Linear Independence Measures for Logarithms of Algebraic Numbers

Michel Waldschmidt

Institut de Mathématiques, Université P. et M. Curie (Paris VI) miw@math.jussieu.fr

Cetraro, June 28 – July 8, 2000 CIME

Let α_{1}, …, α_{n} be nonzero algebraic numbers and b_{1}, …, b_{n} rational integers. Assume α_{1}^{b_{1}} ⋯ α_{n}^{b_{n}} ≠ 1. According to Liouville’s inequality (Proposition 1.19), the lower bound

|α_{1}^{b_{1}} ⋯ α_{n}^{b_{n}} − 1| ≥ e^{−cB}

holds with B = max{|b_{1}|, …, |b_{n}|} and with a positive number c depending only on α_{1}, …, α_{n}. A fundamental problem is to prove a sharper estimate.

Transcendence methods lead to linear independence measures, over the field of algebraic numbers, for logarithms of algebraic numbers. Such measures are nothing else than lower bounds for numbers of the form

Λ = β_{0} + β_{1} log α_{1} + ⋯ + β_{n} log α_{n},

where β_{0}, …, β_{n} are algebraic numbers, α_{1}, …, α_{n} are nonzero algebraic numbers, while log α_{1}, …, log α_{n} are logarithms of α_{1}, …, α_{n} respectively.

In the special case where β_{0} = 0 and β_{1}, …, β_{n} are rational integers, writing b_{i} for β_{i}, we have

Λ = b_{1} log α_{1} + ⋯ + b_{n} log α_{n},

which is the so-called homogeneous rational case. The importance of this special case is due to the fact that for |Λ| ≤ 1/2 we have

\frac{1}{2}|Λ| ≤ |e^{Λ} − 1| ≤ 2|Λ|

with

e^{Λ} − 1 = α_{1}^{b_{1}} ⋯ α_{n}^{b_{n}} − 1.

Hence we are back to the problem of estimating from below the distance between 1 and a number of the form α_{1}^{b_{1}} ⋯ α_{n}^{b_{n}}.

The first three lectures are devoted to the qualitative theory of transcendental numbers, the last three ones to the quantitative theory of Diophantine approximation.

According to Hermite-Lindemann’s Theorem, a number Λ = β − log α, with algebraic α and β, is zero only in the trivial case β = log α = 0. We start by assuming further that α and β are positive integers. It is a nontrivial fact that a positive integer b cannot be the logarithm of another positive integer a. In the first lecture we give two proofs of this result: the first one uses an auxiliary function, the second one uses an interpolation determinant together with a zero estimate. In the second lecture we complete the proof of

http://www.math.jussieu.fr/∼miw/articles/ps/Cetraro.ps             Typeset: 8 January 2002

Hermite-Lindemann’s Theorem in the general case (with algebraic α and β), by means of the interpolation determinant method, but without a zero estimate: this is achieved thanks to an extrapolation argument.

In the third lecture we introduce Baker’s Theorem on the linear independence of logarithms of algebraic numbers. After a brief survey of the available methods, we produce a proof by means of an interpolation determinant involving an extrapolation.

An introduction to Diophantine approximation is given in § 4, where we address the question of estimating from below the distance between b and log a, for a and b positive integers. A conjecture attributed to K. Mahler states that this distance should be at least a negative power of a:

|b − log a| ≥^{?} a^{−c} for a ≥ 2.

So far one does not know how to prove this result with a constant exponent, but only with exponent a constant times log log a (K. Mahler; see (4.4) below):

|b − log a| ≥ a^{−c log log a} for a ≥ 3.

We discuss a proof of this result by means of a method which is inspired by a recent work of M. Laurent and D. Roy [LaR].

The last two sections are devoted to Baker’s method and to the question of measures of linear independence for an arbitrary number of logarithms of algebraic numbers. In the fifth lecture we survey available methods and in the last one we explain how to replace Matveev’s auxiliary function by an interpolation determinant.

Notation. As a general rule we use the notation of [W5]. In particular the absolute logarithmic height is denoted by h. The length of a polynomial P ∈ ℂ[X_{1}, …, X_{n}] (which is nothing else than the sum of the absolute values of its coefficients) is denoted by L(P).

For \underline{z} = (z_{1}, …, z_{n}) ∈ ℂ^{n}, we set

|\underline{z}| = max_{1≤i≤n} |z_{i}| and ∥\underline{z}∥ = |z_{1}| + ⋯ + |z_{n}|.

In § 4.1, ∥ · ∥_{ℤ} denotes the distance of a real number to the nearest integer:

∥x∥_{ℤ} = min_{k∈ℤ} |x − k|.

For x ∈ ℝ we set

log_{+} x = log max{1, x} and |x|_{+} = max{1, |x|}.

The integral part of x is denoted by [x]:

[x] ∈ ℤ, 0 ≤ x − [x] < 1

while ⌈x⌉ denotes the least integer ≥ x:

⌈x⌉ ∈ ℤ, 0 ≤ ⌈x⌉ − x < 1.

For r ≥ 0 we denote by

B_{n}(0, r) = \{\underline{z} ∈ ℂ^{n} ; |\underline{z}| ≤ r\}

the closed polydisk of ℂ^{n} of center 0 and radius r, by |f|_{r} the supremum norm of a continuous function f: B_{n}(0, r) → ℂ and by H_{n}(r) the set of continuous functions f: B_{n}(0, r) → ℂ which are holomorphic in the interior of B_{n}(0, r).

Several differential operators will be used. For \underline{k} = (k_{1}, …, k_{n}) ∈ ℕ^{n} and F a function of n variables z_{1}, …, z_{n}, D\underline{k}F is the derivative

(\frac{∂}{∂z_{1}} )^{k_{1}} ⋯ (\frac{∂}{∂z_{n}} )^{k_{n}} F.

For a function F of a single variable z we write F^{(k)} in place of (d/dz)^{k}F.

The notation \underline{k}! stands for k_{1}! ⋯ k_{n}! and \underline{z}\underline{k} for z_{1}^{k_{1}} ⋯ z_{n}^{k_{n}}. In § 3 for \underline{x} = (x_{0}, …, x_{n}) ∈ ℂ^{n+1} we shall introduce also the notation

D\underline{x} = x_{0} \frac{∂}{∂z_{0}} + ⋯ + x_{n} \frac{∂}{∂z_{n}} ·

In § 4 we shall denote by D_{b} the following derivation, attached to a complex number b,

\frac{∂}{∂X} + bY \frac{∂}{∂Y} ,

on the ring ℂ[X, Y].

For n and k rational integers, the binomial coefficient

\binom{n}{k} = \frac{n!}{k!(n − k)!}

is considered to be 0 unless 0 ≤ k ≤ n.

The symmetric group on {1, …, L} will be denoted by 𝔖_{L}.

Acknowledgements. Many thanks to Stéphane Fischler who carefully checked a preliminary version of the first draft and made a lot of useful comments. It is also a great pleasure to thank Francesco Amoroso, Umberto Zannier and the Fondazione C.I.M.E. (Centro Internazionale Matematico Estivo) for their invitation to deliver these lectures in the superb surroundings provided by Grand Hotel San Michele in Cetraro (Cosenza).

Table of Contents

1. Introduction to Transcendence Proofs

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1 Sketch of Proof 1.2 Tools for the Auxiliary Function 1.3 Proof with an Auxiliary Function and without Zero Estimate 1.4 Tools for the Interpolation Determinant Method 1.5 Proof with an Interpolation Determinant and a Zero Estimate 1.6 Remarks

2. Extrapolation with Interpolation Determinants

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1 Upper Bound for a Determinant in a Single Variable 2.2 Proof of Hermite-Lindemann’s Theorem with an Interpolation

Determinant and without Zero Estimate

3. Linear Independence of Logarithms of Algebraic Numbers

. . . . . . . . . . . . . . . . . . . 30

3.1 Introduction to Baker’s Method 3.2 Proof of Baker’s Theorem 3.3 Further Extrapolation with the Auxiliary Function 3.4 Upper Bound for a Determinant in Several Variables 3.5 Extrapolation with an Interpolation Determinant

4. Introduction to Diophantine Approximation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 On a Conjecture of Mahler 4.2 Fel’dman’s Polynomials 4.3 Output of the Transcendence Argument 4.4 From Polynomial Approximation to Algebraic Approximation 4.5 Proof of Theorem 4.5

5. Measures of Linear Independence of Logarithms of Algebraic Numbers

. . . . . . . . 67

5.1 Introduction 5.2 Baker’s Method with an Auxiliary Function Appendix. From 2 to n Logarithms

6. Matveev’s Theorem with Interpolation Determinants

. . . . . . . . . . . . . . . . . . . . . . . 85

6.1 First Extrapolation 6.2 Using Kummer’s Condition 6.3 Second Extrapolation 6.4 An Approximate Schwarz Lemma for Interpolation Determinants

References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

1 Introduction to Transcendence Proofs

We shall provide two proofs of the following result.

Theorem 1.1. Let a and b be two positive integers. Then e^{b} ≠ a.

This statement is a special case of Hermite-Lindemann’s Theorem:

Theorem 1.2. Let α and β be two nonzero algebraic numbers. Then e^{β} ≠ α.

A proof of Theorem 1.2 will be given in § 2.

1.1 Sketch of Proof

Here are the basic ideas of both proofs of Theorem 1.1. The guest star of these proofs is the exponential function e^{z}. It is a transcendental function: this means that the exponential monomials z^{τ} e^{tz} (τ ≥ 0, t ≥ 0) are linearly independent. Consider the values at a point z = sb with s ∈ ℕ:

(z^{τ} e^{tz})(sb) = (sb)^{τ} (e^{b})^{ts}.

If both b and e^{b} are integers, then this number is also a rational integer.

We need to use a special property for the number e: Theorem 1.1 would not be true if e were replaced by 2 for instance! We take derivatives of our exponential monomials. For σ = 0, 1, …,

(1.3)  (\frac{d}{dz} )^{σ} (z^{τ} e^{tz}) = ∑_{κ=0}^{min\{τ,σ\}} \frac{σ!τ!}{κ!(σ − κ)!(τ − κ)!}z^{τ−κ}t^{σ−κ}e^{tz},

and

(\frac{d}{dz} )^{σ} (z^{τ} e^{tz})(sb) = ∑_{κ=0}^{min\{τ,σ\}} \frac{σ!τ!}{κ!(σ − κ)!(τ − κ)!}(sb)^{τ−κ}t^{σ−κ}(e^{b})^{ts}.

These numbers again belong to the ring ℤ[e^{b}, b].

Starting with these numbers, there are several ways of performing the proof. We indicate two of them.

The first one rests on the construction of an auxiliary polynomial (AP)^{1}. Since the functions z and e^{z} are algebraically independent, if P ∈ ℤ[X, Y] is a nonzero polynomial, the exponential polynomial F (z) = P (z, e^{z}) is not the zero function. Together with its derivatives, it takes values in ℤ[e^{b}, b] at all points sb, s ∈ ℕ. Assuming b and e^{b} are in ℤ, one wants to construct such a nonzero polynomial P for which F is the zero function, which will be a contradiction. Now the numbers F^{(σ)}(sb) are rational integers, hence have absolute value either 0 or at least 1; this is the lower bound (LB). If P is constructed so that many numbers F^{(σ)}(sb) have absolute value < 1, then F will have a lot of zeroes, hence (by a rigidity principle for analytic functions, called Schwarz’ Lemma) |F | will be small on a

1 With Masser’s notation in [Ma].

rather large disk: this is the upper bound (UB). This will enable us to deduce that F^{(σ)}(sb) vanishes for further values of (σ, s). Once we succeed in increasing the number of known zeroes of F, there is an alternative: either we proceed by induction and extrapolate until we get so many zeroes that F has to be the zero function (for instance if all derivatives of F vanish at one point), or else we prove an auxiliary result (the zero estimate, or non-vanishing condition (NV)) which yields the desired conclusion.

One should say a little bit more about the initial construction of P, for which many numbers F^{(σ)}(sb) have absolute value < 1. One solution is to select P so that the first coefficients in the Taylor expansion at the origin of F have small absolute values (see [W5] § 4.5 and [R]). Another (more classical) way is to require that many numbers F^{(σ)}(sb) vanish. Then the existence of P ≠ 0 is clear as soon as the number of equations we consider is smaller than the number of unknowns (the unknowns are the coefficients of P), because the conditions are linear and homogeneous. In the special case we consider here with b a real positive number, it can be proved that when the number of unknowns is the same as the number of equations, then the determinant of the system is not zero (Pólya’s Lemma 1.10). Therefore the extrapolation can be reduced to the minimum. On the other hand for the proof of Theorem 1.1 such an argument is not required.

The second method was suggested by M. Laurent [La1]: instead of solving a system of homogeneous linear equations F^{(σ)}(sb) = 0 for several values of (σ, s), consider the matrix of this system. To be more precise the matrix one considers is the one which arises from the zero estimate: if the zero estimate shows, for a given set of pairs (σ, s), that no nonzero polynomial P (with suitable bounds for its degree) can satisfy all equations F^{(σ)}(sb) = 0, then the matrix of the associated linear system has maximal rank. Consider a maximal nonsingular submatrix and its nonzero determinant Δ. The main observation of M. Laurent is that a sharp upper bound for |Δ| can be reached by means of Schwarz’ Lemma. Since Δ lies in the ring ℤ[e^{b}, b], as soon as the estimate 0 < |Δ| < 1 is established one deduces that one at least of the two numbers b, e^{b} is not a rational integer.

1.2 Tools for the Auxiliary Function

We introduce four main tools for the proof of Theorem 1.1 by means of an auxiliary function: Liouville’s inequality (LB), Schwarz’ Lemma (UB), the Zero Estimate (NV) and Thue-Siegel’s Lemma (AP).

We shall use here only a trivial case of Liouville’s inequality (a more general statement is Proposition 1.19 below): (1.4) For any n ∈ ℤ with n ≠ 0 we have |n| ≥ 1.

The analytic upper bound for our auxiliary function is a consequence of the following Schwarz’ Lemma (see [W5] Exercise 4.3, [G] and also Lemma 1.18 for a quantitative refinement):

Lemma 1.5. Let m, σ_{1}, …, σ_{m} be positive integers and r, R positive real numbers with r ≤ R. Let ζ_{1}, …, ζ_{m} be distinct elements in the disk |ζ| ≤ R and F ∈ H_{1}(R) an analytic function which vanishes at each ζ_{i} with multiplicity ≥ σ_{i} (1 ≤ i ≤ m). Then

|F |_{r} ≤ |F |_{R} ∏_{i=1}^{m} (\frac{R^{2} + r|ζ_{i}|}{R(r + |ζ_{i}|)})^{−σ_{i}} ·

We recall the definition: a function F vanishes at a point ζ with multiplicity ≥ σ if F^{(k)}(ζ) = 0 for 0 ≤ k < σ.

In our applications, we shall introduce a parameter E > 1 such that R ≥ Er and R ≥ E|ζ_{i}| for 1 ≤ i ≤ m. The conclusion yields

|F |_{r} ≤ |F |_{R} (\frac{E^{2} + 1}{2E})^{−N}

where N = σ_{1} + ⋯ + σ_{m} is a lower bound for the number of zeroes of F in the disk |z| ≤ R/E. In practice N will be large, E ≥ e, and E^{−N} will be the main term in the right hand side. In particular |F |_{R} will not be too large, and from the conclusion of Lemma 1.5 we shall infer that |F |_{r} is quite small.

For our first transcendence proof the zero estimate is a very simple one:

• If F is a nonzero analytic function near z_{0}, there exists σ ∈ ℕ such that

F^{(σ)}(z_{0}) ≠ 0.

Our last tool is Thue-Siegel’s Lemma:

Lemma 1.6. Let m and n be positive integers with n > m and a_{ij} (1 ≤ i ≤ n, 1 ≤ j ≤ m) rational integers. Define

A = max\{1, max_{{1≤i≤n \atop 1≤j≤m}} |a_{ij}|\}.

There exist rational integers x_{1}, …, x_{n} which satisfy

0 < max{|x_{1}|, …, |x_{n}|} ≤ (nA)^{m/(n−m)}

and

∑_{i=1}^{n} a_{ij}x_{i} = 0 for 1 ≤ j ≤ m.

For a proof of this result, we refer for instance to [B2] Lemma 1, Chap. 2, [F] Theorem 6.1, Chap. 1 § 6.1, [FN] Theorem 1.10, Chap. 1 § 4.1 or [L1] Lemma 1, Chap. VII § 2.

1.3 Proof with an Auxiliary Function and without Zero Estimate

Here is a first proof of Theorem 1.1. Let b be a positive integer such that e^{b} is also a positive integer. We denote by T_{0}, T_{1}, S_{0} and S_{1} positive integers which we shall choose later: during the proof we shall introduce conditions on these parameters and at the end of the proof we shall check that it is possible to select the parameters so that these conditions are satisfied. Right now let us just say that these integers will be sufficiently large.

We want to deduce from Lemma 1.6 that there exists a nonzero polynomial P ∈ ℤ[X, Y], of degree < T_{0} in X and degree < T_{1} in Y, such that the exponential polynomial F (z) = P (z, e^{z}) has a zero of multiplicity ≥ S_{0} at each point 0, b, 2b, …, (S_{1} −1)b. If this unknown polynomial P is

P (X, Y) = ∑_{τ=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} c_{τt}X^{τ} Y^{t},

then the conditions

F^{(σ)}(sb) = 0 (0 ≤ σ < S_{0}, 0 ≤ s < S_{1})

can be written

∑_{τ=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} c_{τt} (\frac{d}{dz} )^{σ} (z^{τ} e^{tz})(sb) = 0 (0 ≤ σ < S_{0}, 0 ≤ s < S_{1}).

Finding P amounts to solving a system of S_{0}S_{1} linear equations, with rational integers coefficients, in T_{0}T_{1} unknowns c_{τt} (0 ≤ τ < T_{0}, 0 ≤ t < T_{1}). We are going to apply Lemma 1.6 with n = T_{0}T_{1} and m = S_{0}S_{1}. In place of the condition n > m we shall require n ≥ 2m, so that the so-called “Dirichlet’s exponent” m/(n − m) is at most 1. This yields the first main condition on our parameters:

T_{0}T_{1} ≥ 2S_{0}S_{1}.

We need an upper bound for the number A occurring in Lemma 1.6. Consider (1.3). We wish to estimate from above the modulus of the complex number

∑_{κ=0}^{min\{τ,σ\}} \frac{σ!τ!}{κ!(σ − κ)!(τ − κ)!}z^{τ−κ}t^{σ−κ}

for |z| ≤ R with R > 0, and for 0 ≤ t < T_{1}. A first upper bound is given by

∑_{κ=0}^{τ} \frac{τ!}{κ!(τ − κ)!}σ^{κ}R^{τ−κ}T_{1}^{σ−κ} = T_{1}^{σ} (\frac{σ}{T_{1}} + R)^{τ}

and another one is

∑_{κ=0}^{σ} \frac{σ!}{κ!(σ − κ)!}τ^{κ}R^{τ−κ}T_{1}^{σ−κ} = R^{τ} (\frac{τ}{R} + T_{1})^{σ} .

Putting these estimates together yields

sup_{|z|≤R} | ∑_{κ=0}^{min\{τ,σ\}} \frac{σ!τ!}{κ!(σ − κ)!(τ − κ)!}z^{τ−κ}t^{σ−κ}| ≤

(1.7) R^{τ} T_{1}^{σ} min \{(1 + \frac{σ}{T_{1}R})^{τ} ; (1 + \frac{τ}{T_{1}R})^{σ}\} .

Similar estimates are known for more general exponential polynomials, also in several variables: see for instance [W5] Lemmas 4.9 and 13.6.

Here we shall not use the full force of this estimate. Taking z = bs, R = bS_{1} we deduce

A ≤ T_{1}^{S_{0}}(bS_{1} + S_{0})^{T_{0}}e^{bT_{1}(S_{1}−1)}.

Hence a nonzero polynomial P exists, satisfying the required conditions and with

max_{{0≤τ<T_{0} \atop 0≤t<T_{1}}} |c_{τt}| ≤ T_{0}T_{1}A.

We need an upper bound for the length

L(P) = ∑_{τ=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} |c_{τt}|

of P. As soon as T_{0}, T_{1}, S_{0}, S_{1} are sufficiently large, we have

bS_{1} ≤ \frac{1}{4}bS_{0}S_{1}, S_{0} ≤ \frac{1}{4}bS_{0}S_{1}, T_{0}^{2} ≤ 2^{T_{0}}, T_{1}^{2} ≤ e^{bT_{1}},

hence

bS_{1} + S_{0} ≤ \frac{1}{2}bS_{0}S_{1}, T_{0}^{2}(bS_{1} + S_{0})^{T_{0}} ≤ (bS_{0}S_{1})^{T_{0}}

and therefore

L(P) ≤ T_{0}^{2}T_{1}^{2}A ≤ T_{1}^{S_{0}}(bS_{0}S_{1})^{T_{0}}e^{bT_{1}S_{1}}.

Since the two functions z and e^{z} are algebraically independent, the function F is not the zero function: at each point sb with 0 ≤ s < S_{1} its vanishing order is finite (and ≥ S_{0}, by construction). We denote by S′_{0} the minimum of these orders. In other terms S′_{0} is the largest integer such that the conditions

F^{(σ)}(sb) = 0 (0 ≤ σ < S′_{0}, 0 ≤ s < S_{1})

hold and therefore there is an integer s′ in the range 0 ≤ s′ < S_{1} with

F^{(S′_{0})}(s′b) ≠ 0.

An upper bound for S′_{0} follows from Lemma 1.10 below, namely:

S′_{0}S_{1} ≤ T_{0}T_{1},

but this estimate will not be used in the present proof: we need only the lower bound S′_{0} ≥ S_{0}.

By assumption the number F^{(S′_{0})}(s′b) is a nonzero rational integer, hence has absolute value ≥ 1. We shall deduce from Cauchy’s inequalities and Schwarz’ Lemma 1.5 an upper bound for this number in terms of the parameters T_{0}, T_{1}, S_{0} and S_{1}. It will then suffice to check that the parameters can be selected so that this upper bound is less than 1 and the contradiction will follow.

Cauchy’s inequalities yield

|F^{(S′_{0})}(s′b)| ≤ S′_{0}!|F |_{r}

for any r ≥ s′b + 1. We take r = 2bS_{1}. Next we apply Lemma 1.5 with R = Er, where E > 1 is a new parameter which we are free to choose. As we shall see a suitable choice is E = S′_{0}/b; notice that E, which is selected at this stage of the proof, is allowed to depend on S′_{0}, while S′_{0} in turn depends on T_{0}, T_{1}, S_{0} and S_{1}.

Define m = S_{1}, σ_{1} = ⋯ = σ_{m} = S′_{0} and ζ_{i} = (i − 1)b (1 ≤ i ≤ m). Since max_{1≤i≤m} |ζ_{i}| ≤ r, by Lemma 1.5 we have

|F |_{r} ≤ (\frac{E^{2} + 1}{2E})^{−S′_{0}S_{1}} |F |_{R}.

It remains to bound |F |_{R} from above:

|F |_{R} ≤ ∑_{τ=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} |c_{τt}| sup_{|z|=R}|z^{τ} e^{tz}|
≤ T_{1}^{S_{0}}(bS_{0}S_{1})^{T_{0}}e^{bT_{1}S_{1}}R^{T_{0}}e^{T_{1}R}
≤ T_{1}^{S_{0}}(bES_{0}S_{1})^{2T_{0}}e^{3bET_{1}S_{1}}.

Hence

|F |_{r} ≤ (E/2)^{−S′_{0}S_{1}}T_{1}^{S_{0}}(bES_{0}S_{1})^{2T_{0}}e^{3bET_{1}S_{1}}.

This explains our second main condition on the parameters: taking into account the inequality S′_{0}!|F |_{r} ≥ 1, we shall deduce the desired contradiction as soon as we are able to check

S′_{0}!T_{1}^{S_{0}}(bES_{0}S_{1})^{2T_{0}}e^{3bET_{1}S_{1}} < (E/2)^{S′_{0}S_{1}}.

Here is an admissible choice for these parameters. Recall that b is a fixed positive integer. We start by selecting a sufficiently large, but fixed, positive integer S_{1} and we set T_{1} = S_{1}. Next let S_{0} be an integer, which is much larger than S_{1}; the required estimates below are easy to check by letting S_{0} → ∞. Now define T_{0} = 3S_{0} and E = S′_{0}/b. With this choice we have

T_{0}T_{1} > 2S_{0}S_{1},
S′_{0}! < (S′_{0})^{S′_{0}} < (E/2)^{S′_{0}S_{1}/4}

because S′_{0} < E^{2} < (E/2)^{S_{1}/4},

T_{1}^{S_{0}} < (E/2)^{S′_{0}S_{1}/4}

because T_{1} < E/2 < (E/2)^{S_{1}/4} and S_{0} ≤ S′_{0},

bES_{0}S_{1} < (S′_{0})^{3} and (S′_{0})^{6T_{0}} < (E/2)^{S′_{0}S_{1}/4},

and finally

3bET_{1}S_{1} = 3S′_{0}T_{1}S_{1} < \frac{1}{4}S′_{0}S_{1} log(E/2).

This completes the proof of Theorem 1.1. □

Remark. To a certain extent this proof of Theorem 1.1 involves an extrapolation: we get more and more derivatives of F vanishing at all points 0, b, …, (S_{1} − 1)b. It is only a matter of presentation: instead of defining S′_{0} as we did, it amounts to the same to check by induction on S′_{0} ≥ S_{0} that F has a zero of multiplicity at least S′_{0} at each point sb with 0 ≤ s < S_{1}. At the end of the induction we get a contradiction.

We could also extrapolate on the points at the same time as on the derivatives. Here is a variant of the proof.

We may assume b ≥ 3^{2}. Fix a large^{3} positive integer N and set

T_{0}(N) = 2N^{2}b[log b], T_{1}(N) = N^{2}[log b],
S_{0}(N) = N^{3}b[log b], S_{1}(N) = N[log b]

and L(N) = T_{0}(N)T_{1}(N), so that

L(N) = 2N^{4}b[log b]^{2} = 2S_{0}(N)S_{1}(N).

The first step in the preceding proof yields a nonzero polynomial P ∈ ℤ[X, Y] of degree < T_{0}(N) in X and < T_{1}(N) in Y, of length bounded by

L(P) ≤ T_{1}^{S_{0}}(bS_{0}S_{1})^{T_{0}}e^{bT_{1}S_{1}}
≤ exp{3L(N)(log N)/N}
≤ exp\{L(N)/\sqrt{N}\},

such that the function F (z) = P (z, e^{z}) satisfies

F^{(σ)}(sb) = 0 for 0 ≤ σ < S_{0}(N) and 0 ≤ s < S_{1}(N).

The second step is an inductive argument: we prove that for any M ≥ N we have

(1.8)  F^{(σ)}(sb) = 0 for 0 ≤ σ < S_{0}(M) and 0 ≤ s < S_{1}(M).

This is true by construction for M = N. Assuming (1.8) is true for M, we deduce it for M + 1 as follows. Let (σ′, s′) ∈ ℕ^{2} satisfy 0 ≤ σ′ < S_{0}(M + 1) and 0 ≤ s′ < S_{1}(M + 1). Combining the induction hypothesis with Lemma 1.5 where we choose

2 For the proof of Theorem 1.1, this involves no loss of generality and the only reason for this assumption is that we prefer to write log b in place of log_{+} b. For the same reason when we shall need to introduce log log b later we shall assume b ≥ 16.

3 We assume that N is larger than some absolute constant (independent of b); here one could assume as well than N is larger than some function of b, but it turns out not to be necessary. The relevance of this fact will appear in § 4 only.

m = S_{1}(M), ζ_{i} = (i − 1)b (1 ≤ i ≤ m), σ_{1} = ⋯ = σ_{m} = S_{0}(M),
r = 2bS_{1}(M + 1) and R = 2er,

we deduce

|F |_{r} ≤ e^{−S_{0}(M)S_{1}(M)}|F |_{R}
≤ e^{−S_{0}(M)S_{1}(M)}L(P)R^{T_{0}(N)}e^{T_{1}(N)R}
≤ e^{−S_{0}(M)S_{1}(M)/2},

because

log L(P) + T_{0}(N) log R + T_{1}(N)R ≤ \frac{1}{2}S_{0}(M)S_{1}(M).

