written in block form
µ
¶
A(v)
0
,
B(v) C(v)
where the matrix A(v) has size δ × λ. Then, E is defined over Q and contains w. Since
A(v) has rank ≤λ, the matrix θn (v), for v ∈ E, has rank ≤λ + d − δ < d. Hence, we get
2
det θn (v) = 0, so v ∈ W . This shows that E is contained in W .
Proof of Corollary 10.2: We apply Theorem 10.1 with d1 = `1 = N = 2n and M =
θn (w). Since the coefficients of θn (w) are
0, ±x1 , . . . , ±xn−1 , xn , ±y1 , . . . , ±yn−1 , yn ,
(10.7)
they all belong to L. As in Lemma 10.3, define X to be the Q-vector space generated by
∗
the columns of θn (w). According to this lemma, if g : Cd1 → Cd1 is a surjective linear
map defined over Q, and if we set U = ker g, then we have
`∗1 := dimQ g(X ) = dimQ (X/(X ∩ U )) ≥ dimC (Cd1 /U ) = d1∗ .
This uses the hypothesis that w does not belong to any vector subspace of Cn defined over
Q and contained in W . Since d1 = `1 , the above inequality shows that the last hypothesis
of Theorem 10.1 is satisfied. Moreover, since w ∈ W , the matrix θn (w) has rank r ≤ N /2.
We deduce that the family (10.7) admits a measure of simultaneous approximation of the
form
C Dr κ+1 (h + log D)r κ+1 (log D)−r κ−1
426
ROY AND WALDSCHMIDT
with
κ = 2/d1 = 2−n+1
and r ≤ d1 /2 = 2n−1 = 1/κ.
2
This yields the desired result.
Proof of Theorem 2.8: Write the polynomial Q in the form
X
ai j X i X j
Q(X 1 , . . . , X n ) =
1≤i, j≤n
with rational numbers a11 , . . . , ann . Define a C-linear map ϕ : Cn → C2n by
Ã
!
n
n
X
X
a1 j z j , . . . ,
an j z j ,
ϕ(z 1 , . . . , z n ) = z 1 , . . . , z n ,
j=1
j=1
and consider the point w = ϕ(λ1 , . . . , λn ). Since (λ1 , . . . , λn ) is a zero of Q in Ln , the
point w belongs to W ∩ L2n , where W is the hypersurface defined in Corollary 10.2. Let
E denote the smallest vector subspace of Cn , which is defined over Q, and contains w.
Since λ1 , . . . , λn are Q-linearly independent, and since E is defined over Q, E contains
ϕ(Cn ). Moreover, ϕ(Cn ) is not contained in W because the polynomial Q does not vanish
identically on Cn . Hence, E is not contained in W , and the conclusion follows from
Corollary 10.2 and Proposition 1.3.
2
Remark. One deduces Corollary 10.2 from Theorem 2.8 by considering a basis λ1 , . . . , λm
of the Q-vector space spanned by the 2n numbers x1 , . . . , xn , y1 , . . . , yn : from the hypothesis w ∈ W of Corollary 10.2 we deduce that λ1 , . . . , λm satisfy a nontrivial homogeneous
quadratic dependence relation. So, the two results are, in fact, equivalent.
11.
Proof of Theorem 4.1
Under the assumptions of Theorem 4.1, set α = ℘ (ω1 /2) and β = ℘ 0 (ω1 /2). Let K 0
denote the number field Q(g2 , g3 , α, β), and let c0 , D0 and h 0 be positive integers. Choose
algebraic numbers γ1 , γ2 , γ10 , γ20 , and set
K = K 0 (γ1 , γ2 , γ10 , γ20 ),
h = max{h 0 , h(γ1 ), h(γ2 ), h(γ10 ), h(γ20 )} and
D = max{D0 , [K : Q]}.