Since

log(S_{0}(M + 1)!) ≤ \frac{1}{4}S_{0}(M)S_{1}(M),

Cauchy’s inequalities yield

|F^{(σ′)}(s′b)| < 1

for 0 ≤ σ′ < S_{0}(M + 1) and 0 ≤ s′ < S_{1}(M + 1). Since the left hand side is a rational integer, we deduce F^{(σ′)}(s′b) = 0 and the inductive argument follows.

Plainly we conclude F = 0, which completes this new proof of Theorem 1.1.

Remark. In this inductive argument from M to M + 1, the first step (with M = N) is the hardest one: as soon as M is large with respect to N, the required estimates are easier to check.

1.4 Tools for the Interpolation Determinant Method

Some tools which have already been introduced above will be needed for the proof involving interpolation determinants. For instance Liouville’s inequality is just the same (1.4). On the other hand in place of Schwarz’ Lemma 1.5 we shall use M. Laurent’s fundamental observation that interpolation determinants have a small absolute value (see [La1] § 6.3, Lemma 3 and [La2] § 6, Lemma 6). The following estimate ([W5] Lemma 2.8) is a consequence of the case m = 1, ζ_{1} = 0 of Lemma 1.5.

Lemma 1.9. Let φ_{1}, …, φ_{L} be entire functions in ℂ, ζ_{1}, …, ζ_{L} elements of ℂ, σ_{1}, …, σ_{L} nonnegative integers and 0 < r ≤ R real numbers, with |ζ_{μ}| ≤ r (1 ≤ μ ≤ L). Then the absolute value of the determinant

Δ = det(φ_{λ}^{(σ_{μ})}(ζ_{μ}))_{1≤λ,μ≤L}

is bounded from above by

|Δ| ≤ (\frac{R}{r})^{−(L(L−1)/2)+σ_{1}+···+σ_{L}} L! ∏_{λ=1}^{L} max_{1≤μ≤L} sup_{|z|=R} |φ_{λ}^{(σ_{μ})}(z)| .

The zero estimate we need is the following result due to G. Pólya ([W5] Corollary 2.3):

Lemma 1.10. Let w_{1}, …, w_{n} be pairwise distinct real numbers, x_{1}, …, x_{m} also pairwise distinct real numbers and τ_{1}, …, τ_{n}, σ_{1}, …, σ_{m} nonnegative integers, with

τ_{1} + ⋯ + τ_{n} = σ_{1} + ⋯ + σ_{m}.

Choose any ordering for the pairs (τ, i) with 0 ≤ τ < τ_{i} and 1 ≤ i ≤ n and any ordering for the pairs (σ, j) with 0 ≤ σ < σ_{j} and 1 ≤ j ≤ m. Then the square matrix

((\frac{d}{dz} )^{σ}(z^{τ} e^{w_{i}z})(x_{j}))_{{(τ,i) \atop (σ,j)}}

is nonsingular.

We call this result a zero estimate because it can be stated as follows: if c_{τi} are complex numbers (0 ≤ τ < τ_{i}, 1 ≤ i ≤ n), not all of which are zero, then the exponential polynomial

f(z) = ∑_{i=1}^{n} ∑_{τ=0}^{τ_{i}−1} c_{τi}z^{τ} e^{w_{i}z}

cannot vanish at each x_{j} with multiplicity ≥ σ_{j} (1 ≤ j ≤ m).

One main characteristic of Laurent’s interpolation determinant method is that there is no need of Thue-Siegel’s Lemma 1.6.

1.5 Proof with an Interpolation Determinant and a Zero Estimate

Here is another proof of Theorem 1.1.

We start with a positive real number b ≥ 1, without any other assumption. We introduce auxiliary parameters T_{0}, T_{1}, S_{0} and S_{1}, which are positive integers and E > 1 a real number. These parameters will be specified later, but it is convenient to assume T_{0}, T_{1}, S_{0} and S_{1} are all ≥ 2.

Consider the matrix

M = ((\frac{d}{dz} )^{σ}(z^{τ} e^{tz})(sb))_{{0≤τ<T_{0}, 0≤t<T_{1} \atop 0≤σ<S_{0}, 0≤s<S_{1}}} ,

with T_{0}T_{1} rows labeled with (τ, t) and S_{0}S_{1} columns labeled with (σ, s). Here we shall work with a square matrix, which means that we require

T_{0}T_{1} = S_{0}S_{1}.

We denote by L this number, so that M is a square L × L matrix. By Lemma 1.10 with n = T_{1},

w_{i} = i − 1, τ_{i} = T_{0} (1 ≤ i ≤ n),

m = S_{1} and

x_{j} = (j − 1)b, σ_{j} = S_{0} (1 ≤ j ≤ m),

it follows that M is nonsingular. Let Δ be the determinant of M. By Lemma 1.9 with

{φ_{1}, …, φ_{L}} = {z^{τ} e^{tz} ; 0 ≤ τ < T_{0}, 0 ≤ t < T_{1}},
{(σ_{1}, ζ_{1}), …, (σ_{L}, ζ_{L})} = {(σ, sb) ; 0 ≤ σ < S_{0}, 0 ≤ s < S_{1}}

and r = bS_{1}, R = Er, we have

|Δ| ≤ E^{−L(L−1−S_{0})/2}L! ∏_{τ=0}^{T_{0}−1} ∏_{t=0}^{T_{1}−1} max_{0≤σ<S_{0}} sup_{|z|=R} |(\frac{d}{dz} )^{σ} (z^{τ} e^{tz})| .

Since, for 0 ≤ σ < S_{0} and |z| = R, we have by (1.7) and (1.3)

\begin{aligned}|(\frac{d}{dz} )^{σ} (z^{τ} e^{tz})| & ≤ T_{1}^{S_{0}}(R + S_{0})^{T_{0}}e^{T_{1}R} \\ & ≤ T_{1}^{S_{0}}(bES_{0}S_{1})^{T_{0}}e^{bET_{1}S_{1}}, \end{aligned}

we deduce

\begin{aligned}|Δ| & ≤ E^{−L(L−1−S_{0})/2}L!T_{1}^{LS_{0}}(bES_{0}S_{1})^{LT_{0}}e^{bELT_{1}S_{1}} \\ & ≤ E^{−L^{2}/2}L^{L}(ET_{1})^{LS_{0}}(bES_{0}S_{1})^{LT_{0}}e^{bELT_{1}S_{1}}. \end{aligned}

This estimate holds unconditionally. If we can select our parameters so that |Δ| < 1, then this will prove that the nonzero number Δ cannot be a rational integer, hence one at least of b and e^{b} is not a rational integer. Therefore the proof of Theorem 1.1 will be completed if we show that our parameters may be selected so that

E^{L/2} > L(ET_{1})^{S_{0}}(bES_{0}S_{1})^{T_{0}}e^{bET_{1}S_{1}}.

Here is an admissible choice: let N be a sufficiently large positive integer (independent of b). Assuming b ≥ 3, define E = e,

T_{0} = N^{2}b[log b], T_{1} = N^{2}[log b],
S_{0} = N^{3}b[log b], S_{1} = N[log b],

so that L = N^{4}b[log b]^{2}.

This completes the proof of Theorem 1.1. □

1.6 Remarks

In this last proof of Theorem 1.1, we did not need to assume that b and e^{b} are integers: Liouville’s inequality (1.4) is used at the very end and provides the conclusion. More precisely it is plain that the interpolation determinant method of § 1.5 yields the following explicit result.

Proposition 1.11. Let b be a positive real number. Let T_{0}, T_{1}, S_{0}, S_{1} and L be positive integers satisfying

L = T_{0}T_{1} = S_{0}S_{1}.

Let E be a positive number, E > 1. Then there exists a polynomial f ∈ ℤ[Z_{1}, Z_{2}], of degree < LT_{1}S_{1} in Z_{1} and < LT_{0} in Z_{2}, of length bounded by

L(f) ≤ L!T_{1}^{LS_{0}}(S_{0}S_{1})^{LT_{0}},

such that

0 < |f(e^{b}, b)| ≤ E^{−L^{2}/2}L!(ET_{1})^{LS_{0}}(|b|_{+}ES_{0}S_{1})^{LT_{0}}e^{|b|ELT_{1}S_{1}}.

We now explain how to modify the first proof (in § 1.3) involving an auxiliary function and deduce the following variant of Proposition 1.11.

Proposition 1.12. Let b be a positive real number. Let T_{0}, T_{1}, S_{0}, S_{1} and L be positive integers such that

L = T_{0}T_{1} = S_{0}S_{1}.

Let E, U, V, W be positive real numbers satisfying

E ≥ e, W ≥ 12 log E,
(1.13)  U ≥ log L + T_{0} log(|b|_{+}ES_{1}) + |b|_{+}ET_{1}S_{1}

and

(1.14)  4(U + V + W)^{2} ≤ LW log E.

There exists a nonzero polynomial f ∈ ℤ[Z_{1}, Z_{2}] of degree < T_{1}S_{1} in Z_{1} and < T_{0} in Z_{2}, of length

L(f) ≤ Le^{W} T_{1}^{S_{0}}(S_{0} + S_{1})^{T_{0}},

such that

0 < |f(e^{b}, b)| ≤ S_{0}!e^{−V} .

Remark. It is interesting to compare the two estimates provided by Propositions 1.11 and 1.12. Condition (1.14) is satisfied with^{4}

4 This choice yields a weak upper bound for the length of f. From this point of view a better choice is for instance U = V = (1/20N)L log E and W = U/N with N ≥ 5.

U = V = W = \frac{1}{36}L log E.

Up to terms of smaller order (when T_{0}, T_{1}, S_{0} and S_{1} are all sufficiently large), the estimates one deduces from Proposition 1.11 for the degrees, the logarithm of the length and the logarithm of the absolute value are L times the corresponding ones in Proposition 1.12.

For all practical purposes, Proposition 1.12, which is obtained by the auxiliary function method, is much sharper that Proposition 1.11. This fact has been an obstacle during a while to develop the interpolation determinant method. For instance it took several years before proofs of algebraic independence results could be achieved by means of Laurent’s interpolation determinant method. A nice solution has been provided by M. Laurent and D. Roy in [LaR], who point out that the polynomial f given by the proof of Proposition 1.11 has a further quite interesting property: its first derivatives

(\frac{∂}{∂Z_{1}} )^{k_{1}} (\frac{∂}{∂Z_{2}} )^{k_{2}} f

with (k_{1}, k_{2}) ∈ ℕ^{2} satisfying, say, k_{1} + k_{2} < L/2, also have a small absolute value at the point (e^{b}, b). We shall develop this argument later (see Theorem 4.10).

Our proof of Proposition 1.12 uses an auxiliary function. Since we do not assume b and e^{b} are integers, we cannot apply Lemma 1.6 as we did in § 1.3. Instead of solving linear equations, we select (again by means of Dirichlet’s box principle) the coefficients c_{τt} of the auxiliary polynomial P so that a set of inequalities is satisfied. There are several possibilities. Here we shall use the following auxiliary function.

Lemma 1.15. Let L be a positive integer, U, V, W, R, r positive real numbers and φ_{1}, …, φ_{L} functions in H_{1}(R). Assume

U + V + W ≥ 12, e ≤ \frac{R}{r} ≤ e^{(U+V +W)/6}, ∑_{λ=1}^{L} |φ_{λ}|_{R} ≤ e^{U}

and

4(U + V + W)^{2} ≤ LW log(R/r).

Then there exist rational integers p_{1}, …, p_{L}, with

0 < max_{1≤λ≤L} |p_{λ}| ≤ e^{W} ,

such that the function F = p_{1}φ_{1} + ⋯ + p_{L}φ_{L} satisfies

|F |_{r} ≤ e^{−V} .

We do not give the proof of Lemma 1.15 (see [W5] Proposition 4.10, which provides a similar statement in several variables). It suffices to say that it combines Dirichlet’s box principle (Lemma 1.16) with an interpolation formula (Lemma 1.17).

Here is Lemma 4.12 of [W5].

Lemma 1.16. Let ν, μ, X be positive integers, U, V positive real numbers and u_{ij} (1 ≤ i ≤ ν, 1 ≤ j ≤ μ) complex numbers. Assume

∑_{i=1}^{ν} |u_{ij}| ≤ e^{U}, (1 ≤ j ≤ μ)

and

(\sqrt{2}Xe^{U+V} + 1)^{2μ} ≤ (X + 1)^{ν}.

Then there exists (ξ_{1}, …, ξ_{ν}) ∈ ℤ^{ν} satisfying

0 < max_{1≤i≤ν} |ξ_{i}| ≤ X

and

max_{1≤j≤μ} |∑_{i=1}^{ν} u_{ij}ξ_{i}| ≤ e^{−V} .

The next result is Lemma 4.13 of [W5] (in case r = 0 we agree that r^{∥\underline{k}∥} = 1 for \underline{k} = \underline{0}). For the proof of Lemma 1.15, the case n = 1 suffices, but we shall need the general case in § 4.4.

Lemma 1.17. Let n, K be positive integers, r and R real numbers satisfying 0 ≤ r < R and F an entire function in ℂ^{n}. Then

|F |_{r} ≤ (1 + \sqrt{K}) (\frac{r}{R})^{K} |F |_{R} + ∑_{∥\underline{k}∥<K} |D\underline{k}F (0)|\frac{r^{∥\underline{k}∥}}{\underline{k}!} ·

Proof of Proposition 1.12. We apply Lemma 1.15 to the functions

{φ_{1}, …, φ_{L}} = {z^{τ} e^{tz} ; 0 ≤ τ < T_{0}, 0 ≤ t < T_{1}}

with

r = |b|_{+}S_{1}, R = Er.

From hypothesis (1.13) we derive

∑_{λ=1}^{L} |φ_{λ}|_{R} = ∑_{τ=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} sup_{|z|=R}|z^{τ} e^{tz}| ≤ LR^{T_{0}}e^{T_{1}R} ≤ L(|b|_{+}ES_{1})^{T_{0}}e^{|b|_{+}ET_{1}S_{1}} ≤ e^{U}.

We deduce the existence of a nonzero polynomial

P (X, Y) = ∑_{τ=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} c_{τt}X^{τ} Y^{t} ∈ ℤ[X, Y],

of degree < T_{0} in X and < T_{1} in Y, with integer coefficients bounded in absolute value by e^{W}, such that the function F (z) = P (z, e^{z}) satisfies

|F |_{r} ≤ e^{−V} .

By Lemma 1.10 there is a nonzero element γ in the set

{F^{(σ)}(sb) ; 0 ≤ σ < S_{0}, 0 ≤ s < S_{1}}.

From Cauchy’s inequality, and since r ≥ s|b| + 1, we deduce the upper bound

|γ| ≤ S_{0}!|F |_{r} ≤ S_{0}!e^{−V} .

Writing

γ = F^{(σ)}(sb),

define f ∈ ℤ[Z_{1}, Z_{2}] by

∑_{τ=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} c_{τt} ∑_{κ=0}^{min\{τ,σ\}} \frac{σ!τ!}{κ!(σ − κ)!(τ − κ)!}s^{τ−κ}t^{σ−κ}Z_{1}^{ts}Z_{2}^{τ−κ}

so that, using (1.3), we can write

γ = f(e^{b}, b).

The degrees of f plainly satisfy the required conditions in the conclusion of Proposition 1.12, and finally the length of f is bounded thanks to (1.7):

L(f) ≤ ∑_{τ=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} |c_{τt}| ∑_{κ=0}^{min\{τ,σ\}} \frac{σ!τ!}{κ!(σ − κ)!(τ − κ)!}s^{τ−κ}t^{σ−κ} ≤ Le^{W} T_{1}^{S_{0}}(S_{0} + S_{1})^{T_{0}}.

□

One can prove a variant of Proposition 1.12 by constructing the auxiliary function F (z) = P (z, e^{z}) in a slightly different way. One applies Lemma 1.16 again, but now we require that many values at points sb of the function F and of its first derivatives have a small absolute value^{5}. A rigidity principle for analytic functions (Lemma 1.18) enables us to deduce that |F |_{r} is rather small for a suitable parameter r. We are back to the situation of our first proof: we invoke Pólya’s Lemma 1.10 and produce a nonzero value of F (or of one of its derivatives). This nonzero number is the value at the point (e^{b}, b) of a polynomial f ∈ ℤ[Z_{1}, Z_{2}] which satisfies the desired conclusion.

Lemma 1.17 is quite simple, since only one point z = 0 is involved. For functions of a single variable one can consider an arbitrary finite set of points^{6}. Here is Lemma 5.1 of [R].

5 Lemma 1.15 is proved in [W5] by constructing P so that the first Taylor coefficients of F at the origin have a small absolute value; hence it may be considered as a variant of this approach, which consists in taking only s = 0 at this stage of the proof - the number b does not occur in this case.

6 Such a statement is called an “approximate Schwarz’ Principle” in [P2] § 3.a.

Lemma 1.18. Let ℓ be a positive integer, w_{1}, …, w_{ℓ} pairwise distinct complex numbers and m_{1}, …, m_{ℓ} positive integers. Put

L = ∑_{j=1}^{ℓ} m_{j},  ϱ = max_{1≤j≤ℓ} max {1, |w_{j}|},
δ_{1} = min_{1≤j≤ℓ} ∏_{{1≤j′≤ℓ \atop j′≠j}} |w_{j} − w_{j′}|^{m_{j′} /L},  δ_{2} = min \{1 , min_{{1≤j,j′≤ℓ \atop j≠j′}} |w_{j} − w_{j′}|^{m_{j′} /L}\},

with the convention that δ_{1} = δ_{2} = 1 when ℓ = 1. Then, for any pair of real numbers r and R with R ≥ 2r and r ≥ 2ϱ and for any function F ∈ H_{1}(R), we have

|F |_{r} ≤ (\frac{6r}{δ_{1}δ_{2}} )^{L} max_{{1≤j≤ℓ \atop 0≤κ<m_{j}}} \frac{1}{κ!} |F^{(κ)}(w_{j})| + (\frac{6r}{R} )^{L}|F |_{R}.

Each of the two Propositions 1.11 and 1.12 yields the real case of Hermite-Lindemann’s Theorem 1.2: in place of the trivial Liouville’s inequality (1.4) we have used so far, it suffices to invoke the next result, which is Proposition 3.14 of [W5]:

Proposition 1.19 (Liouville’s Inequality). Let 𝕂 be a number field of degree D, v an Archimedean absolute value of 𝕂 and ν_{1}, …, ν_{ℓ} positive integers. For 1 ≤ i ≤ ℓ, let γ_{i1}, …, γ_{iν_{i}} be elements of 𝕂. Further, let f be a polynomial in ν_{1} + ⋯ + ν_{ℓ} variables, with coefficients in ℤ, which does not vanish at the point \underline{γ} = (γ_{ij})_{1≤j≤ν_{i},1≤i≤ℓ}. Assume f has total degree at most N_{i} with respect to the ν_{i} variables corresponding to γ_{i1}, …, γ_{iν_{i}}. Then

log |f(\underline{γ})|_{v} ≥ −(D − 1) log L(f) − D ∑_{i=1}^{ℓ} N_{i}h(1: γ_{i1}: ⋯: γ_{iν_{i}}).

Finally the complex case of Hermite-Lindemann’s Theorem 1.2 can also be proved easily by the same arguments, either with an auxiliary function or with an interpolation determinant. The only new feature is to replace Pólya’s Lemma 1.10 by another zero estimate, for instance Lemma 4.7. We refer to [B2] Chap. 1 § 3, [F] Chap. 1 § 9, [FN] Chap. 2 § 2 and [W1] § 3.1 for proofs of the Hermite-Lindemann’s Theorem by means of an auxiliary function and to [W5] Chap. 2 for the interpolation determinant method with a zero estimate.

In the next section we provide a new proof of Hermite-Lindemann’s Theorem 1.2 by means of an interpolation determinant but without any zero estimate: we shall extrapolate like in § 1.3.

2 Extrapolation with Interpolation Determinants

The proof given in § 1.3 rests on an auxiliary function and involves an extrapolation; this extrapolation enabled us to conclude without using the zero estimate Lemma 1.10. We explain here how to perform an extrapolation by means of the interpolation determinant method of § 1.5.

2.1 Upper Bound for a Determinant in a Single Variable

We are looking for an upper bound for an interpolation determinant. Lemma 1.9 is proved by M. Laurent in [La2] § 6 (also in [W5] Lemma 2.8) by means of Schwarz’ Lemma 1.5 for the function

Φ : z ⟼ det(φ_{λ}^{(σ_{μ})}(zζ_{μ}))_{1≤λ,μ≤L}

which has a zero at the origin of multiplicity at least

0 + 1 + ⋯ + (L − 1) − (σ_{1} + ⋯ + σ_{L}) = \frac{L(L − 1)}{2} − ∑_{μ=1}^{L} σ_{μ}.

Example. (See Masser’s Lecture 1 in [Ma]). Define

φ(z) = z + z^{2} + z^{4} + z^{8} + ⋯ = ∑_{m=0}^{∞} z^{2^{m}}.

Set L = 6 and take for φ_{1}, …, φ_{6} the functions

1, z, φ(z), zφ(z), φ^{2}(z), zφ^{2}(z).

Further set

σ_{1} = 0, σ_{2} = 1, σ_{3} = 2, σ_{4} = 3, σ_{5} = 4, σ_{6} = 0

and

ζ_{1} = ζ_{2} = ζ_{3} = ζ_{4} = ζ_{5} = 0, ζ_{6} = 1.

The function

Φ(z) ∏_{μ=1}^{6} \frac{1}{σ_{μ}!} = det(\frac{1}{σ_{μ}!}φ_{λ}^{(σ_{μ})}(zζ_{μ}))_{1≤λ,μ≤L}

= det ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & z \\ 0 & 1 & 1 & 0 & 1 & φ(z) \\ 0 & 0 & 1 & 1 & 0 & zφ(z) \\ 0 & 0 & 1 & 2 & 1 & φ^{2}(z) \\ 0 & 0 & 0 & 1 & 2 & zφ^{2}(z) \end{matrix} )

= 2zφ^{2}(z) + 4zφ(z) − 3φ^{2}(z) + z − φ(z) has a zero of multiplicity ≥ 5 at the origin; as pointed out by D.W. Masser [Ma], actually the multiplicity is 6.

Back to the general case, we need to take into account further zeroes. Such an upper bound is given in Corollary 2.4 of [W4]; the proof relies on a Schwarz Lemma for Cartesian products (see [W4] Proposition 2.3; see also [G] for a general discussion of this issue). Philippon ([P2] Lemme 4) also gave upper estimates for interpolation determinants and he does not need to deal with Cartesian products: he uses a much more simple inductive argument which suffices for interpolation determinants (but does not seem to extend to Cartesian products). Here we follow his approach.

We first combine Schwarz’ Lemma 1.5 with Cauchy’s inequalities.

Lemma 2.1. Let R, r, ϱ and E be positive real numbers, F an element of H_{1}(R), m a positive integer, ζ_{1}, …, ζ_{m} pairwise distinct complex numbers, ξ a complex number and κ, σ_{1}, …, σ_{m} nonnegative integers. Set

N = σ_{1} + ⋯ + σ_{m}.

Assume

R ≥ max\{r, |ξ| + ϱ\}, r ≥ max_{1≤i≤m} |ζ_{i}| and 1 ≤ E ≤ \frac{R^{2} + r(|ξ| + ϱ)}{R(r + |ξ| + ϱ)}.

Assume also that F satisfies

F^{(σ)}(ζ_{i}) = 0 for 0 ≤ σ < σ_{i} and 1 ≤ i ≤ m.

Then

|F^{(κ)}(ξ)| ≤ κ!ϱ^{−κ}E^{−N}|F |_{R}.

Proof. By Cauchy’s inequalities

|F^{(κ)}(ξ)| ≤ κ!ϱ^{−κ}|F |_{|ξ|+ϱ}.

Since F has at least N zeroes (counting multiplicities) in the disk B_{1}(0, r), we deduce from Schwarz’ Lemma 1.5:

|F |_{|ξ|+ϱ} ≤ E^{−N}|F |_{R}.

Hence the result. □

The next result is an extension of Corollary 2.4 of [W4] where we include multiplicities.

Proposition 2.2. Let R be a positive real number, φ_{1}, …, φ_{L} elements of H_{1}(R) with L ≥ 1, ξ_{1}, …, ξ_{L} complex numbers in B_{1}(0, R) and κ_{1}, …, κ_{L} nonnegative integers. Consider the determinant

Δ = det (φ_{λ}^{(κ_{μ})}(ξ_{μ}))_{1≤λ,μ≤L} .

Further let m_{1}, …, m_{L} be nonnegative integers and, for 1 ≤ μ ≤ L and 1 ≤ i ≤ m_{μ}, let ζ_{μi} be a complex number and σ_{μi} a nonnegative integer. We assume that for each μ = 1, …, L, the m_{μ} numbers ζ_{μ1}, …, ζ_{μm_{μ}} are pairwise distinct. Set

N_{μ} = ∑_{i=1}^{m_{μ}} σ_{μi} (1 ≤ μ ≤ L).

For 1 ≤ μ ≤ L, let r_{μ}, R_{μ}, ϱ_{μ} and E_{μ} be positive real numbers satisfying

R ≥ R_{μ} ≥ max{r_{μ}, |ξ_{μ}| + ϱ_{μ}}, r_{μ} ≥ max_{1≤i≤m_{μ}} |ζ_{μi}|

and

1 ≤ E_{μ} ≤ \frac{R_{μ}^{2} + r_{μ}(|ξ_{μ}| + ϱ_{μ})}{R_{μ}(r_{μ} + |ξ_{μ}| + ϱ_{μ})}·

Denote by Φ the analytic mapping

(φ_{1}, …, φ_{L}): B_{1}(0, R) → ℂ^{L}.

Assume that for any (μ, i, κ) ∈ ℕ^{3} satisfying 1 ≤ μ ≤ L, 1 ≤ i ≤ m_{μ} and 0 ≤ κ < σ_{μi}, the μ vectors

(2.3)  Φ^{(κ_{1})}(ξ_{1}), …, Φ^{(κ_{μ−1})}(ξ_{μ−1}), Φ^{(κ)}(ζ_{μi})

in ℂ^{L} are linearly dependent. Then

|Δ| ≤ L! max_{τ∈𝔖_{L}} ∏_{μ=1}^{L}(κ_{μ}!ϱ_{μ}^{−κ_{μ}}E_{μ}^{−N_{μ}}|φ_{τ(μ)}|_{R_{μ}}).

Proof. We prove Proposition 2.2 by induction on L. For L = 1 we have Φ = φ_{1},

Δ = Φ^{(κ_{1})}(ξ_{1}), 1 ≤ E_{1} ≤ \frac{R_{1}^{2} + r_{1}(|ξ_{1}| + ϱ_{1})}{R_{1}(r_{1} + |ξ_{1}| + ϱ_{1})}

and hypothesis (2.3) reads

Φ^{(κ)}(ζ_{1i}) = 0 for 0 ≤ κ < σ_{1i} and 1 ≤ i ≤ m_{1}.

From Lemma 2.1 we deduce

|Δ| ≤ κ_{1}!ϱ_{1}^{−κ_{1}}E_{1}^{−N_{1}}|φ_{1}|_{R_{1}}.

Hence Proposition 2.2 is true in case L = 1.

Assume now that the conclusion is true for L replaced by L − 1. Define F ∈ B_{1}(0, R) by

F (z) = det (Φ^{(κ_{1})}(ξ_{1}), …, Φ^{(κ_{L−1})}(ξ_{L−1}), Φ(z)) .

By assumption (2.3) with μ = L, for 1 ≤ i ≤ m_{L} and 0 ≤ κ < σ_{Li}, we have

F^{(κ)}(ζ_{Li}) = 0.

Since

Δ = F^{(κ_{L})}(ξ_{L}),

we deduce from Lemma 2.1

|Δ| ≤ κ_{L}!ϱ_{L}^{−κ_{L}}E_{L}^{−N_{L}}|F |_{R_{L}}.

We expand the determinant F with respect to the last column: define, for 1 ≤ λ ≤ L,

Φ_{λ} = (φ_{1}, …, φ_{λ−1}, φ_{λ+1}, …, φ_{L}): B_{1}(0, R) → ℂ^{L−1}

and let Δ_{λ} denote the determinant of the (L − 1) × (L − 1) matrix

(Φ_{λ}^{(κ_{1})}(ξ_{1}), …, Φ_{λ}^{(κ_{L−1})}(ξ_{L−1})) ,

so that

F (z) = ∑_{λ=1}^{L}(−1)^{L−λ}φ_{λ}(z)Δ_{λ}.

Hence

|F |_{R_{L}} ≤ L max_{1≤λ≤L} |φ_{λ}|_{R_{L}}|Δ_{λ}|

and therefore

|Δ| ≤ κ_{L}!Lϱ_{L}^{−κ_{L}}E_{L}^{−N_{L}} max_{1≤λ≤L} |φ_{λ}|_{R_{L}}|Δ_{λ}|.

We fix an index λ^{0} ∈ {1, …, L} such that

|Δ| ≤ κ_{L}!Lϱ_{L}^{−κ_{L}}E_{L}^{−N_{L}}|φ_{λ^{0}}|_{R_{L}}|Δ_{λ^{0}}|.

Using the induction hypothesis, we deduce that there exists a bijective map t from {1, …, L − 1} to {1, …, λ^{0} − 1, λ^{0} + 1, …, L} such that

|Δ_{λ^{0}}| ≤ (L − 1)! ∏_{μ=1}^{L−1} (κ_{μ}!ϱ_{μ}^{−κ_{μ}}E_{μ}^{−N_{μ}}|φ_{t(μ)}|_{R_{μ}}) .

Define τ ∈ 𝔖_{L} by τ(μ) = t(μ) for 1 ≤ μ < L and τ(L) = λ^{0}. Proposition 2.2 follows. □

We shall use a special case of Proposition 2.2.

We consider a finite sequence (ζ_{0}, …, ζ_{N}) of complex numbers, which are not supposed to be pairwise distinct. We define the associated multiplicity sequence (σ_{0}, …, σ_{N}) as follows:

σ_{ν} = Card{i; 0 ≤ i < ν , ζ_{i} = ζ_{ν}} (0 ≤ ν ≤ N).

If ζ_{0}, …, ζ_{N} are pairwise distinct then σ_{0} = ⋯ = σ_{N} = 0. In general, (ζ_{0}, …, ζ_{N}) consists of ℓ distinct complex numbers w_{1}, …, w_{ℓ}, where w_{j} is repeated m_{j} times (1 ≤ j ≤ ℓ), so that

∏_{ν=0}^{N}(z − ζ_{ν}) = ∏_{j=1}^{ℓ}(z − w_{j})^{m_{j}} and m_{1} + ⋯ + m_{ℓ} = N + 1.

Then for an analytic function F the N + 1 equations

F^{(σ_{ν})}(ζ_{ν}) = 0 for 0 ≤ ν ≤ N

are nothing else than

F^{(κ)}(w_{j}) = 0 for 0 ≤ κ < m_{j} and 1 ≤ j ≤ ℓ.

The pairs (σ_{ν}, ζ_{ν}) (0 ≤ ν ≤ N) are pairwise distinct and for each ν = 0, …, N and each σ = 0, …, σ_{ν} there is an index μ in the range 0 ≤ μ ≤ ν with (σ_{μ}, ζ_{μ}) = (σ, ζ_{ν}).

Corollary 2.4. Let L, N be integers with 1 ≤ L ≤ N + 1, R a positive real number and (ζ_{0}, …, ζ_{N}) a sequence of N+1 elements in B_{1}(0, R). Denote by (σ_{0}, …, σ_{N}) the associated multiplicity sequence. Let 0 = ν_{0} ≤ ν_{1} < ⋯ < ν_{L} ≤ N be integers and φ_{1}, …, φ_{L} elements of H_{1}(R). Consider the determinant

Δ = det (φ_{λ}^{(σ_{νμ})}(ζ_{ν_{μ}}))_{1≤λ,μ≤L} .

For 1 ≤ μ ≤ L, let r_{μ}, R_{μ}, ϱ_{μ} and E_{μ} be positive real numbers satisfying

R ≥ R_{μ} ≥ max{r_{μ}, |ζ_{ν_{μ}}| + ϱ_{μ}}, r_{μ} ≥ max_{0≤ν<ν_{μ}} |ζ_{ν}|

and

1 ≤ E_{μ} ≤ \frac{R_{μ}^{2} + r_{μ}(|ζ_{ν_{μ}}| + ϱ_{μ})}{R_{μ}(r_{μ} + |ζ_{ν_{μ}}| + ϱ_{μ})}·

Denote by Φ the analytic mapping

(φ_{1}, …, φ_{L}): B_{1}(0, R) → ℂ^{L}.

Assume that for any (μ, ν) ∈ ℕ^{2} satisfying 1 ≤ μ ≤ L and ν_{μ−1} ≤ ν < ν_{μ}, the μ vectors

Φ^{(σ_{ν_{1}} )}(ζ_{ν_{1}}), …, Φ^{(σ_{ν_{μ−1}} )}(ζ_{ν_{μ−1}}), Φ^{(σ_{ν})}(ζ_{ν})

in ℂ^{L} are linearly dependent. Then

|Δ| ≤ L! max_{τ∈𝔖_{L}} ∏_{μ=1}^{L}(σ_{ν_{μ}}!ϱ_{μ}^{−σ_{νμ}} E_{μ}^{−ν_{μ}}|φ_{τ(μ)}|_{R_{μ}}).

Proof. We apply Proposition 2.2 with

ξ_{μ} = ζ_{ν_{μ}} and κ_{μ} = σ_{ν_{μ}} (1 ≤ μ ≤ L).

We define m_{μ}, ζ_{μi} and σ_{μi} as follows: for 1 ≤ μ ≤ L, we denote by m_{μ} the number of distinct elements in the sequence (ζ_{0}, …, ζ_{ν_{μ}−1}), by ζ_{μi} these distinct elements and by σ_{μi} the number of ν in the range 0 ≤ ν < ν_{μ} such that ζ_{ν} = ζ_{μi}. Therefore

N_{μ} = σ_{μ1} + ⋯ + σ_{μm_{μ}} = ν_{μ}.
□

Remark. Corollary 2.4 includes Corollary 2.4 of [W4]: if some σ_{ν_{μ}} is zero, then ϱ_{μ}^{−σ_{νμ}} = 1 even if we replace ϱ_{μ} by 0 in the definitions of R_{μ} and E_{μ}. Another special case of Corollary 2.4 is related to Lemma 2.5 of [W5], which is nothing else than the case σ_{1} = ⋯ = σ_{L} = 0 of Lemma 1.9. Indeed we can take in Corollary 2.4

ζ_{0}, …, ζ_{L−1} pairwise distinct, N = L − 1, σ_{0} = ⋯ = σ_{L−1} = 0,
ν_{μ} = μ − 1, R_{μ} = R, r_{μ} = r, E_{μ} = (1 ≤ μ ≤ L),

with

R ≥ r ≥ max{|ζ_{0}|, …, |ζ_{L−1}|}.

Since ν_{1} + ⋯ + ν_{L} = L(L − 1)/2, we deduce from Corollary 2.4

|det(φ_{λ}(ζ_{μ}))_{1≤λ,μ≤L}| ≤ (\frac{R^{2} + r^{2}}{2rR} )^{−L(L−1)/2} L! ∏_{λ=1}^{L} sup_{|z|=R} |φ_{λ}(z)| .

Apart from the quantity (R^{2} + r^{2})/2rR which replaces R/r, this is the estimate of Lemma 2.5 of [W5].

It is not clear to me whether Proposition 2.2 contains the general case of Lemma 1.9 (without the restriction σ_{1} = ⋯ = σ_{L} = 0).

2.2 Proof of Hermite-Lindemann’s Theorem with an Interpolation Determinant and without Zero Estimate

Thanks to Corollary 2.4, one can modify the proof of § 1.5 involving an interpolation determinant so that Lemma 1.10 (zero estimate) is not required any more. In this section we explain how to extrapolate and to increase either the number of derivatives, or the number of points, or both.

Proof of Hermite-Lindemann’s Theorem 1.2. Let α and β be two complex numbers with β ≠ 0 and α = e^{β}.

Step 1. Introducing the Parameters

Consider two nondecreasing sequences (S_{0}(N))_{N≥0} and (S_{1}(N))_{N≥0} of nonnegative integers with S_{0}(0) = S_{1}(0) = 0 and such that the sequence (S_{0}(N)S_{1}(N))_{N≥0} is increasing.

We construct a sequence ζ_{0}, ζ_{1}, … as follows. For each N ≥ 0, the sequence

(ζ_{S_{0}(N)S_{1}(N)}, …, ζ_{S_{0}(N+1)S_{1}(N+1)−1})

consists of

• each element sβ with 0 ≤ s < S_{1}(N) repeated S_{0}(N + 1) − S_{0}(N) times, and

• each element sβ with S_{1}(N) ≤ s < S_{1}(N + 1) repeated S_{0}(N + 1) times. Denote by (σ_{0}, σ_{1}, …, σ_{ν}, …) the associated multiplicity sequence.

For each ν ≥ 0 denote by N_{ν} the least integer N ≥ 1 for which ν < S_{0}(N)S_{1}(N). Hence we have N_{0} = 1 and for ν ≥ 0 (2.5) S_{0}(N_{ν} − 1)S_{1}(N_{ν} − 1) ≤ ν < S_{0}(N_{ν})S_{1}(N_{ν}), σ_{ν} < S_{0}(N_{ν}) and |ζ_{ν}| < S_{1}(N_{ν})|β|.

We also introduce two sufficiently large integers T_{0} and T_{1} and we set L = T_{0}T_{1}.

Step 2. The Matrix M and the Determinant Δ

Consider the matrix with L rows and infinitely many columns

M = (C_{0}, C_{1}, …, C_{ν}, …),

where C_{ν} is the column vector (with L = T_{0}T_{1} rows)

((\frac{d}{dz} )^{σ_{ν}} (z^{τ} e^{tz})(ζ_{ν}))_{0≤τ<T_{0}, 0≤t<T_{1}} .

We claim that the rank of M is L. Indeed a linear relation between the rows

∑_{τ_{0}=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} c_{τt}(\frac{d}{dz} )^{σ_{ν}} (z^{τ} e^{tz})(ζ_{ν}) = 0 for ν ≥ 0

would mean that the exponential polynomial

F (z) = ∑_{τ_{0}=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} c_{τt}z^{τ} e^{tz}

satisfies

F^{(σ_{ν})}(ζ_{ν}) = 0 for ν ≥ 0.

These relations can also be written

F^{(σ)}(sb) = 0 for σ ≥ 0 and s ≥ 0,

and they plainly imply c_{τt} = 0 for 0 ≤ τ < T_{0} and 0 ≤ t < T_{1}.

We select L columns of M as the minimal ones in the lexicographic ordering such that we obtain a nonsingular matrix. Concretely we define ν_{1}, …, ν_{L} as follows:

ν_{1} = min{ν ≥ 0, C_{ν} ≠ 0},

and for 2 ≤ μ ≤ L

ν_{μ} = min{ν > ν_{μ−1} ; C_{ν_{1}}, …, C_{ν_{μ−1}}, C_{ν} are linearly independent}.

Hence we have 0 = ν_{1} < ν_{2} < ⋯ < ν_{L} and the matrix

(C_{ν_{1}}, …, C_{ν_{L}})

is nonsingular. We denote its determinant by Δ.

Remark. We may assume ζ_{0} = 0 and σ_{0} = 0; in this case the first column has T_{1} components 1 (those with index (0, t) such that 0 ≤ t < T_{1}) and (T_{0} − 1)T_{1} components 0 (the other ones, with index (τ, t) such that 1 ≤ τ < T_{0} and 0 ≤ t < T_{1}).

Step 3. Upper Bound for |Δ|

We apply Corollary 2.4 with ϱ_{μ} = 1, r_{μ} = |β|_{+}S_{1}(N_{ν_{μ}}), R_{μ} = 3er_{μ}, E_{μ} = e and

{φ_{1}, …, φ_{L}} = {z^{τ} e^{tz} ; 0 ≤ τ < T_{0}, 0 ≤ t < T_{1}}.

We deduce

|Δ| ≤ L! ∏_{μ=1}^{L}(e^{−ν_{μ}}σ_{ν_{μ}}!R_{μ}^{T_{0}}e^{T_{1}R_{μ}})

≤ L! exp ∑_{μ=1}^{L}(−ν_{μ} + log(S_{0}(N_{ν_{μ}})!) + T_{0} log(3e|β|_{+}S_{1}(N_{ν_{μ}})) + 3e|β|_{+}T_{1}S_{1}(N_{ν_{μ}})).

Step 4. Lower Bound for |Δ|

The number Δ is not zero and lies in the ring ℤ[α, β]: there is a polynomial f ∈ ℤ[Z_{1}, Z_{2}] such that

Δ = f(α, β),

the degree of f in Z_{1} and Z_{2} respectively is at most

∑_{μ=1}^{L} T_{1}S_{1}(N_{ν_{μ}}) and LT_{0},

while the length of f is bounded by

L(f) ≤ L! ∏_{μ=1}^{L} T_{1}^{S_{0}(N_{νμ})}(S_{0}(N_{ν_{μ}}) + S_{1}(N_{ν_{μ}}))^{T_{0}}.

Assume now that α and β are both algebraic. Then we may use Liouville’s inequality Proposition 1.19:

|Δ| ≥ L!^{−D+1}e^{−LDT_{0}h(β)}

∏_{μ=1}^{L}(T_{1}^{−(D−1)S_{0}(N_{νμ})}(S_{0}(N_{ν_{μ}}) + S_{1}(N_{ν_{μ}}))^{−(D−1)T_{0}}e^{−DT_{1}S_{1}(N_{νμ})h(α)}).

Step 5. Choice of parameters

Define, for N ≥ 0,

S_{0}(N) = N^{3} and S_{1}(N) = N.

By (2.5) we have for ν ≥ 1

(N_{ν} − 1)^{4} ≤ ν < N_{ν}^{4},

hence

S_{0}(N_{ν}) ≤ (ν^{1/4} + 1)^{3} ≤ 8ν^{3/4}, S_{1}(N_{ν}) ≤ ν^{1/4} + 1 ≤ 2ν^{1/4}.

Fix a sufficiently large integer T_{0} (larger than some constant depending only on α and β) and define T_{1} = T_{0}, so that L = T_{0}^{2}.

For 1 ≤ μ ≤ L the following estimates are plain:

D log L + DT_{0}h(β) + (D − 1)T_{0} log(S_{0}(N_{ν_{μ}})S_{1}(N_{ν_{μ}})) + T_{0} log(3e|β|_{+}S_{1}(N_{ν_{μ}})) ≤ (c_{1} + c_{2} log ν_{μ})T_{0},

(D − 1)S_{0}(N_{ν_{μ}}) log(T_{1}S_{0}(N_{ν_{μ}})) ≤ c_{3}ν_{μ}^{3/4} log(T_{0}ν_{μ})

and

T_{1}S_{1}(N_{ν_{μ}})(Dh(α) + 3e|β|_{+}) ≤ c_{4}T_{0}ν_{μ}^{1/4},

where c_{1}, …, c_{4} are positive real numbers which depend only on α and β. Therefore we have

D log L + DT_{0}h(β) + (D − 1)T_{0} log(S_{0}(N_{ν_{μ}})S_{1}(N_{ν_{μ}})) + T_{0} log(3e|β|_{+}S_{1}(N_{ν_{μ}}))

+(D − 1)S_{0}(N_{ν_{μ}}) log T_{1} + S_{0}(N_{ν_{μ}}) log(S_{0}(N_{ν_{μ}})) + T_{1}S_{1}(N_{ν_{μ}})(Dh(α) + 3e|β|_{+}) ≤ Q_{μ},

where

Q_{μ} = c_{5}T_{0}ν_{μ}^{1/4} + c_{6}ν_{μ}^{3/4} log(T_{0}ν_{μ})

and again c_{5}, c_{6} depend only on α and β.

Step 6. Conclusion

We claim

(2.6)  ∑_{μ=1}^{L}(ν_{μ} − Q_{μ}) > 0.

Indeed, since μ ↦ Q_{μ} is increasing, we have

∑_{1≤μ≤L/2} Q_{μ} ≤ ∑_{L/2<μ≤L} Q_{μ}.

The estimate

ν_{μ} ≥ 0 for 1 ≤ μ ≤ L/2

is trivial, while for L/2 < μ ≤ L the lower bound

ν_{μ} ≥ μ − 1 ≥ (L − 1)/2 = (T_{0}^{2} − 1)/2

implies

ν_{μ}^{3/4} > 4c_{5}T_{0} and ν_{μ}^{1/4} > 4c_{6} log(T_{0}ν_{μ}),

hence

ν_{μ} > 2Q_{μ} for L/2 < μ ≤ L.

Therefore our claim (2.6) is vindicated.

According to steps 5 and 6, the conclusions of steps 3 and 4 are not compatible, hence one at least of the two numbers α, β is transcendental.

This completes the proof of Hermite-Lindemann’s Theorem 1.2. □

3 Linear Independence of Logarithms of Algebraic Numbers

The main result, due to A. Baker ([B1] and [B2] Th. 1.2), is the following:

Theorem 3.1. Let α_{1}, …, α_{n} be nonzero algebraic numbers. For each i = 1, …, n, let λ_{i} ∈ ℂ satisfy e^{λ_{i}} = α_{i}. Assume the n numbers λ_{1}, …, λ_{n} are linearly independent over ℚ. Then the n + 1 numbers 1, λ_{1}, …, λ_{n} are linearly independent over the field \overline{ℚ} of algebraic numbers.

We shall use the notation log α_{i} in place of λ_{i}. One should keep in mind that this notation may be troublesome: for instance Theorem 3.1 can be applied with

α_{1} = α_{2} = 2, λ_{1} = log 2, λ_{2} = log 2 + 2iπ,

and the conclusion shows that the three numbers 1, log 2, π are linearly independent over \overline{ℚ}. However the same conclusion can be obtained by taking α_{1} = 2 and α_{2} = −1 for instance.

By the way, when α_{1}, …, α_{n} are nonzero complex numbers, for any choice (λ_{1}, …, λ_{n}) ∈ ℂ^{n} with e^{λ_{i}} = α_{i} (1 ≤ i ≤ n), the following conditions are clearly equivalent:

(i) The numbers α_{1}, …, α_{n} are multiplicatively independent, which means that any re-

lation

α_{1}^{a_{1}} ⋯ α_{n}^{a_{n}} = 1

with (a_{1}, …, a_{n}) ∈ ℤ^{n} implies a_{1} = ⋯ = a_{n} = 0. (ii) The n + 1 complex numbers 2πi, λ_{1}, …, λ_{n} are linearly independent over ℚ.

Hence, given complex numbers λ_{1}, …, λ_{n}, the multiplicative subgroup of ℂ^{×} generated by e^{λ_{1}}, …, e^{λ_{n}} has rank (as a ℤ-module) equal to r − 1, where r is the dimension of the ℚ-vector space spanned by 2πi, λ_{1}, …, λ_{n}. In particular, if λ_{1}, …, λ_{n} are ℚ-linearly independent, then this rank is

\{ \begin{matrix}n & if 2πi, λ_{1}, …, λ_{n} are linearly independent over ℚ, \\ n − 1 & if 2πi, λ_{1}, …, λ_{n} are ℚ-linearly dependent. \end{matrix}

Proofs of Baker’s Theorem 3.1 on linear independence of logarithms are given in [B2] Chap. 2, [F] Chap. 10 § 1, [FN] Chap. 4, § 1.3, [W1] Chap. 8 and [W5] § 10.1. A “dual” argument (extension of Schneider’s method, while Baker’s method is an extension of Gel’fond’s method) is worked out in [W5] Chap. 6 (for the homogeneous case) and § 9.1 (for the nonhomogeneous case). See also [W5] § 4.2 for a proof, following Bertrand and Masser, which rests on Schneider-Lang’s Criterion for Cartesian products (involving again Gel’fond’s method).

Here we consider only Baker’s method. In § 3.1 we explain why Baker’s method can be introduced by means of functions of either one or several variables. A proof of Theorem 3.1 by means of Baker’s method, involving an auxiliary function with an extrapolation argument, is given in § 3.2, which includes also a sketch of proof with an interpolation determinant but without any extrapolation. For both proofs the zero estimate which is used there is due to Philippon [P1]. In § 3.3 we show how to replace this zero estimate by a much simpler one, due to R. Tijdeman, by means of a further extrapolation with the auxiliary function. Our ultimate goal in this third lecture is to extrapolate with an interpolation determinant (§ 3.5), in order to complete the proof of Theorem 3.1 without any auxiliary function, and with Tijdeman’s zero estimate in place of Philippon’s one. This is achieved thanks to a generalization (in § 3.4) in several variables of the results of § 2.1 giving upper bounds for interpolation determinants.

3.1 Introduction to Baker’s Method

We explain the basic ideas of the proof of Theorem 3.1 by means of Baker’s method with an auxiliary function involving an extrapolation.

Assume α_{1}, …, α_{n} are nonzero algebraic numbers, log α_{1}, …, log α_{n} are linearly independent over ℚ, β_{0}, …, β_{n−1} are algebraic numbers, and

(3.2)  log α_{n} = β_{0} + β_{1} log α_{1} + ⋯ + β_{n−1} log α_{n−1}.

We shall eventually reach a contradiction.

From now on α^{z} stands for exp(z log α), which has a meaning as soon as a complex number λ = log α has been selected with e^{λ} = α.

Hence relation (3.2) implies

α_{n} = e^{β_{0}}α_{1}^{β_{1}} ⋯ α_{n−1}^{β_{n−1}},

and more generally for z ∈ ℂ

α_{n}^{z} = e^{β_{0}z}α_{1}^{β_{1}z} ⋯ α_{n−1}^{β_{n−1}z}.

To each polynomial P ∈ ℤ[Y_{0}, Y_{1}, …, Y_{n}] we associate analytic functions of 1, n and n + 1 complex variables. The proof of Baker’s qualitative Theorem 3.1 on linear independence of logarithms of algebraic numbers requires a Schwarz’ Lemma, while the quantitative refinements (measures of linear independence; see § 5) will require an approximate Schwarz’ Lemma. A fundamental fact is that one needs such auxiliary results only for functions of a single variable: even if we introduce functions of several variables, we shall consider only the values of our functions at multiples of a single point (but derivatives are taken in several directions); therefore it would be possible to avoid completely the introduction of several variables, but we use them only to explain the role of certain differential operators.

3.1.a Using a Single Variable

Consider the entire functions

z, α_{1}^{z}, …, α_{n}^{z}.

To the auxiliary polynomial P ∈ ℤ[\underline{Y}] is attached the exponential polynomial

G(z) = P (z, α_{1}^{z}, …, α_{n}^{z}),

which can be written also

G(z) = P (z, α_{1}^{z}, …, α_{n−1}^{z}, e^{β_{0}z}α_{1}^{β_{1}z} ⋯ α_{n−1}^{β_{n−1}z}).

In order to take (3.2) into account, we consider derivatives of G. We avoid difficulties (related with Liouville’s inequality) arising from the unwanted transcendental numbers log α_{i} (1 ≤ i < n) by writing the derivatives of G as polynomials in log α_{1}, …, log α_{n−1}, and the coefficients of these polynomials are themselves exponential polynomials with algebraic coefficients.

We start with the first derivative G′(z) = (d/dz)G(z) of G: this is the value, at (z, α_{1}^{z}, …, α_{n}^{z}), of the polynomial

(\frac{∂}{∂Y_{0}} + ∑_{k=1}^{n}(log α_{k})\frac{∂}{∂Y_{k}} ) P = (∂_{0} + ∂_{1} log α_{1} + ⋯ + ∂_{n−1} log α_{n−1})P,

where ∂_{0}, …, ∂_{n−1} are the differential operators

∂_{0} = \frac{∂}{∂Y_{0}} + β_{0}Y_{n} \frac{∂}{∂Y_{n}} ,  ∂_{k} = Y_{k} \frac{∂}{∂Y_{k}} + β_{k}Y_{n} \frac{∂}{∂Y_{n}}  (1 ≤ k ≤ n − 1)

on the ring ℂ[\underline{Y}] We now take higher derivatives. For \underline{σ} = (σ_{0}, …, σ_{n−1}) ∈ ℕ^{n} we write ∂\underline{σ} in place of

∂_{0}^{σ_{0}} ⋯ ∂_{n−1}^{σ_{n−1}}.

Since

(∂_{0} + ∂_{1} log α_{1} + ⋯ + ∂_{n−1} log α_{n−1})^{k} = ∑_{∥\underline{σ}∥=k} \frac{k!}{\underline{σ}!}(log α_{1})^{σ_{1}} ⋯ (log α_{n−1})^{σ_{n−1}}∂\underline{σ},

we have

G^{(k)}(z) = ∑_{∥\underline{σ}∥=k} \frac{k!}{\underline{σ}!}(log α_{1})^{σ_{1}} ⋯ (log α_{n−1})^{σ_{n−1}}G\underline{σ}(z),

where

G\underline{σ}(z) = P\underline{σ}(z, α_{1}^{z}, …, α_{n}^{z}) and P\underline{σ} = ∂\underline{σ}P ∈ ℂ[\underline{Y}].

Now if we compose derivations we easily deduce for \underline{σ} and \underline{λ} in ℕ^{n}

∂^{\underline{σ}+\underline{λ}} = ∂\underline{σ} ∘ ∂\underline{λ},

which yields the fundamental relation for \underline{σ} ∈ ℕ^{n} and ℓ ∈ ℕ

(3.3)  G(ℓ) ; \underline{σ}(z) = ∑_{∥\underline{λ}∥=ℓ} \frac{ℓ!}{\underline{λ}!}(log α_{1})^{λ_{1}} ⋯ (log α_{n−1})^{λ_{n−1}}G_{\underline{σ}+\underline{λ}}(z).

As a consequence, if S_{0} and S_{1} are positive integers for which

(3.4)  G\underline{σ}(s) = 0 for ∥\underline{σ}∥ < S_{0} and 0 ≤ s < S_{1},

then for any \underline{σ} ∈ ℕ^{n} with ∥\underline{σ}∥ < S_{0} the function G\underline{σ} has a zero at s = 0, …, S_{1} − 1 of multiplicity ≥ S_{0} − ∥\underline{σ}∥.

This is really the main point in Baker’s method [B1 I] p.212, which has no counterpart in the dual method of [W5] Chap. 6 (there is no efficient extrapolation so far when one deals with functions of several variables). By means of the one dimensional approximate Schwarz’ Lemma 1.18, one deduces from (3.4) a sharp upper bound for |G\underline{σ}|_{r} and gets more equations like (3.4): this is the extrapolation.

Remark. If we were to replace (3.2) by an algebraic relation between logarithms of algebraic numbers, for instance

log α_{n} = A(log α_{1}, …, log α_{n−1})

where A is a polynomial of total degree > 1, then one could also write the derivatives of G as polynomials in (log α_{1}, …, log α_{n−1}), but there are no nice relations like (3.3) between the corresponding exponential polynomials replacing the G\underline{σ}.

3.1.b Introducing n Variables

As we said, from a strict logical point of view, introducing functions of several variables is not required. But it may help to understand better the meaning of the differential operators ∂_{k} (0 ≤ k < n).

To the auxiliary polynomial P is also associated an analytic function of n complex variables

Φ(z_{0}, z_{1} …, z_{n−1}) = P (z_{0}, e^{z_{1}}, …, e^{z_{n−1}}, e^{β_{0}z_{0}+β_{1}z_{1}+···+β_{n−1}z_{n−1}}).