We will show that the hypothesis
©
ª
|ω1 − γ1 | + |ω2 − γ2 | + |η1 − γ10 | + |η2 − γ20 | ≤ exp −2c014 D 3/2 (h + log D)3/2
leads to a contradiction if c0 , D0 and h 0 are sufficiently large, independently of the choice
γ1 , γ2 , γ10 , γ20 , and from this the conclusion will follow.
Let E be the elliptic curve over K 0 which is associated with ℘, and let σ be the Weierstraß
sigma function associated with ℘. It is known that an extension G 1 of E by the additive group
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
427
Ga admits an embedding in P5 . Explicit formulas for the exponential map of G 1 (C) are
given by Hindry in [12]. These formulas identify the tangent space TG 1 (C) with C2 . Here,
we consider the nonisotrivial extension G 1 whose exponential map expG 1 : C2 → G 1 (C)
is given by the entire functions ϕ0 , . . . , ϕ5 , with
ϕ0 (z 1 , z 2 ) = σ (z 2 )3 ,
ϕ2 /ϕ0 (z 1 , z 2 ) = ℘ 0 (z 2 ),
ϕ1 /ϕ0 (z 1 , z 2 ) = ℘ (z 2 ),
1
ϕ4 /ϕ0 (z 1 , z 2 ) = ℘ (z 2 )(z 1 + ζ (z 2 )) + ℘ 0 (z 2 ),
2
ϕ3 /ϕ0 (z 1 , z 2 ) = z 1 + ζ (z 2 ),
ϕ5 /ϕ0 (z 1 , z 2 ) = ℘ 0 (z 2 )(z 1 + ζ (z 2 )) + 2℘ 2 (z 2 ).
We shall achieve the required contradiction by applying Theorem 2.1 of [33] to the product
G = Ga × G 1 . In the notations of this theorem, we thus have d0 = 1, d1 = 0, d2 = 2,
d = 3, n = 1 and δ1 = 2. The group G is defined over K 0 and therefore is defined over K .
Moreover, if we identify TG (C) with C × TG 1 (C) = C3 , then TG (K ) becomes identified
with K 3 . We choose
`0 = 1
and
w10 = w1 = (1, 0, 1).
Then, W = K w1 is a subspace of TG (K ) and we have W 0 = Cw10 . We will also take
V 0 = W 0 , thus r = r3 = 1 and r1 = r2 = 0. Define parameters A, B, . . . , V by
log A = c09 (h + log D),
log B = c0 (h + log D),
£
¤
T0 = S0 = c011 D 1/2 (h + log D)1/2
£
¤
T1 = c03 D 1/2 (h + log D)1/2
£
¤
S1 = c04 D 1/2 (h + log D)1/2
U = c013 D 3/2 (h + log D)3/2 ,
A1 = A,
B1 = B2 = B,
E =e
and
V = c0 U.
It is easy to check that they satisfy all the hypotheses in Section 2f) of [33]. Essentially,
one verifies
DT0 log B = DS0 log B ≤ U,
DT1 log A ≤ U,
and
B ≥ T0 + T1 + 3S0 .
The main condition in Section 2g) of [33] which imposes bounds on the value H (G; T0 , T1 )
of the Hilbert function of G follows from the inequalities
1 4
c U < T0 T12 ≤ c04 U.
2 0
We put M = S12 and we choose for η10 , . . . , η0M ∈ V 0 the following multiples of w10
((s1 + 1/2)ω1 + s2 ω2 , 0, (s1 + 1/2)ω1 + s2 ω2 ),
(0 ≤ s1 , s2 < S1 ).
428
ROY AND WALDSCHMIDT
Accordingly, we take
6 = {expG (η1 ), . . . , expG (η M )} ⊂ G(K )
where η1 , . . . , η M denote the points
((s1 + 1/2)γ1 + s2 γ2 , (s1 + 1/2)(γ10 − η1 ) + s2 (γ20 − η2 ), (s1 + 1/2)ω1 + s2 ω2 ),
with 0 ≤ s1 , s2 < S1 . If we compute the image of such a point under the exponential map
of G, we find that its projection on the factor Ga is
(s1 + 1/2)γ1 + s2 γ2 ,
and that its projection on G 1 is
(1 : α : β : (s1 + 1/2)γ10 + s2 γ20 : α((s1 + 1/2)γ10 + s2 γ20 )
+ β/2 : β((s1 + 1/2)γ10 + s2 γ20 ) + 2α 2 ).