We take derivatives of Φ with respect to the n variables, and consider the values of these derivatives at the point

\underline{v} = (1, log α_{1}, …, log α_{n−1}) ∈ ℂ^{n}.

Obviously we have

Φ(z\underline{v}) = G(z)

for z ∈ ℂ, but what is more interesting is the connection between the derivatives. From the definition of the differential operators ∂_{0}, …, ∂_{n−1}, it is plain that for 0 ≤ k ≤ n − 1 we have

\frac{∂}{∂z_{k}} Φ(z_{0}, z_{1} …, z_{n−1}) = (∂_{k}P)(z_{0}, e^{z_{1}}, …, e^{z_{n−1}}, e^{β_{0}z_{0}+β_{1}z_{1}+···+β_{n−1}z_{n−1}}).

Hence for \underline{σ} = (σ_{0}, …, σ_{n−1}) ∈ ℕ^{n} and z ∈ ℂ we have

G\underline{σ}(z) = (D\underline{σ}Φ)(z\underline{v})

where

D\underline{σ} = (\frac{∂}{∂z_{0}} )^{σ_{0}} ⋯ (\frac{∂}{∂z_{n−1}} )^{σ_{n−1}} .

3.1.c Introducing n + 1 Variables

Instead of working with n variables it is sometimes convenient (for instance for the zero estimate) to consider n + 1 variables: define

F (z_{0}, z_{1} …, z_{n}) = P (z_{0}, e^{z_{1}}, …, e^{z_{n}}).

The point

\underline{u} = (1, log α_{1}, …, log α_{n}) ∈ ℂ^{n+1}

lies in the hyperplane W of equation

z_{n} = β_{0}z_{0} + β_{1}z_{1} + ⋯ + β_{n−1}z_{n−1}.

A basis of W is w = (\underline{w}_{0}, …, \underline{w}_{n−1}) where

\underline{w}_{k} = (δ_{k0}, …, δ_{k,n−1}, β_{k}) (0 ≤ k ≤ n − 1).

These elements \underline{w}_{0}, …, \underline{w}_{n−1} are the n column vectors of the matrix

( \begin{matrix}1 & 0 & ⋯ & 0 \\ 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & 1 \\ β_{0} & β_{1} & ⋯ & β_{n−1} \end{matrix} ) = ( \begin{matrix}& I_{n} \\ β_{0} & ⋯ & β_{n−1} \end{matrix} ) .

Since

z_{0}\underline{w}_{0} + ⋯ + z_{n−1}\underline{w}_{n−1} = (z_{0}, …, z_{n−1}, β_{0}z_{0} + ⋯ + β_{n−1}z_{n−1}),

we have

Φ(z_{0}, …, z_{n−1}) = F (z_{0}\underline{w}_{0} + ⋯ + z_{n−1}\underline{w}_{n−1}),

and one may view Φ as the restriction of F to the hyperplane W, equipped with the basis w, by means of the isomorphism

\begin{matrix}ℂ^{n} & ⟶ & W \\ (z_{0}, …, z_{n−1}) & ⟼ & z_{0}\underline{w}_{0} + ⋯ + z_{n−1}\underline{w}_{n−1}. \end{matrix}

To take the derivatives of Φ in all n directions amounts to taking the derivatives of F in the directions of W. More precisely, for \underline{x} = (x_{0}, …, x_{n}) ∈ ℂ^{n+1}, define

D\underline{x} = x_{0} \frac{∂}{∂z_{0}} + ⋯ + x_{n} \frac{∂}{∂z_{n}} ·

For instance

D_{\underline{w}_{k}} = \frac{∂}{∂z_{k}} + β_{k} \frac{∂}{∂z_{n}} for 0 ≤ k ≤ n − 1,

hence for 0 ≤ k ≤ n − 1

D_{\underline{w}_{k}}F (z_{0}, …, z_{n}) = (∂_{k}P)(z_{0}, e^{z_{1}}, …, e^{z_{n}}).

Define also, for \underline{σ} = (σ_{0}, …, σ_{n−1}) ∈ ℕ^{n},

D\underline{σ} ; w = D_{\underline{w}_{0}}^{σ_{0}} ⋯ D_{\underline{w}_{n−1}}^{σ_{n−1}}.

Then

D\underline{σ} ; wF (z_{0}, …, z_{n}) = (∂\underline{σ}P)(z_{0}, e^{z_{1}}, …, e^{z_{n}})

and

D\underline{σ} ; wF (z_{0}, …, z_{n−1}, β_{0}z_{0} + ⋯ + β_{n−1}z_{n−1}) = (\frac{∂}{∂z_{0}} )^{σ_{0}} ⋯ (\frac{∂}{∂z_{n−1}} )^{σ_{n−1}} Φ(z_{0}, …, z_{n−1}).

In particular

G\underline{σ}(z) = D\underline{σ} ; wF (z\underline{u}).

3.2 Proof of Baker’s Theorem

We prove Theorem 3.1 following basically [W1] Chap. 8. One main difference is that we shall use a sharper zero estimate than Theorem 6.1.1 of [W1], and therefore we do not need a long extrapolation like in [W1]: here a single step will be sufficient. This explains why the interpolation determinant method could easily be used in [W5] § 10.1. In § 3.3 we shall explain how a longer extrapolation enables one to use a weaker zero estimate.

Denote by 𝕂 the number field

𝕂 = ℚ(α_{1}, …, α_{n}, β_{0}, …, β_{n−1}).

Let T_{0}, T_{1}, …, T_{n}, S_{0} and S_{1} be sufficiently large positive integers. Explicit conditions on these parameters will occur along the proof, and we shall discuss them later, but it may help the reader to know that a suitable choice is

(3.5)  T_{0} = 2[𝕂 : ℚ]N^{2n+1}, T_{1} = ⋯ = T_{n} = N^{2n−1},  S_{0} = N^{2n+1}, S_{1} = N.

We also set L = T_{0}T_{1} ⋯ T_{n} and T = T_{1} + ⋯ + T_{n}.

3.2.a Construction of the Auxiliary Polynomial

By means of Thue-Siegel’s Lemma 1.6, we show the existence of a nonzero auxiliary polynomial P ∈ ℤ[Y_{0}, …, Y_{n}], with degree < T_{i} in Y_{i} (0 ≤ i ≤ n), such that the equations (3.4) hold.

These conditions amount to a homogeneous linear system of equations with coefficients in 𝕂, where the unknowns are the coefficients of P.

Our first condition on the parameters will be

2[𝕂 : ℚ]\binom{S_{0} + n − 1}{n}S_{1} ≤ L,

so that the number of equations is at most half the number of unknowns^{7}.

The coefficients of the linear system are the numbers

7 One could construct P with coefficients in the field 𝕂, and then omit the factor [𝕂 : ℚ].

(3.6)  γ_{τ\underline{t}}^{(\underline{σ},s)} = ∂\underline{σ}(Y_{0}^{τ} Y_{1}^{t_{1}} ⋯ Y_{n}^{t_{n}})(s, α_{1}^{s}, …, α_{n}^{s})

with 0 ≤ τ < T_{0}, 0 ≤ t_{i} < T_{i} (1 ≤ i ≤ n) and \underline{σ} ∈ ℕ^{n}, ∥\underline{σ}∥ < S_{0}, 0 ≤ s < S_{1}. We write them explicitly by computing the derivatives of

z_{0}^{τ} e^{t_{1}z_{1}} ⋯ e^{t_{n−1}z_{n−1}}e^{t_{n}(β_{0}z_{0}+···+β_{n−1}z_{n−1})}.

One obtains easily

γ_{τ\underline{t}}^{(\underline{σ},s)} = ∑_{κ=0}^{min\{τ,σ_{0}\}} \frac{σ_{0}!τ!}{κ!(σ_{0} − κ)!(τ − κ)!}s^{τ−κ}(β_{0}t_{n})^{σ_{0}−κ} ∏_{i=1}^{n−1}(t_{i} + t_{n}β_{i})^{σ_{i}} ∏_{j=1}^{n} α_{j}^{t_{j}s}.

One should use a variant of Lemma 1.6 taking into account the fact that the coefficients of our linear system are in 𝕂 rather than in ℤ, but anyway a rough estimate shows that we end up with a nonzero polynomial P of length at most

L(P) ≤ exp{c_{1}(T_{0} + S_{0}) log L + c_{2}T S_{1}}.

Here and below, c_{1}, …, c_{16} denote positive numbers which do not depend on T_{0}, T_{1}, …, T_{n}, S_{0}, S_{1}. For instance with our choice (3.5) we get

L(P) ≤ exp{c_{3}N^{2n+1} log N}.

Remark. The whole point in this argument is that (3.2) allows us to consider values of polynomials in n − 1 variables at the point (log α_{1}, …, log α_{n−1}) in place of values of polynomials in n variables at the point (log α_{1}, …, log α_{n}). Without (3.2) it would be necessary to replace

\binom{S_{0} + n − 1}{n} with \binom{S_{0} + n}{n + 1}

and the only difference with the present proof is that no choice of parameters would be admissible!

3.2.b Extrapolation on Integral Points

We introduce further parameters S′_{0} and S′_{1} which are positive integers with S′_{0} < S_{0} and S′_{1} > S_{1}, and we are going to prove

(3.7)  G\underline{σ}(s) = 0 for (\underline{σ}, s) ∈ ℕ^{n} × ℕ with ∥\underline{σ}∥ < S′_{0} and 0 ≤ s < S′_{1}.

With the choice of parameters (3.5) we shall take

S′_{0} = [S_{0}/2], S′_{1} = N^{2}.

Fix \underline{σ} ∈ ℕ^{n} with ∥\underline{σ}∥ < S′_{0}. The function

G\underline{σ}(z) = ∂\underline{σ}P (z, α_{1}^{z}, …, α_{n}^{z})

has a zero at each point s = 0, …, S_{1} − 1 of multiplicity ≥ S_{0} − S′_{0}. The one variable Schwarz’ Lemma 1.5 with r = S′_{1}, E = e, R = 2er provides the following upper bound:

|G\underline{σ}|_{r} ≤ e^{−(S_{0}−S′_{0})S_{1}}|G\underline{σ}|_{R}.

This yields an upper bound for |G\underline{σ}(s)| with s ∈ ℤ in the range 0 ≤ s < S′_{1} which is not compatible with Liouville’s lower bound provided that

c_{4}(T_{0} + S_{0}) log L + c_{5}T S′_{1} < (S_{0} − S′_{0})S_{1}.

Hence (3.7).

3.2.c Using Philippon’s Zero Estimate

The next auxiliary Lemma is a very special case of Philippon’s zero estimate [P1] (see also Chap. 8 of [W5] by D. Roy).

Proposition 3.8. Let α_{1}, …, α_{n} be nonzero complex numbers which generate a multiplicative subgroup of ℂ^{×} of rank ≥ n − 1 and let β_{0}, …, β_{n−1} be complex numbers. Assume that 1, β_{1}, …, β_{n−1} are linearly independent over ℚ. Let T_{0}, T_{1}, …, T_{n}, S_{0}, S_{1} and L be positive integers satisfying the following conditions:

L = T_{0}T_{1} ⋯ T_{n}, T_{1} ≤ T_{2} ≤ ⋯ ≤ T_{n},

S_{0} > (n + 1)T_{n}, S_{0}S_{1} > \frac{1}{2}n!(n + 1)! max\{T_{0}, 2T_{n}\},

and

(3.9)  \binom{S_{0} + n − 1}{n}S_{1} > (n + 1)!T_{0}T_{1} ⋯ T_{n}.

Assume also

• either β_{0} ≠ 0

• or else n ≥ 2 and

(3.10)  \binom{S_{0} + n − 2}{n − 1}S_{1} > (n + 1)!T_{1} ⋯ T_{n}.

For τ ∈ ℕ, \underline{t} ∈ ℕ^{n}, \underline{σ} ∈ ℕ^{n} and s ∈ ℕ, consider the number γ_{τ\underline{t}}^{(\underline{σ}s)} given by (3.6) and build up the matrix:

M = (γ_{τ\underline{t}}^{(\underline{σ}s)})_{{(τ,\underline{t}) \atop (\underline{σ},s)}}

where the index of rows (τ, \underline{t}) runs over the elements in ℕ×ℕ^{n} with 0 ≤ τ < T_{0}, 0 ≤ t_{i} < T_{i} (1 ≤ i ≤ n), while the index of columns (\underline{σ}, s) runs over the elements in ℕ^{n} × ℕ with ∥\underline{σ}∥ < (n + 1)S_{0} and 0 ≤ s < (n + 1)S_{1}. Then M has rank L.

Remark. Using (3.9) it is easily checked that (3.10) can be replaced by

nT_{0} > S_{0} + n − 1.

Remark. Given ℚ-linearly independent complex numbers λ_{1}, …, λ_{n}, the n numbers α_{i} = e^{λ_{i}} (1 ≤ i ≤ n) generate a multiplicative subgroup of ℂ^{×} of rank ≥ n − 1. Conversely, if α_{1}, …, α_{n} are nonzero elements of ℂ which generate a multiplicative subgroup of ℂ^{×} of rank ≥ n − 1, then there exist ℚ-linearly independent complex numbers λ_{1}, …, λ_{n} such that α_{i} = e^{λ_{i}} for 1 ≤ i ≤ n.

Proposition 3.8 is essentially Proposition 10.2 of [W5], with a few differences:

• We do not assume β_{0} ≠ 0 here. At the same time our points are (s, α_{1}^{s}, …, α_{n}^{s}) in place^{8} of (sβ_{0}, α_{1}^{s}, …, α_{n}^{s}).

• We work with polynomials in Y_{0}, …, Y_{n} of degree < T_{i} in Y_{i} (0 ≤ i ≤ n), while in [W5] we considered polynomials in X_{0}, X_{1}^{±1}, …, X_{n}^{±1} of degree ≤ T_{0} in X_{0} and degree ≤ T_{1} in each of the variables X_{i}^{±1}. Also here we consider nonnegative integers s with 0 ≤ s < (n + 1)S_{1}, while in [W5] we had s ∈ ℤ with |s| ≤ (n + 1)S_{1}. Also here we use strict inequalities for ∥\underline{σ}∥.

These changes introduce few modifications in the proof of Proposition 10.2 of [W5], but for the convenience of the reader we provide the details.

Proof. Consider the algebraic groups G_{0} = 𝔾_{a}, G_{1} = 𝔾_{m}^{n}, G = G_{0} × G_{1}, of dimensions d_{0} = 1, d_{1} = n and d = n + 1 respectively. Let W be the hyperplane in ℂ^{n+1} of equation

β_{0}z_{0} + β_{1}z_{1} + ⋯ + β_{n−1}z_{n−1} = z_{n}.

Introduce also the set

Σ = {(s, α_{1}^{s}, …, α_{n}^{s}) ; s ∈ ℕ, 0 ≤ s < S_{1}} ⊂ G(ℂ) = ℂ × (ℂ^{×})^{n}.

If the rank of the matrix M is less than L, then there exists a nonzero polynomial P in ℂ[Y_{0}, Y_{1}, …, Y_{n}], of degree < T_{i} in Y_{i} (0 ≤ i ≤ n), for which the functions G\underline{σ}(z) = ∂\underline{σ}P (z, α_{1}^{z}, …, α_{n}^{z}) satisfy

G\underline{σ}(s) = 0 for (\underline{σ}, s) ∈ ℕ^{n} × ℕ with ∥\underline{σ}∥ < (n + 1)S_{0} and 0 ≤ s < (n + 1)S_{1}.

According to Philippon’s zero estimate there exists an algebraic subgroup G^{∗} of G of dimension d^{∗} ≤ n and codimension d′ = n + 1 − d^{∗} such that

(3.11)  \binom{S_{0} + ℓ_{0} − 1}{ℓ_{0}}Card (\frac{Σ + G^{∗}}{G^{∗}} ) H(G^{∗}; \underline{T}) ≤ H(G; \underline{T}),

where

ℓ_{0} = \{ \begin{matrix}d′ − 1 & if T_{e}(G^{∗}) ⊂ W, \\ d′ & otherwise. \end{matrix}

The notation H(G^{∗}; \underline{T}) stands for a multihomogeneous Hilbert-Samuel polynomial (see [W5], § 5.2.3); for instance H(G; \underline{T}) = (n + 1)!L.

We first check that this inequality (3.11) is not satisfied with G^{∗} = {e}: indeed when G^{∗} = {e} we have

8 This was an oversight in Proposition 10.2 of [W5]!

d^{∗} = 0, ℓ_{0} = n, Card (\frac{Σ + G^{∗}}{G^{∗}} ) = Card(Σ) = S_{1}, H(G^{∗}; \underline{T}) = 1,

so that by (3.9)

\binom{S_{0} + ℓ_{0} − 1}{ℓ_{0}}Card (\frac{Σ + G^{∗}}{G^{∗}} ) H(G^{∗}; \underline{T}) > (n + 1)!L.

Therefore d^{∗} ≥ 1.

Write G^{∗} = G_{0}^{∗} × G_{1}^{∗} where G_{0}^{∗} is an algebraic subgroup of G_{0} and G_{1}^{∗} an algebraic subgroup of G_{1}. Denote by d_{0}^{∗} and d_{1}^{∗} the dimensions of G_{0}^{∗} and G_{1}^{∗} respectively, and by d′_{0} and d′_{1} their codimensions:

d_{0}^{∗} + d′_{0} = d_{0} = 1, d_{1}^{∗} + d′_{1} = d_{1} = n.

Assume first T_{e}(G^{∗}) ⊂ W. Since 1, β_{1}, …, β_{n−1} are linearly independent over ℚ, the hyperplane of ℂ^{n} of equation

β_{1}z_{1} + ⋯ + β_{n−1}z_{n−1} = z_{n}

does not contain any nonzero element of ℚ^{n}. Since T_{e}(G_{1}^{∗}) is a subspace of ℂ^{n} which is rational over ℚ, we deduce G_{1}^{∗} = {e}, hence G^{∗} = 𝔾_{a} × {e}, d^{∗} = 1, d′ = n, ℓ_{0} = n − 1 and H(G^{∗}; \underline{T}) = T_{0}. Now the condition T_{e}(G^{∗}) ⊂ W implies β_{0} = 0, hence (3.10) gives n ≥ 2 and

\binom{S_{0} + n − 2}{n − 1}S_{1} > (n + 1)!T_{1} ⋯ T_{n}.

Since n ≥ 2 and since α_{1}, …, α_{n} generate a subgroup of ℂ^{×} of rank ≥ n − 1, we have

Card (\frac{Σ + G^{∗}}{G^{∗}} ) = Card\{(α_{1}^{s}, …, α_{n}^{s}) ; s ∈ ℕ, 0 ≤ s < S_{1}\} = S_{1}.

Hence (3.11) does not hold and we get a contradiction.

Therefore T_{e}(G^{∗}) ⊄ W and ℓ_{0} = d′.

Consider the case G_{0}^{∗} = {0}. We have d_{0}^{∗} = 0, d^{∗} = d_{1}^{∗}, d′ = n + 1 − d_{1}^{∗},

H(G^{∗}; \underline{T}) ≥ (d_{1}^{∗} + 1)!T_{1} ⋯ T_{d_{1}^{∗}} and Card (\frac{Σ + G^{∗}}{G^{∗}} ) = S_{1}.

Therefore (3.11) implies

\binom{S_{0} + d′ − 1}{d′}S_{1} ≤ \frac{(n + 1)!}{(n + 2 − d′)!}T_{0}T_{d_{1}^{∗}+1} ⋯ T_{n},

hence

S_{0}^{d′}S_{1} ≤ \frac{(n + 1)!d′!}{(n + 2 − d′)!}T_{0}T_{d_{1}^{∗}+1} ⋯ T_{n}.

However we have S_{0} > T_{n}, 1 ≤ d′ ≤ n and

\frac{d′!}{(n + 2 − d′)!} ≤ \frac{1}{2}n!,

hence we get a contradiction with the inequality

S_{0}S_{1} > \frac{1}{2}n!(n + 1)!T_{0}.

So we have d_{0}^{∗} = 1, d^{∗} = d_{1}^{∗} + 1, d′ = n − d_{1}^{∗} and

H(G^{∗}; \underline{T}) ≥ (d_{1}^{∗} + 1)!T_{0}T_{1} ⋯ T_{d_{1}^{∗}}.

Now (3.11) gives

\binom{S_{0} + d′ − 1}{d′}Card (\frac{Σ + G^{∗}}{G^{∗}} ) ≤ \frac{(n + 1)!}{(n + 1 − d′)!}T_{d_{1}^{∗}+1} ⋯ T_{n}

from which we deduce

S_{0}^{d′}Card (\frac{Σ + G^{∗}}{G^{∗}} ) ≤ \frac{(n + 1)!d′!}{(n + 1 − d′)!}T_{d_{1}^{∗}+1} ⋯ T_{n}.

Using the estimates

S_{0} > T_{n}, 1 ≤ d′ ≤ n, \frac{d′!}{(n + 1 − d′)!} ≤ n!

and

S_{0}S_{1} > n!(n + 1)!T_{n}

we obtain

Card (\frac{Σ + G^{∗}}{G^{∗}} ) < S_{1},

which means Σ ∩ G^{∗} ≠ {e}. The assumption on the rank of the subgroup of ℂ^{∗} generated by α_{1}, …, α_{n} then implies d_{1}^{∗} = n − 1, d′ = 1 and we get the estimate

S_{0} ≤ (n + 1)T_{n}

which is not compatible with our assumptions. □

3.2.d End of the Proof of Baker’s Theorem 3.1 We apply Proposition 3.8 with S_{0} and S_{1} replaced respectively by

⌈S′_{0}/(n + 1)⌉ and ⌈S′_{1}/(n + 1)⌉.

From (3.5) and (3.7), using the estimates

S′_{0} > (n + 1)^{2}T, S′_{0}S′_{1} > n!(n + 1)!(n + 1)^{2} max \{\frac{1}{2}T_{0}, T \} ,

nT_{0} > S_{0} + n − 1 and S′_{0}^{n}S′_{1} > n!(n + 1)!L,

we deduce a contradiction, which completes the proof of Baker’s Theorem 3.1. □

Remark. The basic ideas for a proof of Baker’s Theorem with an interpolation determinant in [W5] § 10.1.4 are essentially the same. Instead of using Dirichlet’s pigeonhole principle to solve a system of linear equations, we only consider the matrix of this linear system. More precisely the relevant matrix M is the one occurring in the zero estimate (Proposition 3.8): it has maximal rank, and enables one to start with a nonzero determinant Δ. As usual the required lower bound for |Δ| is given by Liouville’s estimate (Proposition 1.19). On the other hand the argument occurring above turns out to be perfectly adaptable to yield an upper bound for |Δ| which gives just what we need.

The difference between the proof with an auxiliary function and the proof with an interpolation determinant is that in the latter Dirichlet’s box principle is not required. However there is a substitute to the auxiliary function, which is the (explicit) exponential polynomial given by a determinant

det(C_{1}, …, C_{L−1}, Φ(z)),

where C_{1}, …, C_{L−1} are L−1 vector columns of M, while the last column vector Φ : ℂ → ℂ^{L} is given by

(z^{τ} α_{1}^{t_{1}z} ⋯ α_{n}^{t_{n}z})_{0≤τ<T_{0}, 0≤t_{i}<T_{i} (1≤i≤n)}.

3.3 Further Extrapolation with the Auxiliary Function

In this section we shall explain how to replace, in the previous proof, Philippon’s zero estimate (Proposition 3.8) by a simpler one. The idea is to extrapolate further and to prove by induction on j = 0, 1, …,

G\underline{σ}(s) = 0 for (\underline{σ}, s) ∈ ℕ^{n} × ℕ with ∥\underline{σ}∥ < S_{0}/2^{j} and 0 ≤ s < S_{1}^{(j)},

with S_{1}^{(0)} = S_{1} and S_{1}^{(1)} = S′_{1}. One cannot continue such an induction forever (in any case one needs 2^{j} ≤ S_{0}). On the other hand, obviously, when the number of equations increases, it is easier to derive a contradiction by means of a zero estimate. This is not only of historical interest: our motivation is related with the problem of linear independence measures, where a short extrapolation yields weaker estimates than a longer one (see § 5).

Baker used a variety of arguments for concluding his proofs, including clever nonvanishing results for certain determinants. In the real case one could just appeal to Pólya’s Lemma 1.10. Dealing with the general case of complex algebraic numbers α_{i} and β_{j}, we shall use Tijdeman’s zero estimate for exponential polynomials in one variable ([T]; see also [F] Chap. 9, § 4, Lemma 8.9, [B2] Chap. 12, § 2, Lemma 6 and [W1] Chap. 6).

Lemma 3.12. Let a_{1}, …, a_{n} be polynomials in ℂ[z], not all of which are zero, of degrees d_{1}, …, d_{n}. Let w_{1}, …, w_{n} be pairwise distinct complex numbers. Define

Ω = max{|w_{1}|, …, |w_{n}|}.

Then for R > 0 the number of zeroes (counting multiplicities) of the function

F (z) = ∑_{i=1}^{n} a_{i}(z)e^{w_{i}z}

in the disk B_{1}(0, R) is at most 2(d_{1} + ⋯ + d_{n} + n − 1) + 5RΩ.

We complete the proof of Baker’s Theorem 3.1 as follows. We repeat the argument of § 3.2.b and perform an induction on j with 1 ≤ j ≤ J. We introduce further parameters S_{0}^{(j)} and S_{1}^{(j)} which are positive integers with

S_{0}^{(0)} = S_{0}, S_{1}^{(0)} = S_{1}, S_{0}^{(1)} = S′_{0}, S_{1}^{(1)} = S′_{1},
S_{0}^{(j)} < S_{0}^{(j−1)} and S_{1}^{(j)} > S_{1}^{(j−1)} (1 ≤ j ≤ J).

One may keep in mind the following picture:

{S_{0}^{(j−1)} \atop S_{0}^{(j)}}_{0  S_{1}^{(j−1)}  S_{1}^{(j)}}

We want to prove, for 0 ≤ j ≤ J,

(3.13)  G\underline{σ}(s) = 0 for ∥\underline{σ}∥ < S_{0}^{(j)} and 0 ≤ s < S_{1}^{(j)}.

This is true by construction (§ 3.2.a) for j = 0, and by the first extrapolation (§ 3.2.b) for j = 1. Relations (3.13) for j + 1 follow from those for j provided that

c_{6}(T_{0} + S_{0}) log(LS_{1}^{(j+1)}) + c_{7}T S_{1}^{(j+1)} < (S_{0}^{(j)} − S_{0}^{(j+1)})S_{1}^{(j)}.

Assuming (3.5), let us choose

(3.14)  J = 2n^{2}, S_{0}^{(j)} = [S_{0}/2^{j}], S_{1}^{(j)} = N^{j+1} (1 ≤ j ≤ J).

At the end of the induction (j = J) we get an exponential polynomial G with a zero at each point s with 0 ≤ s < S_{1}^{(J)} of multiplicity ≥ S_{0}^{(J)}. Since

S_{0}^{(J)}S_{1}^{(J)} > 2L + c_{8}T S_{1}^{(J)},

we get a contradiction with Lemma 3.12.

3.4 Upper Bound for a Determinant in Several Variables

Let F ∈ H_{n}(R) be a function with a zero of multiplicity ≥ S_{0} at S_{1} distinct points of B_{n}(0, r) ∩ ℂ\underline{v} with r ≤ R. Then G(z) = F (\underline{v}z) (which may be viewed as the restriction of F to ℂ\underline{v}) is a function of a single variable with S_{0}S_{1} zeroes in B_{1}(0, r/|\underline{v}|)^{9}. From Schwarz’ Lemma 1.5 we deduce

sup_{\underline{z}∈B_{n}(0,r)∩ℂ\underline{v}} |F (\underline{z})| = |G|_{r/|\underline{v}|} ≤ (\frac{R^{2} + r^{2}}{2rR} )^{−S_{0}S_{1}} |G|_{R/|\underline{v}|} ≤ (\frac{R^{2} + r^{2}}{2rR} )^{−S_{0}S_{1}} |F |_{R}.

Unfortunately there is no similar upper bound for |F |_{r} and we cannot use Cauchy’s inequalities to bound derivatives of F in other directions than ℂ\underline{v}. For instance taking n = 2, \underline{v} = (1, 0), F (z_{1}, z_{2}) = z_{2}^{L}, we have G(z) = F (z, 0) = 0, but

(\frac{∂}{∂z_{2}} )^{L} F (z, 0) = L!.

According to Baker’s remark (see § 3.1.a), for ∥\underline{σ}∥ < S_{0}, the one variable function G\underline{σ}(z) = D\underline{σ}F (\underline{v}z) has at least (S_{0} − ∥\underline{σ}∥)S_{1} zeroes in B_{1}(0, r/|\underline{v}|). Hence

sup_{\underline{z}∈B_{n}(0,r)∩ℂ\underline{v}} |D\underline{σ}F (\underline{z})| ≤ (\frac{R^{2} + r^{2}}{2rR} )^{−(S_{0}−∥\underline{σ}∥)S_{1}} |D\underline{σ}F |_{R}.

This is the key point which explains why, in Baker’s extrapolation argument, the order of derivation needs to decrease; one compensates by increasing the set of points.

Assuming for simplicity ∥\underline{σ}∥ ≤ S_{0}/2, the exponent of 2rR/(R^{2} + r^{2}) is ≥ S_{0}S_{1}/2. For a function of n variables with S_{1} zeroes of multiplicity ≥ S_{0}, one should expect only an exponent S_{0}S_{1}^{1/n} (up to a small absolute multiplicative constant), but the point here is that these zeroes lie on a complex line, and this explains why the exponent can be as large as a constant multiple of S_{0}S_{1}. It is interesting to compare with the interpolation determinant method: the exponent which arises more naturally (see [W5] § 10.1.4) is S_{0}S_{1}^{1/n} and this is sufficient to achieve nontrivial estimates, but a refinement can be included (see [W5] Proposition 10.5), so that one reaches the same exponent (namely S_{0}S_{1}, up to a constant) as with the auxiliary function.

9 Here we do not use all the information: a zero of multiplicity ≥ S_{0} for F involves \binom{S_{0}+n−1}{n} conditions, while for G it involves only S_{0} conditions.

We extend Corollary 2.4 to the situation arising in Baker’s method. We deal with derivatives of functions of several variables, but the points we consider lie on a complex line V = ℂ\underline{v} ⊂ ℂ^{n}.

We first give a variant of Lemma 2.1 for functions of several variables.

Lemma 3.15. Let n ≥ 1 be an integer, V a complex subspace of dimension 1 of ℂ^{n}, R, r, ϱ and E be positive real numbers, F an element of H_{n}(R), \underline{ζ}_{1}, …, \underline{ζ}_{m} pairwise distinct elements of V, \underline{v} an element of V, σ_{1}, …, σ_{m} nonnegative integers and \underline{κ} an element of ℕ^{n}. Set

M = ∑_{i=1}^{m} max\{0, σ_{i} − ∥\underline{κ}∥\}.

Assume

R ≥ ϱ + max\{r, |\underline{v}|\}, r ≥ max_{1≤i≤m} |\underline{ζ}_{i}| and 1 ≤ E ≤ \frac{(R − ϱ)^{2} + r|\underline{v}|}{(R − ϱ)(r + |\underline{v}|)}.

Assume also that F satisfies

D\underline{σ}F (\underline{ζ}_{i}) = 0 for \underline{σ} ∈ ℕ^{n} with ∥\underline{σ}∥ < σ_{i} and 1 ≤ i ≤ m.

Then

|D\underline{κ}F (\underline{v})| ≤ \underline{κ}!ϱ^{−∥\underline{κ}∥}E^{−M}|F |_{R}.

Proof. Let \underline{v}_{0} ∈ V satisfy |\underline{v}_{0}| = 1. Define a function G ∈ H_{1}(R) of a single variable z by

G(z) = D\underline{κ}F (\underline{v}_{0}z).

Define also ζ_{1}, …, ζ_{m} in B_{1}(0, r) by \underline{ζ}_{i} = \underline{v}_{0}ζ_{i} (1 ≤ i ≤ m). The formula

G^{(k)}(z) = ∑_{∥\underline{τ}∥=k} \frac{k!}{\underline{τ}!}\underline{v}\underline{τ} ; 0D^{\underline{κ}+\underline{τ}} F (\underline{v}_{0}z) (k ≥ 0)

shows that G has a zero of multiplicity ≥ max\{0, σ_{i} − ∥\underline{κ}∥\} at ζ_{i} for 1 ≤ i ≤ m. According to Lemma 1.5 (with r replaced by |\underline{v}| and R by R − ϱ) we have

|G|_{|\underline{v}|} ≤ E^{−M}|G|_{R−ϱ}.

We derive the conclusion from Cauchy’s inequalities:

|G|_{R−ϱ} = sup_{|z|=R−ϱ} |D\underline{κ}F (\underline{v}_{0}z)| ≤ \underline{κ}!ϱ^{−∥\underline{κ}∥}|F |_{R}.

□

Remark. For n = 1 Lemma 3.15 contains Lemma 1.5, but not Lemma 2.1 when κ > 0.

The next result is a variant of Proposition 2.2 for functions of several variables.

Proposition 3.16. Let n ≥ 1 be an integer, V a complex line in ℂ^{n}, R a positive real number, φ_{1}, …, φ_{L} elements of H_{n}(R), \underline{ξ}_{1}, …, \underline{ξ}_{L} elements of V ∩B_{n}(0, R) and \underline{κ}_{1}, …, \underline{κ}_{L} elements of ℕ^{n}. Consider the determinant

Δ = det (D^{\underline{κ}^{μ}}φ_{λ}(\underline{ξ}_{μ}))_{1≤λ,μ≤L} .

Let m_{1}, …, m_{L} be nonnegative integers and, for 1 ≤ μ ≤ L and 1 ≤ i ≤ m_{μ}, let \underline{ζ}_{μi} be an element of V and σ_{μi} a nonnegative integer. We assume that for each μ = 1, …, L, the m_{μ} elements \underline{ζ}_{μ1}, …, \underline{ζ}_{μm_{μ}} are pairwise distinct. Set

M_{μ} = ∑_{i=1}^{m_{μ}} max\{0, σ_{μi} − ∥\underline{κ}_{μ}∥\} (1 ≤ μ ≤ L).

For 1 ≤ μ ≤ L, let r_{μ}, R_{μ}, ϱ_{μ} and E_{μ} be positive real numbers satisfying

R ≥ R_{μ} ≥ ϱ_{μ} + max\{r_{μ}, |\underline{ξ}_{μ}|\}, r_{μ} ≥ max_{1≤i≤m_{μ}} |\underline{ζ}_{μi}|

and

1 ≤ E_{μ} ≤ \frac{(R_{μ} − ϱ_{μ})^{2} + r_{μ}|\underline{ξ}_{μ}|}{(R_{μ} − ϱ_{μ})(r_{μ} + |\underline{ξ}_{μ}|)}·

Denote by Φ the analytic mapping

(φ_{1}, …, φ_{L}): B_{n}(0, R) → ℂ^{L}.

Assume that for any (μ, i, \underline{κ}) ∈ ℕ^{2+n} satisfying 1 ≤ μ ≤ L, 1 ≤ i ≤ m_{μ} and ∥\underline{κ}∥ < σ_{μi}, the μ vectors

(3.17)  D^{\underline{κ}^{1}}Φ(\underline{ξ}_{1}), …, D^{\underline{κ}^{μ−1}}Φ(\underline{ξ}_{μ−1}), D\underline{κ}Φ(\underline{ζ}_{μi})

in ℂ^{L} are linearly dependent. Then

|Δ| ≤ L! max_{τ∈𝔖_{L}} ∏_{μ=1}^{L}(\underline{κ}_{μ}!ϱ_{μ}^{−∥\underline{κ}_{μ}∥}E_{μ}^{−M_{μ}}|φ_{τ(μ)}|_{R_{μ}}).

Proof. We prove Proposition 3.16 by induction on L. For L = 1 we have Φ = φ_{1},

Δ = D^{\underline{κ}^{1}}Φ(\underline{ξ}_{1}), 1 ≤ E_{1} ≤ \frac{(R_{1} − ϱ_{1})^{2} + r_{1}|\underline{ξ}_{1}|}{(R_{1} − ϱ_{1})(r_{1} + |\underline{ξ}_{1}|)}

and hypothesis (3.17) reads

D\underline{κ}Φ(\underline{ζ}_{1i}) = 0 for ∥\underline{κ}∥ < σ_{1i} and 1 ≤ i ≤ m_{1}.

From Lemma 3.15 we deduce

|Δ| ≤ \underline{κ}_{1}!ϱ_{1}^{−∥\underline{κ}_{1}∥}E_{1}^{−M_{1}}|φ_{1}|_{R_{1}}.

Hence Proposition 3.16 is true in case L = 1.

Assume now that the conclusion is true for L replaced by L − 1. Define F ∈ B_{n}(0, R) by

(3.18)  F (\underline{z}) = det (D^{\underline{κ}^{1}}Φ(\underline{ξ}_{1}), …, D^{\underline{κ}^{L−1}}Φ(\underline{ξ}_{L−1}), Φ(\underline{z})) .

By assumption (3.17) with μ = L, for 1 ≤ i ≤ m_{L} and ∥\underline{κ}∥ < σ_{Li}, we have

D\underline{κ}F (\underline{ζ}_{Li}) = 0.

Since

Δ = D^{\underline{κ}^{L}}F (\underline{ξ}_{L}),

we deduce from Lemma 3.15

|Δ| ≤ \underline{κ}_{L}!ϱ_{L}^{−∥\underline{κ}_{L}∥}E_{L}^{−M_{L}}|F |_{R_{L}}.

We expand the determinant in the right hand side of (3.18) with respect to the last column: define, for 1 ≤ λ ≤ L,

Φ_{λ} = (φ_{1}, …, φ_{λ−1}, φ_{λ+1}, …, φ_{L}): B_{n}(0, R) → ℂ^{L−1}

and let Δ_{λ} denote the determinant of the (L − 1) × (L − 1) matrix

(D^{\underline{κ}^{1}}Φ_{λ}(\underline{ξ}_{1}), …, D^{\underline{κ}^{L−1}}Φ_{λ}(\underline{ξ}_{L−1})) .

We have

F (\underline{z}) = ∑_{λ=1}^{L}(−1)^{L−λ}φ_{λ}(\underline{z})Δ_{λ},

hence

|F |_{R_{L}} ≤ L max_{1≤λ≤L} |φ_{λ}|_{R_{L}}|Δ_{λ}|

and therefore

|Δ| ≤ \underline{κ}_{L}!Lϱ_{L}^{−∥\underline{κ}_{L}∥}E_{L}^{−M_{L}} max_{1≤λ≤L} |φ_{λ}|_{R_{L}}|Δ_{λ}|.

We fix an index λ^{0} ∈ {1, …, L} such that

|Δ| ≤ \underline{κ}_{L}!Lϱ_{L}^{−∥\underline{κ}_{L}∥}E_{L}^{−M_{L}}|φ_{λ^{0}}|_{R_{L}}|Δ_{λ^{0}}|.

Using the induction hypothesis, we deduce that there exists a bijective map t from {1, …, L − 1} to {1, …, λ^{0} − 1, λ^{0} + 1, …, L} such that

|Δ_{λ^{0}}| ≤ (L − 1)! ∏_{μ=1}^{L−1}(\underline{κ}_{μ}!ϱ_{μ}^{−∥\underline{κ}_{μ}∥}E_{μ}^{−M_{μ}}|φ_{t(μ)}|_{R_{μ}}).

Define τ ∈ 𝔖_{L} by τ(μ) = t(μ) for 1 ≤ λ < L and τ(L) = λ^{0}. Proposition 3.16 follows. □

We shall use a special case of Proposition 3.16.

Given a sequence (\underline{σ}_{ν}, ζ_{ν})_{0≤ν≤N} of elements in ℕ^{n} ×ℂ, an index ν in the range 0 ≤ ν ≤ N and a complex number ζ, we define the weight w_{ν}(ζ) of index ν of ζ in this sequence as follows: w_{ν}(ζ) = 0 if ζ_{i} ≠ ζ for 0 ≤ i ≤ ν and otherwise

(3.19) w_{ν}(ζ) = max\{∥\underline{σ}∥ ; there exists i with 0 ≤ i ≤ ν such that (\underline{σ}_{i}, ζ_{i}) = (\underline{σ}, ζ)\}.

We consider a sequence (\underline{σ}_{ν}, ζ_{ν})_{0≤ν≤N} which satisfies the following property: (3.20) For 0 ≤ ν ≤ N and for any \underline{σ} ∈ ℕ^{n} satisfying ∥\underline{σ}∥ < ∥\underline{σ}_{ν}∥, there exists i with

0 ≤ i < ν such that (\underline{σ}_{i}, ζ_{i}) = (\underline{σ}, ζ_{ν}).

Such a sequence will be called admissible.

In the case n = 1 the sequence (σ_{ν}, ζ_{ν})_{0≤ν≤N} which occurred in the hypotheses of Corollary 2.4 is admissible and satisfies

w_{ν}(ζ_{ν}) = σ_{ν} for 0 ≤ ν ≤ N.

For an admissible sequence (\underline{σ}_{ν}, ζ_{ν})_{0≤ν≤N}, for an analytic function F of n variables and for \underline{v} ∈ ℂ^{n}, the N + 1 conditions

(3.21)  D^{\underline{σ}^{ν}} F (\underline{v}ζ_{ν}) = 0 for 0 ≤ ν ≤ N

imply that the one-variable function G(z) = F (\underline{v}z) has a zero of multiplicity ≥ w_{N}(ζ) at each point ζ ∈ ℂ. Moreover, for any \underline{τ} ∈ ℕ^{n}, the same conditions (3.21) imply that the function G\underline{τ} (z) = D\underline{τ} F (\underline{v}z) has a zero at ζ of multiplicity

≥ max\{0, w_{N}(ζ) − ∥\underline{τ}∥\}.

Corollary 3.22. Let n and L be positive integers, φ_{1}, …, φ_{L} elements of H_{n}(R), 0 = ν_{0} ≤ ν_{1} < ⋯ < ν_{L} ≤ N nonnegative integers, \underline{σ}_{0}, …, \underline{σ}_{N} elements of ℕ^{n}, ζ_{0}, …, ζ_{N} complex numbers and \underline{v} ∈ ℂ^{n} \ \{0\}. Consider the determinant

Δ = det (D^{\underline{σ}^{νμ}} φ_{λ}(\underline{v}ζ_{ν_{μ}}))_{1≤λ,μ≤L} .

Assume (\underline{σ}_{ν}, ζ_{ν})_{0≤ν≤N} is an admissible sequence. For 0 ≤ ν ≤ N define

M_{ν} = ∑_{ζ∈B_{1}(0,R/|\underline{v}|)} max\{0, w_{ν}(ζ) − ∥\underline{σ}_{ν}∥\}.

For 1 ≤ μ ≤ L, let r_{μ}, R_{μ} and ϱ_{μ} be positive real numbers satisfying

R ≥ R_{μ} ≥ ϱ_{μ} + max\{r_{μ}, |\underline{v}ζ_{ν_{μ}}|\} and r_{μ} ≥ max_{0≤ν<ν_{μ}} |\underline{v}ζ_{ν}|.

Let E_{μ} satisfy

1 ≤ E_{μ} ≤ \frac{(R_{μ} − ϱ_{μ})^{2} + r_{μ}|\underline{v}ζ_{ν_{μ}}|}{(R_{μ} − ϱ_{μ})(r_{μ} + |\underline{v}ζ_{ν_{μ}}|)}·

Denote by Φ the analytic mapping

(φ_{1}, …, φ_{L}): B_{n}(0, R) → ℂ^{L}.

Assume that for 0 ≤ μ ≤ L − 1 and for ν_{μ} ≤ ν < ν_{μ+1} the system of μ + 1 vectors

D^{\underline{σ}^{ν_{1}}} Φ(\underline{v}ζ_{ν_{1}}), …, D^{\underline{σ}^{νμ}} Φ(\underline{v}ζ_{ν_{μ}}), D^{\underline{σ}^{ν}} Φ(\underline{v}ζ_{ν})

in ℂ^{L} is linearly dependent. Then

|Δ| ≤ L! max_{τ∈𝔖_{L}} ∏_{μ=1}^{L}(\underline{σ}_{ν_{μ}}!ϱ_{μ}^{−∥\underline{σ}_{νμ} ∥}E_{μ}^{−M_{νμ}} |φ_{τ(μ)}|_{R_{μ}}).

Remark. This statement does not contain Corollary 2.4, because the exponent M_{ν} is usually smaller than

∑_{ζ∈B_{1}(0,R/|\underline{v}|)} w_{ν}(ζ).

It does not seem clear how to get a result containing both the one variable Proposition 2.2 and Corollary 3.22. It might be easier to combine Corollary 3.22 with Proposition 10.5 of [W5].

Proof. We apply Proposition 3.16 with V = ℂ\underline{v},

\underline{ξ}_{μ} = \underline{v}ζ_{ν_{μ}} and \underline{κ}_{μ} = \underline{σ}_{ν_{μ}} (1 ≤ μ ≤ L).

We define m_{μ}, \underline{ζ}_{μi} and σ_{μi} as follows: for 1 ≤ μ ≤ L, we consider the sequence (ζ_{0}, …, ζ_{ν_{μ}}) given by the hypotheses of Corollary 3.22, we denote by m_{μ} the number of distinct elements in this sequence, by ζ_{μi} these distinct elements, by σ_{μi} the weight of index ν_{μ} of ζ_{μi} in this sequence, and we set \underline{ζ}_{μi} = \underline{v}ζ_{μi}. Therefore

∑_{ζ∈B_{1}(0,R/|\underline{v}|)} max\{0, w_{ν_{μ}}(ζ) − ∥\underline{σ}_{ν_{μ}}∥\} ≥ ∑_{i=1}^{m_{μ}} max\{0, σ_{μi} − ∥\underline{κ}_{μ}∥\} (1 ≤ μ ≤ L).

Corollary 3.22 follows. □

3.5 Extrapolation with an Interpolation Determinant

We give a proof of Baker’s Theorem 3.1 by means of an interpolation determinant and an extrapolation; the zero estimate which will enable us to get the conclusion is Tijdeman’s Lemma 3.12.

Let T_{0}, T_{1}, …, T_{n}, J, S_{0}^{(j)} and S_{1}^{(j)} (0 ≤ j ≤ J) be positive integers. Assume that the sequence (S_{0}^{(j)})_{0≤j≤J} is decreasing and that the sequence (S_{1}^{(j)})_{0≤j≤J} increases. We write

S_{0} and S_{1} for S_{0}^{(0)} and S_{1}^{(0)} respectively and we set L = T_{0} ⋯ T_{n}, T = T_{1} + ⋯ + T_{n}. We denote by {φ_{1}, …, φ_{L}} the L exponential monomials

z^{τ} α_{1}^{t_{1}z} ⋯ α_{n}^{t_{n}z} for 0 ≤ τ < T_{0} and 0 ≤ t_{i} < T_{i} (1 ≤ i ≤ n).

For 0 ≤ j ≤ J, define S_{j} as the set of (\underline{σ}, s) in ℕ^{n}×ℕ satisfying ∥\underline{σ}∥ < S_{0}^{(j)} and 0 ≤ s < S_{1}^{(j)}. We need to choose an ordering (\underline{σ}_{ν}, s_{ν})_{0≤ν<N_{J}} on the union S of these sets. Define

N_{−1} = 0, N_{0} = \binom{S_{0} + n − 1}{n}S_{1}

and, for 1 ≤ j ≤ J,

N_{j} = N_{j−1} + \binom{S_{0}^{(j)} + n − 1}{n}(S_{1}^{(j)} − S_{1}^{(j−1)}).

We choose an ordering so that

\{(\underline{σ}_{ν}, s_{ν}) ; 0 ≤ ν < N_{0}\} = \{(\underline{σ}, s) ∈ ℕ^{n} × ℕ ; ∥\underline{σ}∥ < S_{0} and 0 ≤ s < S_{1}\}

and, for 1 ≤ j ≤ J,

\{(\underline{σ}_{ν}, s_{ν}) ; N_{j−1} ≤ ν < N_{j}\} = \{(\underline{σ}, s) ∈ ℕ^{n} × ℕ ; ∥\underline{σ}∥ < S_{0}^{(j)} and S_{1}^{(j−1)} ≤ s < S_{1}^{(j)}\}.

It remains to specify the order inside each interval N_{j−1} ≤ ν < N_{j} (0 ≤ j ≤ J). There is no complete rigidity, but it suffices to say that we choose an order for which the resulting sequence is admissible (see (3.20)).

For each τ, \underline{t}, \underline{σ} and s, consider the algebraic number γ_{τ\underline{t}}^{(\underline{σ}s)} given by (3.6) and build up the matrix

M = (γ_{τ\underline{t}}^{(\underline{σ}_{ν}s_{ν})})_{{(τ,\underline{t}) \atop 0≤ν<N_{J}}}

with L rows indexed by (τ, \underline{t}) with 0 ≤ τ < T_{0} and 0 ≤ t_{i} < T_{i} (1 ≤ i ≤ n), and N_{J} columns indexed by ν with 0 ≤ ν < N_{J}.

We first deduce from Lemma 3.12 that M has rank L. Indeed otherwise there exists a nonzero polynomial P ∈ ℂ[Y_{0}, …, Y_{n}] for which

∂^{\underline{σ}^{ν}} P (s_{ν}, α_{1}^{s_{ν}} , …, α_{n}^{s_{ν}} ) = 0 for 0 ≤ ν < N_{J},

which means

∂\underline{σ}P (s, α_{1}^{s}, …, α_{n}^{s}) = 0 for (\underline{σ}, s) ∈ S.

In particular the function G(z) = P (z, α_{1}^{z}, …, α_{n}^{z}) satisfies

G^{(k)}(s) = 0 for 0 ≤ k < S_{0}^{(J)} and 0 ≤ s < S_{1}^{(J)}.

In order to apply Lemma 3.12, we assume

S_{0}^{(J)}S_{1}^{(J)} > 2L + c_{8}T S_{1}^{(J)}.

Since M has maximal rank L, one can select L columns which produce a nonzero determinant Δ. We select them minimal as in the proof of § 2.2 and we write

Δ = det(γ_{τ\underline{t}}^{(\underline{σ}_{νμ} s_{νμ})})_{{(τ,\underline{t}) \atop 1≤μ≤L}} .

We want to derive from Corollary 3.22 an upper bound for |Δ|. Set

\underline{v} = (1, log α_{1}, …, log α_{n−1}).

We need first to estimate the weights (3.19) related to our sequence (\underline{σ}_{ν}, s_{ν})_{0≤ν<N_{J}}. Let ν be an index with N_{0} ≤ ν < N_{J}; define j in the range 1 ≤ j ≤ J by N_{j−1} ≤ ν < N_{j}. Thanks to the construction of the sequence (\underline{σ}_{ν}, s_{ν})_{0≤ν<N_{J}}, we have

w_{ν}(s) = S_{0}^{(j−1)} − 1 for 0 ≤ s < S_{1}^{(j−1)}.

Since ∥\underline{σ}_{ν}∥ < S_{0}^{(j)}, it follows that the number

M_{ν} = ∑_{ζ∈B_{1}(0,R/|\underline{v}|)} max\{0, w_{ν}(ζ) − ∥\underline{σ}_{ν}∥\}

satisfies

(3.23)  M_{ν} ≥ (S_{0}^{(j−1)} − S_{0}^{(j)})S_{1}^{(j−1)}.

For each μ with 1 ≤ μ ≤ L we define j_{μ} ∈ {0, …, J} by

N_{j_{μ}−1} ≤ ν_{μ} < N_{j_{μ}}.

Since N_{−1} = 0, for ν_{μ} < N_{0} we have j_{μ} = 0. Hence for any μ = 1, …, L, we have

∥\underline{σ}_{ν_{μ}}∥ < S_{0}^{(j_{μ})} and 0 ≤ s_{ν_{μ}} < S_{1}^{(j_{μ})}.

We want to use (3.23) for each ν_{μ} with μ in the range L/2 < μ ≤ L, so we need to check ν_{μ} ≥ N_{0} for these μ. For this reason we require our parameters to satisfy

(3.24)  2\binom{S_{0} + n − 1}{n}S_{1} ≤ L.

Since

Δ = det (D^{\underline{σ}^{νμ}} φ_{λ}(\underline{v}s_{ν_{μ}}))_{1≤λ,μ≤L} ,

we may apply Corollary 3.22 with

r_{μ} = c_{9}S_{1}^{(j_{μ})}, ϱ_{μ} = 1, R_{μ} = 3er_{μ}, E_{μ} = e.

We deduce

\begin{aligned}|Δ| & ≤ L! ∏_{μ=1}^{L}(\underline{σ}_{ν_{μ}}!e^{−M_{νμ}} R_{μ}^{T_{0}}e^{c_{10}T R_{μ}}) \\ & ≤ L! exp \{∑_{μ=1}^{L}(−M_{ν_{μ}} + c_{11}(T_{0} + S_{0}) log(LS_{1}^{(j_{μ})}) + c_{12}T S_{1}^{(j_{μ})})\} \end{aligned}

(recall that S_{0}^{(j)} ≤ S_{0} for 0 ≤ j ≤ J).

Since Δ is a nonzero number in the field 𝕂, Liouville’s Proposition 1.19 produces a lower bound for |Δ|:

|Δ| ≥ L!^{−D+1} exp {∑_{μ=1}^{L}(−c_{13}(T_{0} + S_{0}) log(LS_{1}^{(j_{μ})}) − c_{14}T S_{1}^{(j_{μ})})} .

Define, for 1 ≤ μ ≤ L,

Q_{μ} = c_{15}(T_{0} + S_{0}) log(LS_{1}^{(j_{μ})}) + c_{16}T S_{1}^{(j_{μ})}.

We assume our parameters are selected so that

2Q_{μ} < (S_{0}^{(j_{μ}−1)} − S_{0}^{(j_{μ})}) S_{1}^{(j_{μ}−1)}.

Then we have

M_{ν_{μ}} ≥ 0 for 1 ≤ μ < \frac{L}{2} − 1,

∑_{1≤μ≤L/2} Q_{μ} ≤ ∑_{L/2<μ≤L} Q_{μ}

and by (3.23)

M_{ν_{μ}} > 2Q_{μ} for \frac{L}{2} − 1 ≤ μ ≤ L.

Therefore

∑_{μ=1}^{L}(M_{ν_{μ}} − Q_{μ}) > 0,

and the contradiction follows.

It remains to select our parameters. We take the same values as in § 3.3, namely (3.5) and (3.14):

T_{0} = 2[𝕂 : ℚ]N^{2n+1}, T_{1} = ⋯ = T_{n} = N^{2n−1}, S_{0} = N^{2n+1}, S_{1} = N

and

J = 2n^{2}, S_{0}^{(j)} = [S_{0}/2^{j}], S_{1}^{(j)} = N^{j+1} (1 ≤ j ≤ J).