The height of this last point in P5 (K ) is bounded above by
c0 (log S1 + h) ≤ log A.
Moreover, the height of the point (1 : w1 ) in P3 (K ) as well as the height of the point in
P M (K ) with projective coordinates 1 and (s1 + 1/2)γ1 + s2 γ2 , (0 ≤ s1 , s2 < S1 ), are both
bounded above by log B. Finally, for any real R ≥ 0 and any z ∈ C3 with |z| ≤ R, we have
−H − (R) ≤ log max{|ϕ0 (z)|, . . . , |ϕ5 (z)|}≤H + (R)
with H − (R) = c0 and H + (R) = c0 R 2 . Since all the points η j have norm |η j | ≤ c0 S1
and since
¡ 1/3 ¢
H + c0 S1 = c0 S12 ≤ D log A,
1/3
1/4
we find that all the conditions in Section 2e) of [33] are fulfilled.
Finally, our hypothesis gives |η0j − η j | ≤ e−V for j = 1, . . . , M, and we clearly have
|w10 − w1 | ≤ e−V . Therefore we can use Theorem 2.1 of [33]. It provides a connected
algebraic subgroup G ∗ of G, distinct from G, such that, if we define
W ∗ = (W + TG ∗ (K ))/TG ∗ (K ),
`∗0 = dim K W ∗
and
6 ∗ = (6 + G ∗ (K ))/G ∗ (K ),
M ∗ = Card 6 ∗ ,
then, since deg G 1 = 6 (see [12]), we have
`∗
S0 0 M ∗ H(G ∗ ; T0 , T1 ) ≤ 72T0 T12 .
(11.1)
Since G ∗ is a proper subgroup of G, it must be contained either in {0} × G 1 or in the
subgroup L of G whose tangent space inside TG (C) = C3 is C2 × {0}. This group L is
SIMULTANEOUS APPROXIMATION AND ALGEBRAIC INDEPENDENCE
429
isomorphic to G2a and yields a quotient G/L isomorphic to E. Since w1 does not belong to
{0} × C2 nor to C2 × {0}, we must have `∗0 = 1 in all cases. If G ∗ is contained in L and has
dimension 2, we find
M∗ = 1
and
H(G ∗ ; T0 , T1 ) ≥ 2T0 T1 .
If G ∗ is contained in L and has dimension 1, its tangent space cannot contain both (γ1 , γ10 , 0)
and (γ2 , γ20 , 0) because the determinant γ1 γ20 − γ2 γ10 is close to ω1 η2 − ω2 η1 which itself
is equal to ±2iπ by virtue of Legendre’s relation. Since every η j is congruent modulo the
group of periods of G to a point of the form
η + s1 (γ1 , γ10 , 0) + s2 (γ2 , γ20 , 0)
for some fixed η ∈ C3 , we deduce that, in that case, we have
M ∗ ≥ S1
and H(G ∗ ; T0 , T1 ) ≥ min{T0 , T1 } = T1 .
Finally, if G ∗ is contained in {0} × G 1 , then all elements of 6 are incongruent modulo
G ∗ (K ) and so we have
M ∗ = S12
and
H(G ∗ ; T0 , T1 ) ≥ 1.
Since T0 ≥ S1 ≥ T1 , we thus have, in all three cases, M ∗ H(G ∗ ; T0 , T1 ) ≥ S1 T1 and (11.1)
yields
S0 S1 T1 ≤ 72T0 T12 .
Since S0 = T0 and S1 > (c0 /2)T1 , this is the desired contradiction.
2
Acknowledgement
We are thankful to Deanna Caveny for her comments on a preliminary broken English
version of this text.
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