4 Introduction to Diophantine Approximation

4.1 On a Conjecture of Mahler

Consider the successive powers of e, namely

e, e^{2}, e^{3}, e^{4}, …,

and their distances to the nearest integer

∥e∥_{ℤ}, ∥e^{2}∥_{ℤ}, ∥e^{3}∥_{ℤ}, ∥e^{4}∥_{ℤ}, …

It is an open problem to prove that these numbers are equidistributed^{10} in the interval (0, 1/2). We are interested in estimating Ψ(B) = min_{1≤b≤B} ∥e^{b}∥_{ℤ} as B → ∞ from below, but let us first say one word on upper bounds. Essentially nothing is known: it is not yet proved that Ψ(B) tends to 0 when B → ∞:

(?)  For any ϵ > 0 there exists b ∈ ℕ such that ∥e^{b}∥_{ℤ} < ϵ,

but it is expected that BΨ(B) tends to 0 when B → ∞:

(?)  For any ϵ > 0 there exists b ∈ ℕ such that ∥e^{b}∥_{ℤ} < \frac{ϵ}{b}·

We come back to lower bounds for Ψ(B). In [M1] p. 397, K. Mahler says:

• … one can easily show that

| log a − b| < \frac{1}{a}

for an infinite increasing sequence of positive integers a and suitable integers b. Indeed, this inequality holds for each pair (a, b) of positive integers for which log a < b < log(a + 1).

For a given positive integer B, if the B numbers

∥e^{b}∥_{ℤ} (1 ≤ b ≤ B)

were evenly distributed in the interval (0, 1/2), the smallest of them would not be less than a constant times 1/B. This is likely to be too optimistic for a conjecture, but a more reasonable question is:

• Is there an absolute constant κ ≥ 1 such that

(4.1)  |e^{b} − a| ≥ \frac{1}{b^{κ}}

for any positive integers a and b with b ≥ 2?

10A result of Koksma ([K] Chap. VIII § 3 N^{∘} 12 Satz 16) states that for almost all real θ ≥ 1 (in the sense of Lebesgue measure) the sequence of fractional parts of θ^{k} is uniformly distributed modulo 1. On the other hand there there is no known example of a transcendental number which satisfies this property.

A straightforward computation shows that this inequality holds with κ = 2.25 for 2 ≤ b ≤ 100.

If |e^{b} − a| is small, then b is close to log a, hence

e^{b} − a = a(e^{b−log a} − 1)

is close to a(b − log a). One easily deduces that (4.1) is equivalent to:

• Is there an absolute constant κ ≥ 1 such that

(4.2)  |b − log a| ≥ \frac{1}{a(log a)^{κ}}

for any positive integers a and b with a ≥ 3?

A symmetric refinement of both (4.1) and (4.2) could be formulated by restricting to large values of a and b, say a ≥ a_{0} and b ≥ b_{0}, and by introducing functions x(log x)(log_{2} x) ⋯ (log_{k−1} x)(log_{k} x)^{κ}, where log_{2} = log log and log_{k} is the k-th iterated logarithm; but since (4.1) and (4.2) are not yet known, it seems more reasonable to consider weaker statements rather than stronger ones!

A weaker estimate than (4.1) and (4.2), which is still an open question, is usually attributed to Mahler (for instance in [W5] Chap. 14, open problems and [W6]) but is not explicit in [M1] nor in [M2].

Conjecture 4.3. Does there exist a positive absolute constant c such that, for any positive rational integers a and b with a ≥ 2,

|e^{b} − a| ≥ a^{−c}?

The same argument as above shows that Conjecture 4.3 amounts to the existence of c′ > 1 with

|b − log a| ≥^{?} e^{−c′b}.

If we restrict to sufficiently large a and b, then c′ is essentially c + 1.

The best known estimate in the direction of Conjecture 4.3 is due to Mahler himself [M1]:

(4.4)  |e^{b} − a| ≥ a^{−c log log a} and |b − log a| ≥ b^{−cb}

for a ≥ 3 and b ≥ 2, with an absolute constant c > 0. Assuming a is sufficiently large, K. Mahler in [M1] gave a sharp explicit numerical value for c, namely c = 40 (for both estimates), which he refined in [M2], getting c = 33. A further refinement is due to F. Wielonsky [Wi]: for sufficiently large a, Mahler’s estimates (4.4) hold with c = 20.

Fel’dman proved several lower bounds for |e^{β} − α| and |β − log α| when α and β are algebraic numbers. References are given in [NW], where a refinement including most previously known results was established.

The following result is a slight refinement of the main estimate of [NW] (apart from the fact that our constant c is not explicit).

Theorem 4.5. There is an absolute constant c > 0 with the following property. Let α, β be two nonzero algebraic numbers and λ a logarithm of α. Define 𝕂 = ℚ(α, β) and D = [𝕂: ℚ]. Let A, B and E be positive real numbers satisfying E ≥ e,

log A ≥ max{h(α) , D^{−1} log E , E|λ|/D}

and^{11}

(4.6)  log B ≥ h(β) + log(D log A) + log D + \frac{1}{D} log E.

Then

|β − λ| ≥ exp{−cD^{2}(log A)(log B)(D log D + log E)(log E)^{−2}}.

According to Theorem 5 of [NW], if we replace the condition (4.6) on B by

log B ≥ h(β) + log_{+} log A + log D + log E,

then c = 105 500 is an admissible value. It is not a big challenge to produce a smaller numerical value; for instance, taking D = 1 and E = e, one deduces from [NW] that for any positive rational numbers a and b,

|e^{b} − a| ≥ exp{−1000(h(b) + log log A + 12)(log A + 12)},

where log A = max{1, h(a)}. In particular we recover Mahler’s results (4.4) on |e^{b} − a| and |b − log a| as a consequence of Theorem 4.5.

On the other hand, any further improvement of Theorem 4.5 in terms of either A, B, D or E seems to require a new idea.

For the proof of Theorem 4.5, a refinement of Propositions 1.11 and 1.12 is needed. To begin with, assume for simplicity that α = a and β = b are positive integers. As we shall see, the condition f(e^{b}, b) ≠ 0, which occurs in both Propositions 1.11 and 1.12, does not suffice any more, but we need f(a, b) ≠ 0. This is achieved by the following zero estimate due to Yu.V. Nesterenko.

Let b be a complex number. The derivation D_{b} = (∂/∂X) + bY (∂/∂Y), on the ring of polynomials in two variables X and Y, has the following remarkable property: for a polynomial P ∈ ℂ[X, Y], the derivative of the complex function P (z, e^{bz}) is the exponential polynomial (D_{b}P)(z, e^{bz}).

Here is Lemma 2 of [NW] (see also [W5] Prop. 2.14).

Lemma 4.7. Assume b ≠ 0. Let T_{0}, T_{1}, S_{0} and S_{1} be positive integers satisfying

S_{0}S_{1} > (T_{0} + S_{1} − 1)T_{1}.

11It does not make any difference if one omits the summand log(D log A) in (4.6), provided that one replaces the factor (log B) in the conclusion by (log B + log(D log A)).

Let (ξ_{0}, η_{0}), …, (ξ_{S_{1}−1}, η_{S_{1}−1}) be elements in ℂ × ℂ^{×} with ξ_{0}, …, ξ_{S_{1}−1} pairwise distinct. Then there is no nonzero polynomial P ∈ ℂ[X, Y], of degree < T_{0} in X and of degree < T_{1} in Y, which satisfies

D_{b}^{σ}P (ξ_{s}, η_{s}) = 0 for 0 ≤ s < S_{1} and 0 ≤ σ < S_{0}.

Remark. An obvious necessary condition for the nonexistence of P is

S_{0}S_{1} ≥ T_{0}T_{1}.

Indeed if S_{0}S_{1} < T_{0}T_{1}, then the homogeneous linear system of equations given by the conditions D_{b}^{σ}P (ξ_{μ}, η_{μ}) = 0 has a nontrivial solution, because the number of unknowns (which are the coefficients of P) is larger than the number of equations. On the other hand, if η_{0} = ⋯ = η_{S_{1}−1} and S_{0} < T_{1}, then

(Y − η_{0})^{S_{0}}

is a nontrivial solution.

Hence one cannot replace T_{0} + S_{1} − 1 in the hypothesis of Lemma 4.7 by a smaller number than max{T_{0} , S_{1}}.

Sketch of proof of Theorem 4.5. We first consider the special case where α = a and β = b are positive integers with, say, b ≥ 3. Our goal is to check

(4.8)  |b − log a| ≥ exp{−cb(log b)}

for some absolute constant c > 0 (which we shall not compute explicitly). Recall that when b and log a are close together, then b(log b) is close to (log a)(log b).

Here the parameter E of Theorem 4.5 is not helpful and we shall just take E = e, but we shall keep it for a while to show why we cannot get a better result by taking a large quotient R/r in Schwarz’ Lemma.

If we were following the proof of Proposition 1.12 in § 1.6, we would get b(log b)^{2} instead of b log b in (4.8). Let us explain why.

As we shall see in (4.17), the conclusion will have the shape

|b − log a| ≥ E^{−L}.

Set U = V = W = (1/36)L log E, so that (1.14) holds (for simplicity we forget the upper bound for the length of f). In order to check (1.13) we need the number L log E to be larger than bET_{1}S_{1} and than T_{0} log(bE). Since L = T_{0}T_{1}, we get the conditions

T_{0} log E > bES_{1} and T_{1} log E > log b.

Writing now S_{0} = T_{0}T_{1}/S_{1}, we obtain

S_{0} > b(log b)E(log E)^{−2}.

In particular this explains our choice E = e.

The conclusion of Proposition 1.12 involves S_{0}!, so we shall require L > log(S_{0}!), hence S_{1} > log S_{0} (we discard lower order terms), and therefore L = S_{0}S_{1} should satisfy L > b(log b)^{2}.

From this point of view the choice of parameters in § 1.5 yields the smallest possible value for L in terms of b.

The main term which is responsible for (log b)^{2} is S_{0}!. If we could forget it, we would get only a constant times b(log b)(log log b) for L by taking

T_{0} = N^{2}b, T_{1} = N^{2}[log b], S_{0} = N^{3}b[log b], and S_{1} = N

with a suitable (sufficiently large) integer N. The quantity b(log b)(log log b) arises from the term S_{0} log T_{1} which appears in the estimate for the length of f in Proposition 1.12.

We perform a very simple change of variables^{12} which will allow us to take a smaller value for S_{0}. In place of evaluating the functions z^{τ} e^{tz} at the points z = sb, we consider rather z^{τ} e^{btz} at the points z = s. This means that we replace

(\frac{d}{dz} )^{σ} (z^{τ} e^{tz})(sb) = ∑_{κ=0}^{min\{τ,σ\}} \frac{σ!τ!}{κ!(σ − κ)!(τ − κ)!}(sb)^{τ−κ}t^{σ−κ}e^{tsb}

by

(\frac{d}{dz} )^{σ} (z^{τ} e^{btz})(s) = ∑_{κ=0}^{min\{τ,σ\}} \frac{σ!τ!}{κ!(σ − κ)!(τ − κ)!}s^{τ−κ}(tb)^{σ−κ}e^{tsb}.

While the former number is a positive real number bounded from above by

(bS_{1})^{τ} T_{1}^{σ} min \{(1 + \frac{σ}{bT_{1}S_{1}} )^{τ} ; (1 + \frac{τ}{bT_{1}S_{1}} )^{σ}\} e^{bT_{1}S_{1}},

(see (1.7)), the latter is at most

S_{1}^{τ} (bT_{1})^{σ} min \{(1 + \frac{σ}{bT_{1}S_{1}} )^{τ} ; (1 + \frac{τ}{bT_{1}S_{1}} )^{σ}\} e^{bT_{1}S_{1}}.

Thanks to this change of variables a variant of Proposition 1.12 can be deduced, where the upper bound for the length of f is

Le^{W} S_{1}^{T_{0}}(T_{0} + T_{1})^{S_{0}},

the degree of f in Z_{2} is < S_{0} and (1.13) is replaced by

U ≥ log L + T_{0} log(ES_{1}) + |b|_{+}ET_{1}S_{1}.

A suitable choice for the parameters is then

T_{0} = N^{2}b[log b], T_{1} = N^{2}[log log b],
S_{0} = N^{3}b[log log b], S_{1} = N[log b],

12Compare with the “duality” of [W5] § 13.7.

which may be used to prove (4.8) with

exp{−cb(log b)} replaced by exp{−cb(log b)(log log b)}

(for b ≥ 16, say).

In order to remove the extra factor log log b and to reach the desired estimate (4.8) involving only exp{−cb(log b)}, we need to get rid of the term S_{1}^{T_{0}} in the upper bound for L(f) in Proposition 1.12, so that one could take

T_{0} = N^{2}b[log b], T_{1} = N^{2}, S_{0} = N^{3}b, and S_{1} = N[log b].

This is achieved by means of the so-called Fel’dman’s polynomials. We use here a variant due to E.M. Matveev (see [W5] Lemma 9.8): we replace z^{τ} by the polynomials^{13} △(z; τ, T_{0}^{♯}) which we now define.

4.2 Fel’dman’s Polynomials

For r a nonnegative integer and z ∈ ℂ, consider the binomial (or Fel’dman’s) polynomial

△(z; r) = \frac{z(z + 1) ⋯ (z + r − 1)}{r!}

if r > 0 and △(z; 0) = 1.

Let τ ≥ 0 and T_{0}^{♯} > 0 be two integers. Following Matveev [Mat I] § 7, we define a polynomial △(z; τ, T_{0}^{♯}) ∈ ℚ[z] of degree τ by

△(z; τ, T_{0}^{♯}) = (△(z; T_{0}^{♯}))^{q} · △(z; r),

where q and r are the quotient and remainder of the division of τ by T_{0}^{♯}:

τ = T_{0}^{♯}q + r, 0 ≤ r < T_{0}^{♯}.

For σ ≥ 0, define

△(z; τ, T_{0}^{♯}, σ) = (\frac{d}{dz} )^{σ} △ (z; τ, T_{0}^{♯}).

For any positive integer n, denote by ν(n) the least common multiple of 1, 2, …, n. The estimate

ν(n) ≤ e^{107n/103}

can be deduced from the prime number Theorem (see for instance [W5] § 9.2.1). The next result is due to Matveev [Mat I] Lemma 7.1 (see also [W5] Lemma 9.8).

Lemma 4.9. Let T_{0}^{♯} > 0, τ > 0 and σ ≥ 0 be rational integers. For any integer κ in the interval 0 ≤ κ ≤ σ and any rational integer m ∈ ℕ, the number

13The polynomial denoted here by △(z; τ, T_{0}^{♯}) is denoted δ_{T_{0}^{♯}}(z; τ) in [W5] § 9.2.1, while our notation △(z; τ, T_{0}^{♯}, σ) corresponds to δ_{T_{0}^{♯}}(z; τ, σ) in [W5] § 9.2.1. Here we follow Matveev’s notation in [Mat I] § 7.

ν(T_{0}^{♯})^{σ} · \frac{1}{κ!} △ (m; τ, T_{0}^{♯}, κ)

is a nonnegative rational integer. Moreover, for any complex number z, we have

∑_{κ=0}^{σ} \binom{σ}{κ}|△(z; τ, T_{0}^{♯}, κ)| ≤ σ!e^{τ+T_{0}^{♯}} (\frac{|z|}{T_{0}^{♯}} + 1)^{τ} .

4.3 Output of the Transcendence Argument

Here is the basic estimate which follows from the transcendence proof (by means of the interpolation determinant method) and will enable us to deduce Theorem 4.5. It is worth to notice that Theorem 4.10 does not involve any arithmetic assumption (the proof does not use Liouville’s inequality).

Theorem 4.10. Let b be a nonzero complex number. Let T_{0}, T_{0}^{♯}, T_{1}, S_{0}, S_{1}, K be positive integers and E a real number with E ≥ e. Set L = T_{0}T_{1} and assume 0 ≤ K < L and

S_{0}S_{1} > (T_{0} + S_{1} − 1)T_{1}.

Then there exists a set {f_{1}, …, f_{M}} of polynomials in the ring ℤ[Z_{1}, Z_{2}], each of degree < LT_{1}S_{1} in Z_{1} and degree < LS_{0} in Z_{2}, of length at most

L!ν(T_{0}^{♯})^{LS_{0}}(S_{0}!)^{L}e^{(T_{0}^{♯}+T_{0})L} (\frac{S_{1}}{T_{0}^{♯}} + 1)^{T_{0}L} T_{1}^{S_{0}L},

such that the polynomials {f_{1}(Z_{1}, b), …, f_{M}(Z_{1}, b)} in ℂ[Z_{1}] have no common zero in ℂ^{×} and such that

\frac{1}{L} max_{{\underline{k}∈ℕ^{2} \atop ∥\underline{k}∥<K}} log |\frac{1}{\underline{k}!}D\underline{k}f_{i}(e^{b}, b)| ≤ −\frac{1}{2L}(L−K)(L−K−1) log E+log(8L)+S_{0} log(ν(T_{0}^{♯})S_{0}E)

+T_{0}^{♯} + T_{0} + T_{0} log (\frac{S_{1}E}{T_{0}^{♯}} + 1) + S_{0} log(2T_{1}|b|_{+}) + T_{1}S_{1}E|b|

for 1 ≤ i ≤ M.

In the conclusion D\underline{k} stands for (∂/∂Z_{1})^{k_{1}}(∂/∂Z_{2})^{k_{2}}.

For the proof of Theorem 4.10 we shall need the following refinement of Lemma 1.9:

Lemma 4.11. Let L and L′ be positive integers with L′ ≤ L and φ_{1}, …, φ_{L′} entire functions in ℂ. Further let ζ_{1}, …, ζ_{L} be complex numbers and σ_{1}, …, σ_{L} nonnegative integers. Let S_{0} satisfy S_{0} ≥ max_{1≤μ≤L} σ_{μ}. Furthermore, for L′ + 1 ≤ λ ≤ L and 1 ≤ μ ≤ L let δ_{λμ} be a complex number. For 1 ≤ λ ≤ L′ and 1 ≤ μ ≤ L we define

δ_{λμ} = φ_{λ}^{(σ_{μ})}(ζ_{μ}).

Finally, let E > 1 and Q_{1}, …, Q_{L} be positive real numbers satisfying

Q_{λ} ≥ log sup_{|z|=E} max_{1≤μ≤L} |φ_{λ}^{(σ_{μ})}(zζ_{μ})| (1 ≤ λ ≤ L′),

Q_{λ} ≥ log max_{1≤μ≤L} |δ_{λ,μ}| (L′ + 1 ≤ λ ≤ L).

We consider the determinant

Δ = det(δ_{λμ})_{1≤λ,μ≤L}.

Then we have

log |Δ| ≤ −\frac{1}{2}L′(L′ − 1) log E + L′S_{0} log E + log(L!) + Q_{1} + ⋯ + Q_{L}.

Proof. Lemma 4.11 is essentially the case n = 1 of Lemma 7.5 in [W5] but it includes derivatives like in Proposition 9.13 of [W5].

For 1 ≤ μ ≤ L, we define functions d_{1μ}(z), …, d_{Lμ}(z) by

d_{λμ}(z) = \{ \begin{matrix}φ_{λ}^{(σ_{μ})}(ζ_{μ}z) & for 1 ≤ λ ≤ L′, \\ δ_{λμ} & for L′ < λ ≤ L. \end{matrix}

From Lemma 9.2 of [W5] we deduce that the function

D(z) = det(d_{λμ}(z))_{1≤λ,μ≤L}

has a zero at the origin of multiplicity ≥ (1/2)L′(L′ −1)−L′S_{0}. As D(1) = Δ, we conclude the proof of Lemma 4.11 by using Schwarz’ Lemma 1.5. □

Proof of Theorem 4.10. For each a ∈ ℂ^{×} we shall construct a polynomial f ∈ ℤ[Z_{1}, Z_{2}] (depending on a) satisfying the required conditions and such that

f(a, b) ≠ 0.

Since the degrees and lengths of these polynomials will be bounded (independently of a), this will mean that we end up with a finite set of polynomials {f_{1}, …, f_{M}} in ℤ[Z_{1}, Z_{2}], such that the only possible common zero in ℂ of the polynomials f_{i}(Z_{1}, b) (i = 1, …, M) is Z_{1} = 0.

Define, for 0 ≤ τ < T_{0} and 0 ≤ t < T_{1},

φ_{τt}(z) = △(z; τ, T_{0}^{♯})e^{tbz}.

For σ = 0, 1, …, S_{0} − 1, we have

φ_{τt}^{(σ)}(z) = ∑_{κ=0}^{σ} \binom{σ}{κ} △ (z; τ, T_{0}^{♯}, κ)(tb)^{σ−κ}e^{tbz}.

Define f̃_{τt}^{(σs)} ∈ ℚ[Z_{1}, Z_{2}] by

f̃_{τt}^{(σs)}(Z_{1}, Z_{2}) = ∑_{κ=0}^{σ} \binom{σ}{κ} △ (s; τ, T_{0}^{♯}, κ)(tZ_{2})^{σ−κ}Z_{1}^{ts},

so that^{14}

(4.12)  f̃_{τt}^{(σs)}(e^{b}, b) = φ_{τt}^{(σ)}(s)

and

f̃_{τt}^{(σs)}(a, b) = D_{b}^{σ}(△(X; τ, T_{0}^{♯})Y^{t})(s, a^{s}).

This polynomial f̃_{τt}^{(σs)} has rational coefficients. In order to get a polynomial with integer coefficients, we multiply it by a denominator: using Lemma 4.9, define f_{τt}^{(σs)} ∈ ℤ[Z_{1}, Z_{2}] by

f_{τt}^{(σs)}(Z_{1}, Z_{2}) = ν(T_{0}^{♯})^{σ}f̃_{τt}^{(σs)}(Z_{1}, Z_{2}).

In place of the exponent σ for ν(T_{0}^{♯}) we could as well put the exponent min{σ, τ}, because △(z; τ, T_{0}^{♯}, κ) = 0 for κ > τ. Notice also that we use only the fact that

ν(T_{0}^{♯})^{σ} △ (s; τ, T_{0}^{♯}, κ)

is an integer; indeed, according to Lemma 4.9, it is a multiple of κ!, but we do not use this fact^{15}.

The polynomial f_{τt}^{(σs)} has degree ≤ ts in Z_{1}, degree ≤ σ in Z_{2} and length at most

ν(T_{0}^{♯})^{σ} ∑_{κ=0}^{σ} \binom{σ}{κ} △ (s; τ, T_{0}^{♯}, κ)t^{σ−κ} ≤ ν(T_{0}^{♯})^{S_{0}}S_{0}!e^{T_{0}^{♯}+T_{0}} (\frac{S_{1}}{T_{0}^{♯}} + 1)^{T_{0}} T_{1}^{S_{0}}.

Consider the matrix

M = (f_{τt}^{(σs)}(a, b))_{{(τ,t) \atop (σ,s)}}

with L rows indexed by (τ, t), (0 ≤ τ < T_{0}, 0 ≤ t < T_{1}) and S_{0}S_{1} columns indexed by (σ, s) with 0 ≤ σ < S_{0}, 0 ≤ s < S_{1}.

We claim that M has rank L = T_{0}T_{1}. Indeed consider a linear dependence relation between the rows:

∑_{τ=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} c_{τt}f_{τt}^{(σs)}(a, b) = 0 for 0 ≤ σ < S_{0} and 0 ≤ s < S_{1}.

Define

14The upper index (σs) in the notation f̃_{τt}^{(σs)} is not a derivation, while φ_{τt}^{(σ)} = (d/dz)^{σ}φ_{τt}. 15Maybe one should, in order to remove the extra log log A which arises in log B + log log A…

P (X, Y) = ∑_{τ=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} c_{τt} △ (X; τ, T_{0}^{♯})Y^{t}.

Then we have

D_{b}^{σ}P (s, a^{s}) = ∑_{τ=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} c_{τt}f̃_{τt}^{(σs)}(a, b),

hence

D_{b}^{σ}P (s, a^{s}) = 0 for 0 ≤ σ < S_{0} and 0 ≤ s < S_{1}.

According to Lemma 4.7 with

ξ_{s} = s, η_{s} = a^{s} (0 ≤ s < S_{1})

and thanks to the assumption

S_{0}S_{1} > (T_{0} + S_{1} − 1)T_{1},

we deduce c_{τt} = 0 for 0 ≤ τ < T_{0} and 0 ≤ t < T_{1}.

Hence M has rank L = T_{0}T_{1}. Select L columns of M with indices (σ_{μ}, s_{μ}) (1 ≤ μ ≤ L), which are linearly independent. This means that the polynomial

f(Z_{1}, Z_{2}) = det(f_{τt}^{(σ_{μ}s_{μ})}(Z_{1}, Z_{2}))_{{(τ,t) \atop 1≤μ≤L}}

does not vanish at the point (a, b). The degree of f in Z_{1} is < LT_{1}S_{1} and in Z_{2} is < LS_{0}. We bound from above the length of f as follows: write

f_{τt}^{(σ_{μ}s_{μ})}(Z_{1}, Z_{2}) = Z_{1}^{ts_{μ}} ∑_{j=0}^{σ_{μ}} q_{jτtμ}Z_{2}^{j}

with

q_{jτtμ} = ν(T_{0}^{♯})^{σ_{μ}} \binom{σ_{μ}}{j} △ (s_{μ}; τ, T_{0}^{♯}, σ_{μ} − j)t^{j}.

By Lemma 4.9 we have

L(f_{τt}^{(σ_{μ}s_{μ})}) = ∑_{j=0}^{σ_{μ}} |q_{jτtμ}|

≤ ν(T_{0}^{♯})^{S_{0}} ∑_{j=0}^{σ_{μ}} \binom{σ_{μ}}{j} △ (s_{μ}; τ, T_{0}^{♯}, σ_{μ} − j)t^{j}

≤ ν(T_{0}^{♯})^{S_{0}}S_{0}!e^{T_{0}^{♯}+T_{0}} (\frac{S_{1}}{T_{0}^{♯}} + 1)^{T_{0}} T_{1}^{S_{0}}.

Therefore

L(f) ≤ L!ν(T_{0}^{♯})^{LS_{0}}S_{0}!^{L}e^{L(T_{0}^{♯}+T_{0})} (\frac{S_{1}}{T_{0}^{♯}} + 1)^{LT_{0}} T_{1}^{LS_{0}}.

These crude estimates could be slightly improved, but this is irrelevant for our purpose, since we do not pay too much attention to the absolute constants.

We want to estimate from above the number

max_{{\underline{k}∈ℕ^{2} \atop ∥\underline{k}∥<K}} log |\frac{1}{\underline{k}!}D\underline{k}f(e^{b}, b)| .

Fix \underline{k} ∈ ℕ^{2} with ∥\underline{k}∥ < K. We have

\frac{1}{\underline{k}!}D\underline{k}f(e^{b}, b) = ∑_{\underline{\underline{κ}}} Δ\underline{\underline{κ}} with Δ\underline{\underline{κ}} = det(\frac{1}{\underline{κ}_{τt}!}D^{\underline{κ}^{τt}}f_{τt}^{(σ_{μ}s_{μ})}(e^{b}, b))_{{(τ,t) \atop 1≤μ≤L}} ,

where \underline{\underline{κ}} ranges over the set of elements

(4.13)  (\underline{κ}_{τt})_{{0≤τ<T_{0} \atop 0≤t<T_{1}}} ∈ (ℕ^{2})^{L} with ∑_{τ=0}^{T_{0}−1} ∑_{t=0}^{T_{1}−1} \underline{κ}_{τt} = \underline{k}.

For \underline{k} = (k_{1}, k_{2}) the number of such \underline{\underline{κ}} is

\binom{k_{1} + L − 1}{L − 1}\binom{k_{2} + L − 1}{L − 1} < 2^{K+2L} < 2^{3L}.

Fix \underline{\underline{κ}} satisfying (4.13) and define

Δ̃\underline{\underline{κ}} = det(\frac{1}{\underline{κ}_{τt}!}D^{\underline{κ}^{τt}} f̃_{τt}^{(σ_{μ}s_{μ})}(e^{b}, b))_{{(τ,t) \atop 1≤μ≤L}} ,

so that

Δ\underline{\underline{κ}} = Δ̃\underline{\underline{κ}} ν(T_{0}^{♯})^{σ_{1}+···+σ_{L}}.

It remains to check the hypotheses of Lemma 4.11 with ζ_{μ} = s_{μ}, L′ = L − K and

Q_{τt} = log(S_{0}!) + T_{0} + T_{0}^{♯} + T_{0} log (\frac{S_{1}E}{T_{0}^{♯}} + 1) + S_{0} log(2T_{1}|b|_{+}) + T_{1}S_{1}E|b|

for 0 ≤ τ < T_{0} and 0 ≤ t < T_{1}.

From (4.13) it follows that there are at least L − K indices (τ, t) with 0 ≤ τ < T_{0} and 0 ≤ t < T_{1} for which \underline{κ}_{τt} = (0, 0); for these indices we use (4.12). We conclude by means of the estimates

max_{1≤μ≤L} sup_{|z|=E} |φ_{τt}^{(σ_{μ})}(s_{μ}z)| ≤ e^{Q_{τt}}

and, for \underline{κ}_{τt} = (κ_{1}, κ_{2}),

max_{1≤μ≤L} |\frac{1}{\underline{κ}_{τt}!}D^{\underline{κ}^{τt}} f̃_{τt}^{(σ_{μ}s_{μ})}(e^{b}, b)| ≤ max_{1≤μ≤L} ∑_{κ=0}^{σ_{μ}} \binom{σ_{μ}}{κ} △ (s_{μ}; τ, T_{0}^{♯}, κ)t^{σ_{μ}−κ}

· \binom{σ_{μ} − κ}{κ_{2}}|b|σ_{μ}−κ−κ_{2}_{+} \binom{ts_{μ}}{κ_{1}}e^{|b|(ts_{μ}−κ_{1})}

≤ S_{0}!e^{T_{0}+T_{0}^{♯}} (\frac{S_{1}}{T_{0}^{♯}} + 1)^{T_{0}} (2|b|_{+}T_{1})^{S_{0}}(2e^{|b|})^{T_{1}S_{1}}

≤ e^{Q_{τt}}.

□

4.4 From Polynomial Approximation to Algebraic Approximation

Let \underline{θ} ∈ ℂ^{n} and \underline{γ} ∈ \overline{ℚ}^{n}. We want to estimate from below |\underline{θ} − \underline{γ}|. Our strategy is as follows (see [W5] Proposition 15.3 and Exercise 15.3). Assume there is a polynomial f ∈ ℤ[\underline{X}] such that f(\underline{γ}) ≠ 0 while |f(\underline{θ})| is sufficiently small. Liouville’s inequality (Proposition 1.19) yields a lower bound for |f(\underline{γ})| (depending on the degrees and length of f, as well as the degrees and heights of the components of γ). Since |f(\underline{θ}) − f(\underline{γ})| can be estimated from above in terms of |\underline{θ} − \underline{γ}|, we deduce the desired lower bound for |\underline{θ} − \underline{γ}|.

A refined estimate can be obtained when not only f(\underline{θ}), but also the values of several derivatives of f at \underline{θ}, have a small absolute value. The required upper bound for |f(\underline{θ})−f(\underline{γ})| in terms of |\underline{θ} − \underline{γ}| will be a consequence of the interpolation formula (Lemma 1.17).

Lemma 4.14. Let ℓ, K, ν_{1}, …, ν_{ℓ}, N_{1}, …, N_{ℓ} be positive integers. Define

m = ν_{1} + ⋯ + ν_{ℓ}.

Let 𝕂 be a number field of degree D = [𝕂: ℚ]. For 1 ≤ i ≤ ℓ, write

\underline{θ}_{i} = (θ_{i1}, …, θ_{iν_{i}}) ∈ ℂ^{ν_{i}} and \underline{γ}_{i} = (γ_{i1}, …, γ_{iν_{i}}) ∈ 𝕂^{ν_{i}}.

Set

\underline{θ} = (\underline{θ}_{1}, …, \underline{θ}_{ℓ}) ∈ ℂ^{m} and \underline{γ} = (\underline{γ}_{1}, …, \underline{γ}_{ℓ}) ∈ 𝕂^{m}

and assume

|\underline{θ} − \underline{γ}| ≤ \frac{1}{2m}.

Let f be a polynomial with integer coefficients in m variables X_{ij} (1 ≤ j ≤ ν_{i}, 1 ≤ i ≤ ℓ), of total degree ≤ N_{i} with respect to the variables X_{i1}, …, X_{iν_{i}}, (1 ≤ i ≤ ℓ), such that

f(\underline{γ}) ≠ 0.

Define

ϵ = \frac{1}{2}L(f)^{−D} exp \{−D ∑_{i=1}^{ℓ} N_{i}h(1: γ_{i1}: ⋯: γ_{iν_{i}})\}

64 Michel Waldschmidt

and assume

(4.15)  max_{{\underline{k}∈ℕ^{m} \atop ∥\underline{k}∥<K}} \frac{1}{\underline{k}!}|D\underline{k}f(\underline{θ})| ≤ \frac{1}{2}ϵL(f).

Then

|\underline{θ} − \underline{γ}|^{K} ≥ ϵ(1 + \sqrt{K})^{−1} ∏_{i=1}^{ℓ}(1 + |\underline{θ}_{i}|)^{−N_{i}}.

Proof . We use the interpolation formula of Lemma 1.17 with F (\underline{z}) = f(\underline{θ} + \underline{z}), r = |\underline{θ} − \underline{γ}|, R = 1. We estimate |F |_{R} from above as follows:

|F |_{R} ≤ L(f) ∏_{i=1}^{ℓ}(1 + |\underline{θ}_{i}|)^{N_{i}}.

Since

∑_{{\underline{k}∈ℕ^{m} \atop ∥\underline{k}∥<K}} r^{∥\underline{k}∥} ≤ ∑_{\underline{k}∈ℕ^{m}} r^{∥\underline{k}∥} = (1 − r)^{−m}

and since

(1 − \frac{1}{2m})^{−m} ≤ 2,

by the assumption r ≤ 1/(2m) we have:

∑_{{\underline{k}∈ℕ^{m} \atop ∥\underline{k}∥<K}} r^{∥\underline{k}∥} ≤ 2.

Therefore

|f(\underline{γ})| ≤ |\underline{θ} − \underline{γ}|^{K}(1 + \sqrt{K})L(f) ∏_{i=1}^{ℓ}(1 + |\underline{θ}_{i}|)^{N_{i}} + 2 max_{∥\underline{k}∥<K} \frac{1}{\underline{k}!}|D\underline{k}f(\underline{θ})|.

By Liouville’s inequality (Proposition 1.19), we deduce from the assumption f(\underline{γ}) ≠ 0:

|f(\underline{γ})| ≥ L(f)^{1−D} ∏_{i=1}^{ℓ} e^{−DN_{i}h(1:γ_{i1}:···:γ_{iν_{i}})}.

Hence

L(f)^{1−D} ∏_{i=1}^{ℓ} e^{−DN_{i}h(1:γ_{i1}:···:γ_{iν_{i}})} ≤

(4.16) |\underline{θ} − \underline{γ}|^{K}(1 + \sqrt{K})L(f) ∏_{i=1}^{ℓ}(1 + |\underline{θ}_{i}|)^{N_{i}} + 2 max_{∥\underline{k}∥<K} \frac{1}{\underline{k}!}|D\underline{k}f(\underline{θ})|.

We now use the definition of ϵ and the assumption (4.15): the left hand side of (4.16) is ≥ 2ϵL(f) and the second term on the right hand side is at most ϵL(f), hence

|\underline{θ} − \underline{γ}|^{K}(1 + \sqrt{K}) ∏_{i=1}^{ℓ}(1 + |\underline{θ}_{i}|)^{N_{i}} ≥ ϵ.

□

Remark. In the special case m = 1 (hence ℓ = ν_{1} = 1, θ ∈ ℂ, and, say, N = N_{1}), here is a slightly different estimate, suggested by Exercise 15.3.a of [W5]:

|\underline{θ} − \underline{γ}|^{K} ≥ \frac{ϵ}{K\binom{N}{K}(1 + |θ|)^{N−K}} ·

The proof is quite similar, but one replaces Lemma 1.17 by Taylor’s expansion

f(γ) = ∑_{k=0}^{K−1} \frac{1}{k!}f^{(k)}(θ)(γ − θ)^{k} + \frac{1}{(K − 1)!} ∫_{θ}^{γ} f^{(K)}(t)(γ − t)^{K−1}dt.

The conclusion follows from the estimates

∑_{k=0}^{K−1} \frac{1}{k!}|f^{(k)}(θ)| |γ − θ|^{k} ≤ ϵL(f), |f(γ)| ≥ 2ϵL(f)

and

\frac{1}{(K − 1)!} sup \{|f^{(K)}(t)| ; t = θ(1 − u) + γu, 0 ≤ u ≤ 1\} ≤ K\binom{N}{K}L(f)(1 + |θ|)^{N−K}.

4.5 Proof of Theorem 4.5

We start by selecting a sufficiently large absolute constant N. Next we assume the hypotheses of Theorem 4.5 are satisfied. From the assumptions we deduce log B ≥ (1/D)+log D ≥ 1, hence

log max{B, N} ≤ (log N)(log B).

Replacing if necessary c by c log N, we may assume B ≥ N.

Next from Liouville’s inequality (see [W5] Exercise 3.7) we deduce

|β| ≥ e^{−Dh(β)},

hence there is no loss of generality to assume |β| ≤ 2|λ| (and consequently λ ≠ 0). Define first T_{1}, S_{1} and T_{0}^{♯} by

T_{1} = [N^{2} \frac{D log D + log E}{log E} ] , S_{1} = [N \frac{D log B}{log E} ] , T_{0}^{♯} = [log B],

and then T_{0} and S_{0} by

T_{0} = [N \frac{D log A}{log E}] S_{1}, S_{0} = 2 [N \frac{D log A}{log E}] T_{1}.

Plainly the number L = T_{0}T_{1} is also equal to (1/2)S_{0}S_{1}. Up to terms of lower order^{16}, L log E is

N^{4}D^{2}(log A)(log B)(D log D + log E)(log E)^{−2},

which is the main term in the final estimate of the conclusion of Theorem 4.5. Denoting by c_{1}, …, c_{12} positive absolute constants, the following estimates are plain:

T_{1}S_{1} ≤ \frac{c_{1}L log E}{ND log A}, S_{0} ≤ \frac{c_{2}L log E}{ND log B},

T_{0}^{♯}S_{0} ≤ \frac{c_{3}}{ND} L log E, log S_{0} ≤ \frac{c_{4}}{ND} S_{1} log E,

log T_{1} ≤ \frac{c_{5}}{ND} S_{1} log E and T_{1}S_{1}E|β| ≤ \frac{c_{6}}{N} L log E.

We apply Theorem 4.10 with b = β and K = [L/2]. Since the polynomials f_{i}(Z_{1}, β) (1 ≤ i ≤ n) have no common zero in ℂ^{×}, one of f_{1}, …, f_{M}, say f, has f(α, β) ≠ 0. This polynomial has degree at most

N_{1} = \frac{c_{7}L^{2} log E}{ND log A} and N_{2} = \frac{c_{8}L^{2} log E}{ND log B}

in Z_{1} and Z_{2} respectively, and length at most

exp (\frac{c_{9}L^{2} log E}{ND} ) .

Moreover, if we set ℓ = m = 2, ν_{1} = ν_{2} = 1, (θ_{1}, θ_{2}) = (e^{β}, β) and (γ_{1}, γ_{2}) = (α, β), then the left hand side of (4.15) is at most

E^{−c_{10}L^{2}}.

Since

ϵ ≥ E^{−c_{11}L^{2}/N}

we deduce from Lemma 4.14

|e^{β} − α|^{L/2} ≥ E^{−c_{12}L^{2}/N}.

Therefore

(4.17)  |β − log α| ≥ E^{−L}

and the conclusion of Theorem 4.5 follows. □

16Beware of integral parts!

5 Measures of Linear Independence of Logarithms of Algebraic Numbers

5.1 Introduction

The last two lectures are devoted to the question of measures of linear independence for logarithms of algebraic numbers. Here are a few references on this topic.

A simple proof for a homogeneous measure of linear independence of an arbitrary number of logarithms is given in Chap. 7 of [W3], using an extension of Schneider’s method. A refined estimate (relying on the same ideas) is given in Chap. 7 of [W5]. The sharpest known estimate (in the general case, homogeneous or not) arising from this method is established in Chap. 9 of [W5].

Baker’s method is explained in [B2] Chap. 3, [L1] Chap. 8, 10 and 11, [F] Chap. 10,

[Sp] Chap. 3, [FN] Chap. 4 and [W5] § 10.1 (see also the introduction of [Mat I] for a historical survey). While these proofs involve an auxiliary function, a measure of linear independence for logarithms of algebraic numbers is obtained in § 10.2 of [W5] by means of an interpolation determinant (without extrapolation).

A comparative discussion of these methods can be found in [W5] (see § 14.4), where a more general estimate is established (the so-called quantitative version of the linear subgroup Theorem).

The state of the art including references to the sharpest known measures of ℚ-linear independence for logarithms of algebraic numbers is given in [W5] § 10.4.6. Now one should add to this picture Matveev’s recent result in [Mat II] (see Theorem 6.1; we refer also to Nesterenko’s lectures).

Here is the main result for the rational case.

Theorem 5.1. For each positive integer n there exists a positive constant C(n) with the following property. Let α_{1}, …, α_{n} be nonzero algebraic numbers and log α_{1}, …, log α_{n} logarithms of α_{1}, …, α_{n} respectively. Assume that the numbers log α_{1}, …, log α_{n} are ℚ-linearly independent. Let b_{1}, …, b_{n} be rational integers. Denote by D the degree of the number field ℚ(α_{1}, …, α_{n}) over ℚ. Further, let W, E, E^{∗} be positive real numbers, each ≥ e and let V_{1}, …, V_{n} be positive real numbers. Assume

V_{j} ≥ max \{h(α_{j}), \frac{E| log α_{j}|}{D}, \frac{log E}{D} \} (1 ≤ j ≤ n),

log E^{∗} ≥ max \{ \frac{1}{D} log E, log (\frac{D}{log E} )\}

and W ≥ log E^{∗}. Further, assume b_{n} ≠ 0 and

e^{W} ≥ max_{1≤j≤n−1} \{\frac{|b_{n}|}{V_{j}} + \frac{|b_{j}|}{V_{n}}\} · \frac{log E}{D} ·

Then the number

Λ = b_{1} log α_{1} + ⋯ + b_{n} log α_{n}

has absolute value bounded from below by

|Λ| > exp{−C(n)D^{n+2}W V_{1} ⋯ V_{n}(log E^{∗})(log E)^{−n−1}}.

For several applications (especially for solving explicitly Diophantine equations) it is quite important to produce an estimate with a small numerical constant C(n), even to the cost of relaxing the dependence in some parameters. A striking example arises with the case n = 2 where a very small value of C(2) can be reached, provided that W is replaced by W^{2} (see Theorem 5.22 below). We shall explain in an appendix to this § 5 how such an estimate can be used also for the case n ≥ 2 in order to get a lower bound for |Λ| which involves a small constant – we need to replace W by e^{κW} for some κ > 0, but the point is that an admissible value for κ is < 1, hence the result is not trivial.

Apart from the explicit value for C, Theorem 5.1 includes essentially all known estimates on this number |Λ|.

Admissible numerical values for C are given in [W5] § 10.4.6. Assuming E = e, Matveev proved recently in [Mat II] that C(n) can be replaced by C^{n} where now C is a positive absolute constant (see § 6).

Our goal is to explain some of the main ideas of the proof of Theorem 5.1 by means of Baker’s method. In this lecture we discuss the classical approach following Baker’s method with an auxiliary function and in the last one we shall show how to replace the auxiliary function by an interpolation determinant.

5.2 Baker’s Method with an Auxiliary Function

Our main goal is to introduce the strategy of the proof, with an emphasis on the ideas and tools. We do not produce exact estimates, but we consider only the dominating terms which will explain the choice of parameters and at the same time provide some explanation for the limitation of the present method.

5.2.a Main Conditions on the Parameters

Assume the hypotheses of Theorem 5.1 are satisfied: α_{1}, …, α_{n} are nonzero algebraic numbers, log α_{1}, …, log α_{n} are ℚ-linearly independent, b_{1}, …, b_{n} are rational integers, b_{n} ≠ 0 and

Λ = b_{1} log α_{1} + ⋯ + b_{n} log α_{n}.

As usual we need to introduce parameters: let T_{0}, T_{1}, …, T_{n}, T_{0}^{♯}, S_{0}, S_{1} and L be positive integers with L = T_{0} ⋯ T_{n}. We also introduce another important parameter, U, which is a positive real number and will play the main role: assuming

0 < |Λ| ≤ e^{−U},

we plan to derive a contradiction as soon as U is sufficiently large. Of course all the point is to be explicit on this condition that U is large enough.

The proof will have a lot in common with the transcendence proof of Theorem 3.1 in § 3 (where the assumption was Λ = 0). Instead of setting β_{k} = −b_{k}/b_{n} and β_{0} = 0, it is slightly more convenient to change the definitions of ∂_{0}, …, ∂_{n−1} and to set now

∂_{0} = \frac{∂}{∂Y_{0}} ,  ∂_{k} = b_{n}Y_{k} \frac{∂}{∂Y_{k}} − b_{k}Y_{n} \frac{∂}{∂Y_{n}}  (1 ≤ k ≤ n − 1).

The auxiliary polynomial will have the shape

P (Y_{0}, …, Y_{n}) = ∑_{(τ,\underline{t})} c_{τ,\underline{t}} △ (Y_{0}; τ, T_{0}^{♯})Y_{1}^{t_{1}} ⋯ Y_{n}^{t_{n}}.

In the sum, (τ, \underline{t}) ranges over the set of all elements in ℕ × ℕ^{n} with 0 ≤ τ < T_{0} and 0 ≤ t_{j} < T_{j} (1 ≤ j ≤ n).

We are interested in the algebraic numbers

∂\underline{σ}P (s, α_{1}^{s}, …, α_{n}^{s})

for (\underline{σ}, s) ∈ ℕ^{n} × ℕ satisfying ∥\underline{σ}∥ < S_{0} and 0 ≤ s < S_{1}. They can be explicitly written down as

∑_{(τ,\underline{t})} c_{τ,\underline{t}}γ_{τ\underline{t}}^{(\underline{σ}s)},

where

γ_{τ\underline{t}}^{(\underline{σ}s)} = ∂\underline{σ}(△(Y_{0}; τ, T_{0}^{♯})Y_{1}^{t_{1}} ⋯ Y_{n}^{t_{n}})(s, α_{1}^{s}, …, α_{n}^{s})

and

∂\underline{σ}(△(Y_{0}; τ, T_{0}^{♯})Y_{1}^{t_{1}} ⋯ Y_{n}^{t_{n}}) = △(Y_{0}; τ, T_{0}^{♯}, σ_{0}) ∏_{k=1}^{n−1}(b_{n}t_{k} − b_{k}t_{n})^{σ_{k}} · Y_{1}^{t_{1}} ⋯ Y_{n}^{t_{n}}.

The auxiliary polynomial P is selected so that equations (3.4) hold.

Our first condition on the parameters will be written

(5.2)  \frac{1}{n!}S_{0}^{n}S_{1} < L.

This is a lousy way of requiring that the number of equations is less than the number of unknowns (namely the coefficients c_{τ\underline{t}}). Here we do not pay attention to the exact value of the absolute constants which come into the picture, but we are only interested with the main constraints. For the same reason we do not look at the estimate for the coefficients c_{τ\underline{t}}.

For \underline{σ} ∈ ℕ^{n} define

G\underline{σ}(z) = ∂\underline{σ}P (z, α_{1}^{z}, …, α_{n}^{z}).

According to their definitions, the operators ∂\underline{σ} are related to derivatives along the hyperplane W of equation b_{1}z_{1} + ⋯ + b_{n}z_{n} = 0 in ℂ^{n+1}. The point \underline{v} = (1, log α_{1}, …, log α_{n}) does not lie in this hyperplane (because Λ ≠ 0), but it is not far from it: to a certain extent |Λ| measures the “distance” between \underline{v} and W. If the point \underline{v} were on the hyperplane W, the conditions (3.4) would imply that the one variable entire function G\underline{σ} has a zero of multiplicity ≥ S_{0} − ∥\underline{σ}∥ as each integer s with 0 ≤ s < S_{1}. When |Λ| is small, the first S_{0} − ∥\underline{σ}∥ derivatives of G\underline{σ} at these points have a small absolute value. More precisely define

g\underline{σ}(z) = ∂\underline{σ}P (z, α_{1}^{z}, …, α_{n−1}^{z}, (α_{1}^{b_{1}} ⋯ α_{n−1}^{b_{n−1}})^{−z/b_{n}}).

Then

G\underline{σ}(z) − g\underline{σ}(z) = ∑_{(τ,\underline{t})} c_{τ,\underline{t}} △ (z; τ, T_{0}^{♯}, σ_{0}) ∏_{k=1}^{n−1}(b_{n}t_{k} − b_{k}t_{n})^{σ_{k}} · ∏_{i=1}^{n} α_{i}^{t_{i}z} · (1 − e^{−t_{n}zΛ/b_{n}}).

From the inequality |e^{w} − 1| ≤ |w|e^{|w|} which is valid for any w ∈ ℂ one deduces

|1 − e^{−t_{n}zΛ/b_{n}} | ≤ |t_{n}zΛ/b_{n}|e^{|t_{n}zΛ/b_{n}|},

hence |G\underline{σ}(z) − g\underline{σ}(z)| is quite small for |z| not too large:

|G\underline{σ}(z) − g\underline{σ}(z)| ≤ |Λ|^{1/2} = e^{−U/2}

for all relevant \underline{σ} and z. In particular from the relations G\underline{σ}(s) = 0 for ∥\underline{σ}∥ < S_{0} and 0 ≤ s < S_{1} one deduces

|g(ℓ) ; \underline{σ}(s)| ≤ e^{−U/3}

for ∥\underline{σ}∥ < S_{0} − S′_{0} and 0 ≤ s < S_{1}. Using an approximate Schwarz’ Lemma like Lemma 1.5 one deduces

|g\underline{σ}(s)| ≤ e^{−2U′}

for the same \underline{σ}, s, with

U′ = min{U, (S_{0} − S′_{0})S_{1} log E}.

Hence

|G\underline{σ}(s)| ≤ e^{−U′}

for the same \underline{σ} and s. Combining this estimate with Liouville’s inequality (Proposition 1.19) we deduce that G\underline{σ} satisfies (3.7). This conclusion can be reached only if the parameters satisfy certain conditions. In the estimate arising from Liouville’s inequality (Proposition 1.19), the dominating terms are the following ones:

e^{−DW S_{0}}(E^{∗})^{−DT_{0}} arising from △ (s; T_{0}^{♯}, τ, σ_{0})

and

∏_{j=1}^{n} e^{−DT_{j}V_{j}S′_{1}} arising from ∏_{j=1}^{n} α_{j}^{−t_{j}s}.

The exact conditions which motivate the definitions of V_{1}, …, V_{n}, W and E^{∗} are explained in [W5] Chap. 9 and 10. In particular the difference between the “homogeneous rational case” considered here and the “general case” (where the rational coefficients b_{i}’s are replaced by algebraic numbers β_{i}, and there may be also an extra β_{0}) occurs here: the factor

(b_{n}t_{k} − b_{k}t_{n})^{σ_{k}}

like it stands would require stronger assumptions on W, namely

W ≥ max_{1≤k≤n} log V_{k} and W ≥ max_{1≤k≤n} log |b_{k}|,

so that DS_{0} log T_{i} is bounded by U′ for 1 ≤ i ≤ n. In order to deal with our weaker condition on W, the idea is to replace this factor by

△(b_{n}t_{k} − b_{k}t_{n}, σ_{k}).

See [W5] Lemma 9.11.

Therefore it is reasonable to require

T_{0} ≤ \frac{U′}{D log E^{∗}}, S_{0} ≤ \frac{U′}{DW} and T_{j} ≤ \frac{U′}{nDS′_{1}V_{j}} (1 ≤ j ≤ n).

Let us take

(5.3)  T_{0} = [c_{1} \frac{U}{D log E^{∗}} ] , S_{0} = [c_{2} \frac{U}{DW} ] , T_{j} = [c_{3} \frac{U}{nDS′_{1}V_{j}} ] (1 ≤ j ≤ n),

where c_{1}, c_{2}, c_{3} (as well as c_{4}, …, c_{11} below) denote absolute positive constants, which we are not interested in.

From condition (5.2) we deduce

(5.4)  U > c_{4}S_{1}(S′_{1}/W)^{n}DV_{1} ⋯ V_{n}(log E^{∗}).

At the end of this first extrapolation step we derive (3.7), provided that

(S_{0} − S′_{0})S_{1} log E ≥ U.

As a first try, let us take S′_{0} = [S_{0}/2] and then

(5.5)  S_{1} = [c_{5} \frac{DW}{log E} ] .

We started with \binom{S_{0}+n}{n}S_{1} equations and we end up with \binom{S′_{0}+n}{n}S′_{1} new equations. It is conceivable that no real progress has been achieved unless the number of new equations exceeds the number of old equations, which means essentially that S′_{0}^{n}S′_{1} should not be smaller than S_{0}^{n}S_{1}. Hence we need at least S′_{1} ≥ 2^{n}S_{1}.

There are several possibilities now. The easiest one is to apply immediately a zero estimate. This is possible only if (S′_{0}^{n}/n!)S′_{1} is somewhat larger than L = T_{0}T_{1} ⋯ T_{n} (which is the opposite of (5.2) when S_{0} and S_{1} are replaced by S′_{0} and S′_{1}). So if we wish to conclude immediately by means of the zero estimate we need to require

(5.6)  \binom{S′_{0} + n − 1}{n}S′_{1} > c_{ZE}L,

where c_{ZE} = (n + 1)! is the loss arising from Proposition 3.8 (this c_{ZE} is a notation of [W5] § 13.5). This gives rise to the condition

S′_{0}^{n}S′_{1} > n!^{2}L.

Comparing with (5.2) gives S′_{1} > n!S_{1}, hence (5.4) yields the condition

(5.7)  U > c_{6}^{n}(n!)^{n}D^{n+2}W V_{1} ⋯ V_{n}(log E^{∗})(log E)^{−n−1}.

The estimate (5.7) corresponds to the sharpest available result with respect to V_{1}, …, V_{n}, W, E and E^{∗} (but not with respect to n), and the choice of parameters is given by (5.3) and (5.5).

5.2.b End of the Proof

What we have shown so far is only that the above scheme of proof cannot reach a better estimate than |Λ| ≥ e^{−U} with U satisfying (5.7). It is a different issue to prove that one can indeed reach such an estimate by means of these arguments. However let us now say that there is mainly a single serious difficulty in doing so: (5.6) is only a necessary condition for applying the zero estimate (see (3.9)). More precisely the zero estimate enables one to conclude the proof, unless there is a linear dependence relation between the rational integers b_{1}, …, b_{n} with “small” coefficients. Indeed, starting from 0 < |Λ| ≤ e^{−U}, the transcendence machinery produces a tuple (t_{1}, …, t_{n}) ∈ ℤ^{n} \ {0} such that t_{1}b_{1} + ⋯ + t_{n}b_{n} = 0 together with a sharp upper bound for max_{1≤i≤n} |t_{i}|. This small linear dependence relation is the explanation for the fact that the system of equations (3.7) is somehow degenerate.

At this stage, the idea is to eliminate one b_{i} thanks to this relation and to work with a linear combination of n − 1 logarithms instead of n. This is not the most efficient way; one may proceed by induction indeed, but it is better to repeat the transcendence argument and to take these relations into account. In [PW1] (and also [PW2] for a more general situation dealing with commutative algebraic groups), the induction is done as follows. The zero estimate produces an algebraic subgroup of 𝔾_{a} × 𝔾_{m}^{n} and the strategy is to start the proof from scratch (construction of the auxiliary polynomial) with such an obstructing subgroup. Thanks to this obstructing subgroup one has a better control on the rank of the system of linear equations to which we apply Thue-Siegel’s Lemma. After the extrapolation, one produces a new set of relations (3.7) to which one applies again the zero estimate; the extremality of the initial obstructing subgroup enables one to conclude (by contradiction).

A different way of performing this induction is used by [BWü], but the underlying ideas are basically the same. This induction is somewhat technical but it is well under control now, and we shall not tell more about it. We assume implicitly that there is no “small” linear dependence relation between the b_{i}’s, in which case the zero estimate shows that there is no nonzero polynomial P ∈ ℂ[Y_{0}, …, Y_{n}] for which a set of equalities (3.7) holds for (\underline{σ}, s) ranging over a set with slightly more than T_{0}T_{1} ⋯ T_{n} elements (compare with condition (5.2)).

So our goal is to get more than T_{0}T_{1} ⋯ T_{n} equations (3.7) for P.

5.2.c Baker’s Method with an Interpolation Determinant and without Extrapolation

This § 5.2 is devoted to Baker’s method with an auxiliary function, but we make a small digression to point out that the arguments described in § 5.2.a and § 5.2.b involving an auxiliary function work out perfectly well for the interpolation determinant method. This is the topic of Chap. 10 of [W5].

The basic scheme of proof is the following: Philippon’s zero estimate enables us to produce a nonzero determinant ([W5], Proposition 10.9) which is an algebraic number. Liouville’s estimate provides a lower bound for its absolute value. The upper bound is obtained by analytic means ([W5], Proposition 10.5). In this analytic argument, one takes into account the order of vanishing of an interpolation determinant at the origin only: extrapolation like in § 3.5 is not necessary.

The role of the obstructing subgroup in connection with interpolation determinants is explained in [W5] § 10.2.3.

5.2.d Dependence on n We consider the dependence on n now. The factor (n!)^{n} in (5.7) is very large (it is comparable with the estimate which occurs

in the interpolation determinant method of [W5] Chap. 10 — see § 14.4.3).

Even if one were to replace the condition S′_{1} > n!S_{1} by the weaker S′_{1} > 2^{n}S_{1}, one would get a large “constant” in terms of n, involving 2^{n^{2}}.

In order to refine condition (5.7), an obvious solution would be to improve the zero estimate. However if one could replace (n + 1)! in (3.9) by, say, n^{c_{7}}, then one would still end up with n^{c_{8}n} in place of c_{6}^{n}(n!)^{n} in the right hand side of (5.7)^{17}.

As soon as we wish to increase S′_{1}/S_{1} by a factor λ > 1, we get a factor λ^{n} in U. Hence we should not take S′_{1}/S_{1} larger than an absolute constant if we do not wish to introduce some n^{n}. We shall take S′_{1} = 2S_{1}. Now, since we want to end up with at least as many equations as we started with, we need that S′_{0}^{n}S′_{1} is not less than S_{0}^{n}S_{1}. It is natural to require that S′_{0}/S_{0} is comparable with 2^{−1/n}, which is not far from 1 − (c_{9}/n). Let us take

S′_{0} = S_{0} (1 − \frac{1}{2n}) .

We need to replace (5.5) by

S_{1} = [c_{10}n\frac{DW}{log E} ] .

From (5.4) we obtain the condition

(5.8)  U > c_{11}^{n}n^{n}D^{n+2}W V_{1} ⋯ V_{n}(log E^{∗})(log E)^{−n−1}.

We shall see later (§ 5.2.h) how to remove the coefficient n^{n} in the right hand side of (5.8), but let us continue.

It seems we have not earned much: we started with roughly (S_{0}^{n}/n!)S_{1} equations and we got (S′_{0}^{n}/n!)S′_{1} new ones, which is about the same. The only improvement concerns the term S_{0}S_{1} log E, that we can replace at the end of this extrapolation by S′_{0}S′_{1} log E, which is essentially twice as large.

This procedure may be repeated: we do it n times (this seems to be an optimal choice) with

S_{0}^{(j)} = S_{0} (1 − \frac{j}{2n}) and S_{1}^{(j)} = 2^{j}S_{1}.

There is a small cost: at the end we shall get |Λ| ≥ e^{−2^{n}U} in place of |Λ| ≥ e^{−U}. If one wishes to keep the conclusion |Λ| ≥ e^{−U} then one should replace everywhere else U by U/2^{n}. Indeed this is just the choice of parameters in [W2].

At the end of the n extrapolation steps one has S_{0}^{(n)} = S_{0}/2 and S_{1}^{(n)} = 2^{n}S_{1}. The number of equations is not large enough to apply immediately the zero estimate 3.8.

17This is a significative difference with the “dual” method in Chap. 9 of [W5], dealing with the algebraic group 𝔾_{a}^{n}×𝔾_{m}, where any improvement of the constant in the zero estimate immediately applies to the final estimate of Theorem 5.1.

Since we have been unable to increase the number of equations enough, we shall follow another strategy which originates in works by Baker and Stark and relies on arguments arising from Kummer’s theory. The idea is to introduce division points and prove

(5.9)  ∂\underline{σ}P (s/p, α_{1}^{s/p}, …, α_{n}^{s/p}) = 0

for various \underline{σ} ∈ ℕ^{n}, s ∈ ℤ and p. At an early stage of the theory, p was selected as a sufficiently large prime number. Later, another argument, which we explain in § 5.2.f, enabled us to work with a smaller value for p, until it was realized that p = 2 also works!

We refer to [L1] for several proofs involving (5.9):

• In Chap. 8, § 2 and § 6, for many values (s, p) with \underline{σ} = \underline{0}.

• In Chap. 10, § 2 and § 5, for a single sufficiently large prime number p and several values (\underline{σ}, s).

• In Chap. 11, § 2 and § 4, (5.9) is applied with p = 2 only. In recent works involving Kummer’s theory only p = 2 is used; however it may be instructive to recall briefly what was done earlier with a large prime p.

5.2.e Kummer’s Theory

In this section we denote by n a positive integer, 𝕂 a number field and α_{1}, …, α_{n} nonzero elements of 𝕂.

We first quote Lemma 3 of [BSt].

Lemma 5.10. Let p be a prime. For 1 ≤ j ≤ n denote by α_{j}^{1/p} any p-th root of α_{j}. For 1 ≤ r ≤ n set

𝕂_{r} = 𝕂(α_{1}^{1/p}, …, α_{r}^{1/p}).

Assume

[𝕂_{n}: 𝕂_{n−1}] < p.

Then there exist an element γ ∈ 𝕂^{×} and integers j_{1}, …, j_{n−1} satisfying 0 ≤ j_{ℓ} < p (1 ≤ ℓ ≤ n − 1), such that

(5.11)  α_{n} = α_{1}^{j_{1}} ⋯ α_{n−1}^{j_{n−1}}γ^{p}.

Remark. Lemma 5.10 is proved by induction in [BSt]. When 𝕂 contains the p-th roots of unity it can also be proved by using arguments from Kummer’s theory as follows. Define

G_{0} = {x^{p} ; x ∈ 𝕂^{×}}

and, for 1 ≤ r ≤ n let G_{r} denote the multiplicative subgroup of 𝕂^{×} spanned by G_{0}, α_{1}, …, α_{r}. According to [L2] Chap. 6, § 8 Th. 8.1, the field 𝕂_{r} is an abelian extension of 𝕂 and the degree of this extension is the index of G_{0} in G_{r}. Assume p is such that [𝕂_{n}: 𝕂_{n−1}] < p. Then (G_{n} : G_{n−1}) < p, hence there is a relation

α_{n}^{a_{n}} = α_{1}^{a_{1}} ⋯ α_{n−1}^{a_{n−1}}β^{p},

with some β ∈ 𝕂^{×} and integers a_{1}, …, a_{n} satisfying 0 ≤ a_{ℓ} < p (1 ≤ ℓ ≤ n − 1) and 1 ≤ a_{n} < p. Writing Bézout’s relation a_{n}u+pv = 1 with rational integers u, v, one deduces the desired relation (5.11).

Lemma 5.12. For 1 ≤ i ≤ n select a complex logarithm log α_{j} of α_{j}. Assume that the numbers log α_{1}, …, log α_{n} are linearly independent over ℚ and denote by G the set of λ ∈ ℂ such that

e^{λ} ∈ 𝕂^{×} and λ, log α_{1}, …, log α_{n} are linearly dependent over ℚ.

Then G is a free ℤ-module of rank n, containing ℤ log α_{1} + ⋯ + ℤ log α_{n} as a subgroup of finite index.

Further, if log θ_{1}, …, log θ_{n} is a basis of G over ℤ, then for any prime p for which 𝕂 contains a primitive p-th root of unity, we have

[𝕂(θ_{1}^{1/p}, …, θ_{n}^{1/p}): 𝕂] = p^{n},

where θ_{i}^{1/p} stands for exp{(1/p) log θ_{i}} (1 ≤ i ≤ n).

Remark. The assumption in the last sentence that 𝕂 contains a primitive p-th root of unity cannot be omitted. Here is an example: take 𝕂 = ℚ, n = 1, α_{1} = 1 and log α_{1} = 2iπ. In this case log θ_{1} = ±2iπ and for each odd prime p we have

[𝕂(θ_{1}^{1/p}): 𝕂] = p − 1.

Proof. It will be more convenient to work with

Λ = {(r_{1}, …, r_{n}) ∈ ℚ^{n}; r_{1} log α_{1} + ⋯ + r_{n} log α_{n} ∈ G},

which is a subgroup of ℝ^{n} isomorphic to G under the mapping

\begin{matrix}Λ & ⟶ & G \\ \underline{r} & ⟼ & r_{1} log α_{1} + ⋯ + r_{n} log α_{n}. \end{matrix}

We have ℤ^{n} ⊂ Λ ⊂ ℚ^{n}. We first want to check that Λ is discrete in ℝ^{n}. For X > 0 and for \underline{r} ∈ Λ satisfying |\underline{r}| ≤ X we have

h(α_{1}^{r_{1}} ⋯ α_{n}^{r_{n}}) ≤ X ∑_{i=1}^{n} h(α_{i}),

hence α_{1}^{r_{1}} ⋯ α_{n}^{r_{n}} belongs to a finite subset of 𝕂^{×} as \underline{r} ranges over the elements of Λ ∩ B_{n}(0, X). This means that the image of Λ ∩ B_{n}(0, X) under the mapping

\begin{matrix}Λ & ⟶ & K^{×} \\ \underline{r} & ⟼ & α_{1}^{r_{1}} ⋯ α_{n}^{r_{n}} \end{matrix}

is finite. Hence the proof that Λ ∩ B_{n}(0, X) is finite will be completed if we check that for each \underline{r}^{0} ∈ ℚ^{n} ∩ B_{n}(0, X), the set of \underline{r} ∈ ℚ^{n} ∩ B_{n}(0, X) such that

α_{1}^{r_{1}} ⋯ α_{n}^{r_{n}} = α_{1}^{r_{1}^{0}} ⋯ α_{n}^{r_{n}^{0}}

is finite. Indeed, since log α_{1}, …, log α_{n} are linearly independent over ℚ, there exists \underline{s} ∈ ℚ^{n} such that

\{\underline{r} ∈ ℚ^{n}; α_{1}^{r_{1}} ⋯ α_{n}^{r_{n}} = 1\} = ℤ\underline{s};

we deduce that the set

\{\underline{r} ∈ ℚ^{n} ∩ B_{n}(0, 2X) ; α_{1}^{r_{1}} ⋯ α_{n}^{r_{n}} = 1\}

is finite.

This shows that Λ is a lattice (discrete subgroup of rank n) in ℝ^{n}, hence a free ℤmodule, and therefore G is also a free ℤ-module of rank n.

Let log θ_{1}, …, log θ_{n} be a basis of G over ℤ. Assume

[𝕂(θ_{1}^{1/p}, …, θ_{n}^{1/p}): 𝕂] < p^{n}

for some prime p. Let m be minimal with 1 ≤ m ≤ n such that

[𝕂(θ_{1}^{1/p}, …, θ_{m}^{1/p}): 𝕂] < p^{m}.

We use Lemma 5.10:

θ_{m} = θ_{1}^{j_{1}} ⋯ θ_{m−1}^{j_{m−1}}γ^{p},

for some γ ∈ 𝕂^{×} and 0 ≤ j_{ℓ} < p, (1 ≤ ℓ ≤ m − 1). Define

λ = \frac{1}{p} log θ_{m} − ∑_{ℓ=1}^{m−1} \frac{j_{ℓ}}{p} log θ_{ℓ}.

Since 𝕂 contains the p-th roots of unity, from (e^{λ}/γ)^{p} ∈ 𝕂^{×} we deduce e^{λ} ∈ 𝕂^{×}, hence λ ∈ G, and therefore λ ∈ ℤ log θ_{1} + ⋯ + ℤ log θ_{n}, which is clearly a contradiction. □

In the transcendence proof following Baker’s method, Lemma 5.12 is applied as follows: when we need estimates for the height, we use the algebraic numbers α_{i}, while when we want to apply Kummer’s condition, we use the numbers θ_{i}. This does not make a difference when using interpolation determinants; if we use an auxiliary function, the systems of equations are equivalent, but in order to investigate small values we need estimates for the transition matrix. The index N of ℤ^{n} in Λ plays an important role in Matveev’s paper [Mat II].

5.2.f Kummer’s Theory with a Large Prime p

The results in this subsection are no more used in recent papers dealing with linear independence measures for logarithms of algebraic numbers, but they keep their independent interest.

We denote by n a positive integer and by α_{1}, …, α_{n} nonzero algebraic numbers.

Lemma 5.13. Let 𝕂 be a number field containing α_{1}, …, α_{n} and let p be a sufficiently large prime.

a) Assume α_{1}, …, α_{n} are multiplicatively independent. Denote by α_{i}^{1/p} any p-th root of α_{i} (1 ≤ i ≤ n). Then

[𝕂(α_{1}^{1/p}, …, α_{n}^{1/p}): 𝕂] = p^{n}.

b) Assume α_{1} is a primitive q-th root of unity for some positive integer q, while α_{2}, …, α_{n} are multiplicatively independent. Denote by α_{1}^{1/p} a primitive pq-th root of unity and by α_{i}^{1/p} any p-th root of α_{i} (2 ≤ i ≤ n). Then

[𝕂(α_{1}^{1/p}, …, α_{n}^{1/p}): 𝕂] = (p − 1)p^{n−1}.

Proof.

a) To start with, consider the case n = 1. The assumption is that α = α_{1} is not a root of unity, and the conclusion amounts to say that if α is a p-th power in 𝕂, then p is bounded (depending on α and 𝕂). We prove this claim by using heights: if α = β^{p} for some β ∈ 𝕂, then h(β) = (1/p)h(α). Moreover β has degree ≤ [𝕂 : ℚ] and is not a root of unity. From a result of Kronecker’s (see for instance [W5] § 3.6) h(β) is bounded from below by a positive constant c depending only on [𝕂 : ℚ]. Hence p ≤ c^{−1}h(α).

More generally, assume α_{1}, …, α_{n} are multiplicatively independent. Define

𝕂_{r} = 𝕂(α_{1}^{1/p}, …, α_{r}^{1/p}) (1 ≤ r ≤ n)

and 𝕂_{0} = 𝕂. Let p be a prime number such that [𝕂_{n}: 𝕂] < p^{n}. Denote by r the least integer such that [𝕂_{r}: 𝕂] < p^{r}. We have 1 ≤ r ≤ n, [𝕂_{r−1}: 𝕂] = p^{r−1} and [𝕂_{r}: 𝕂_{r−1}] < p. By Lemma 5.10 there exists a nontrivial relation

(5.14)  α_{r} = α_{1}^{j_{1}} ⋯ α_{r−1}^{j_{r−1}}γ^{p}

with j_{ℓ} ∈ ℤ, 0 ≤ j_{ℓ} < p (1 ≤ ℓ < r) and γ ∈ 𝕂^{×}. From the properties of the height (see for instance [W5] Chap. 3) we deduce

h(γ) ≤ h(α_{1}) + ⋯ + h(α_{r}),

hence γ belongs to a finite subset of 𝕂^{×} which depends only on α_{1}, …, α_{n}. For each fixed γ there is a unique multiplicative dependence relation like (5.14), hence p is bounded. Moreover explicit upper bounds for the exponents in such a multiplicative dependence relation are known, depending only on α_{1}, …, α_{r} and [𝕂 : ℚ] (see for instance [W5] Lemma 7.19). b) Assume now α_{1} is a primitive q-th root of unity, while α_{1}^{1/p} is a primitive pq-th root of unity. For any prime p which does not divide q the number α_{1} is a p-th power of an element in 𝕂: indeed writing up+vq = 1 we get α_{1} = (α_{1}^{u})^{p}. Hence the field 𝕂_{1} = 𝕂(α_{1}^{1/p}) has degree < p over 𝕂. Moreover, since α_{1}^{q} is a primitive p-th root of unity, the field ℚ(α_{1}^{q}) is the cyclotomic field of degree p − 1 and discriminant ±p^{p−2}. Hence, as soon as p does not divide the discriminant of 𝕂, the field 𝕂_{1} has degree p − 1 over 𝕂.

Next we apply part a) of this Lemma 5.13 to the numbers α_{2}, …, α_{n} and the field 𝕂: we deduce

[𝕂(α_{2}^{1/p}, …, α_{n}^{1/p}): 𝕂] = p^{n−1}.

Finally, 𝕂(α_{1}^{1/p}, …, α_{n}^{1/p}) is the compositum of 𝕂_{1} and 𝕂(α_{2}^{1/p}, …, α_{n}^{1/p}), hence has degree (p − 1)p^{n−1} over 𝕂. □

Lemma 5.15. Assume α_{1}, …, α_{n} are multiplicatively independent. Then there exists a positive integer D with the following property: if Q ∈ 𝕂[X_{1}, …, X_{n}] is a nonzero polynomial of degree < D in each variable X_{i} (1 ≤ i ≤ n) and p a prime with p ≥ D, then

Q(α_{1}^{1/p}, …, α_{n}^{1/p}) ≠ 0.

Proof. Using Lemma 5.13, take D sufficiently large so that for p ≥ D,

[𝕂(α_{1}^{1/p}, …, α_{n}^{1/p}): 𝕂] = p^{n}.

For such a p we prove the result by induction on n. For n = 1, since α_{1}^{1/p} has degree p ≥ D > deg Q over 𝕂, and since Q has coefficients in 𝕂, it follows that α_{1}^{1/p} is not a root of Q.

If Lemma 5.15 holds for n − 1, then the polynomial

P (X) = Q(α_{1}^{1/p}, …, α_{n−1}^{1/p}, X) ∈ 𝕂_{n−1}[X]

is nonzero, has degree < D and coefficients in the field 𝕂_{n−1} = 𝕂(α_{1}^{1/p}, …, α_{n−1}^{1/p}), while α_{n}^{1/p} is algebraic of degree p ≥ D over this field 𝕂_{n−1}. Hence α_{n}^{1/p} is not a root of P. □

Lemma 5.16. Assume α_{1}, …, α_{n} are multiplicatively independent. Then there exists a positive integer D with the following property. Let Q ∈ 𝕂[X_{0}, X_{1}, …, X_{n}] be a nonzero polynomial of degree < D_{0} in X_{0} and < D in each variable X_{i} (1 ≤ i ≤ n). Let p be a prime with p > D and S a subset of ℤ with at least D_{0} elements such that (s, p) = 1 for any s ∈ S. Then one at least of the numbers

Q(s/p, α_{1}^{s/p}, …, α_{n}^{s/p}) (s ∈ S)

is not 0.

Proof. For (s, p) = 1 Bézout’s relations imply

ℚ(α, α^{1/p}) = ℚ(α, α^{s/p}).

Therefore the same arguments as for Lemma 5.15 yield Lemma 5.16. □

Lemmas 5.15 and 5.16 occurred in Baker’s method at an earlier stage of the theory (see [L1] pp. 176–177 and 185). Later, similar results taking into account several values of \underline{σ} were required (see [L1], pp. 223–226 and 237–238).

5.2.g Kummer’s Theory with p = 2

Let p be a prime for which the field generated over 𝕂 = ℚ(α_{1}, …, α_{n}) by the p-th roots α_{i}^{1/p} = exp((1/p) log α_{i}) with 1 ≤ i ≤ n has maximal degree p^{n}:

(5.17)  [𝕂(α_{1}^{1/p}, …, α_{n}^{1/p}) : 𝕂] = p^{n}.

This condition (5.17) yields a decomposition of each relation (5.9)

∑_{(τ,\underline{t})} c_{τ,\underline{t}} △ (s/p; τ, T_{0}^{♯}, σ_{0}) ∏_{k=1}^{n−1}(b_{n}t_{k} − b_{k}t_{n})^{σ_{k}} · α_{1}^{t_{1}s/p} ⋯ α_{n}^{t_{n}s/p} = 0

for which s is prime to p into p^{n} equations, where the sum over (τ, \underline{t}) is restricted to the elements with t_{i} ≡ t_{i}^{0} (mod p). For one at least of these equations, say corresponding to some \underline{t}^{0} ∈ ℕ^{n} with 0 ≤ t_{i}^{0} < p, the coefficients c_{τ,\underline{t}} with

t_{i} ≡ t_{i}^{0} (mod p) (1 ≤ i ≤ n)

do not all vanish. We reduce the number of coefficients of P as follows: fix such a \underline{t}^{0}, write t_{i} = t_{i}^{0} + pt′_{i} (1 ≤ i ≤ n) and consider now the new auxiliary polynomial obtained as a chunk of P

∑_{(τ,\underline{t}′)} c_{τ,\underline{t}^{0}+p\underline{t}′} △ (Y_{0}; τ, T_{0}^{♯})Y_{1}^{t′_{1}} ⋯ Y_{n}^{t′_{n}}.

In this process the number of coefficients^{18} of the auxiliary polynomial P decreases (the upper bound T_{i} for the degree in Y_{i} with 1 ≤ i ≤ n is divided by p). It turns out that the number of relations (3.7) will be essentially fixed along the inductive process, but at the end of the extrapolation the number of relations will be higher than the number of coefficients of the last auxiliary polynomial P, which is what we were looking for.

Each time Kummer’s condition (5.17) is used with, say, p = 2, we replace T_{i} by T_{i}/2. If one performs this extrapolation sufficiently far, one ends up (after the double induction) with some T_{j}, say T_{n}, replaced by 0 and the system of equations one gets in this case is easily shown (see Lemma 5.18) to have no nontrivial solution: there is no need to appeal to the zero estimate 3.8.

If one wishes to get T_{n}/2^{J} < 1 (after J steps), we need 2^{J} > T_{n}. It turns out that such a long extrapolation procedure has a cost: there is another factor log V_{n−1} (assuming

18By “the number of coefficients” of P we mean the number T_{0} ⋯ T_{n} where T_{i} is the known upper bound for the degree of P with respect to Y_{i} (1 ≤ i ≤ n). This is a loose way of speaking, but the values of T_{0}, …, T_{n} should be clear from the context.

V_{1} ≤ ⋯ ≤ V_{n−1} ≤ V_{n}) in the final estimate. This is exactly the main result in [W2], which improves earlier results by Baker in [B3]. See also [Mat I], as well as [Y1] for the p-adic case.

Now Proposition 3.8 enables us to perform a shorter extrapolation, which avoids this cost of log V_{n−1}. This is done in [PW1] as well as in [BWü] and [Mat II]. See also [Y2 I] for the p-adic case.

5.2.h A Simple Zero Estimate

As we saw in § 5.2.f, combining the extrapolation with (5.17), one reduces the set of coefficients of the auxiliary polynomial. The next result shows that we reach a contradiction as soon as all coefficients c_{τ\underline{t}} with 1 ≤ t_{n} < T_{n} vanish.

Lemma 5.18. Let T_{0}, …, T_{n−1}, S_{0}, S_{1} be positive integers and α_{1}, …, α_{n−1} nonzero complex numbers. Assume

S_{0}S_{1} ≥ T_{0} and S_{0} ≥ T_{j} for 1 ≤ j < n.

Then the matrix

M = (∂\underline{σ}(Y_{0}^{τ} Y_{1}^{t_{1}} ⋯ Y_{n−1}^{t_{n−1}})(s, α_{1}^{s}, …, α_{n−1}^{s}))_{{(τ,\underline{t}) \atop (\underline{σ},s)}} ,

where the rows are indexed by (τ, \underline{t}) and the columns by (\underline{σ}, s) with

0 ≤ τ < T_{0}, 0 ≤ t_{i} < T_{i} (1 ≤ i ≤ n − 1) and ∥\underline{σ}∥ < S_{0}, 0 ≤ s < S_{1},

has rank T_{0}T_{1} ⋯ T_{n−1}.

Remark. Because of (5.3) and (5.5), the conditions S_{0}S_{1} ≥ T_{0} and S_{0} ≥ T_{j} are responsible for the requirements D log E^{∗} ≥ log E and V_{j} ≥ log E in the hypotheses of Theorem 5.1.

Proof. On the subring ℂ[Y_{0}, Y_{1}, …, Y_{n−1}] of ℂ[Y_{0}, Y_{1}, …, Y_{n}], we have

∂\underline{σ} = (\frac{∂}{∂Y_{0}} )^{σ_{0}} ⋯ (\frac{∂}{∂Y_{n−1}} )^{σ_{n−1}} .

Hence the problem is reduced to a Cartesian product situation which one deals with by induction as follows.

Consider a relation between the rows of M:

∑_{τ=0}^{T_{0}−1} ∑_{t_{1}=0}^{T_{1}−1} ⋯ ∑_{t_{n−1}=0}^{T_{n−1}−1} c_{τ\underline{t}}∂\underline{σ}(Y_{0}^{τ} Y_{1}^{t_{1}} ⋯ Y_{n−1}^{t_{n−1}})(s, α_{1}^{s}, …, α_{n−1}^{s}) = 0

for any (\underline{σ}, s) satisfying ∥\underline{σ}∥ < S_{0} and 0 ≤ s < S_{1}. This means that the polynomial

P (\underline{Y}) = ∑_{τ=0}^{T_{0}−1} ∑_{t_{1}=0}^{T_{1}−1} ⋯ ∑_{t_{n−1}=0}^{T_{n−1}−1} c_{τ\underline{t}}Y_{0}^{τ} Y_{1}^{t_{1}} ⋯ Y_{n−1}^{t_{n−1}}

satisfies

∂\underline{σ}P (s, α_{1}^{s}, …, α_{n−1}^{s}) = 0 for ∥\underline{σ}∥ < S_{0} and 0 ≤ s < S_{1}.

For (σ_{0}, …, σ_{n−2}) ∈ ℕ^{n−1} with σ_{0} + ⋯ + σ_{n−2} < S_{0} and for 0 ≤ s < S_{1}, the polynomial

(\frac{∂}{∂Y_{0}} )^{σ_{0}} ⋯ (\frac{∂}{∂Y_{n−2}} )^{σ_{n−2}} P (s, α_{1}^{s}, …, α_{n−2}^{s}, Y_{n−1}) ∈ ℂ[Y_{n−1}]

has degree < T_{n−1} ≤ S_{0} and a zero at the point α_{n−1}^{s} of multiplicity ≥ S_{0}, hence this polynomial is 0. By induction one shows in the same way that for 1 ≤ k < n, (σ_{0}, …, σ_{k−1}) ∈ ℕ^{k} with σ_{0} + ⋯ + σ_{k−1} < S_{0} and 0 ≤ s < S_{1}, the polynomial

(\frac{∂}{∂Y_{0}} )^{σ_{0}} ⋯ (\frac{∂}{∂Y_{k−1}} )^{σ_{k−1}} P (s, α_{1}^{s}, …, α_{k−1}^{s}, Y_{k}, …, Y_{n−1}) ∈ ℂ[Y_{k}, …, Y_{n−1}]

is zero. Hence for k = 1 we get

(\frac{∂}{∂Y_{0}} )^{σ_{0}} P (s, Y_{1}, …, Y_{n−1}) = 0 for 0 ≤ σ_{0} < S_{0} and 0 ≤ s < S_{1}.

Since S_{0}S_{1} > T_{0} we deduce P = 0, and therefore M has rank T_{0}T_{1} ⋯ T_{n−1}. □

Remark. There is nothing special with the field ℂ: the result is valid for any field of zero characteristic. Also the same result holds if we select another basis than Y_{0}^{τ} Y_{1}^{t_{1}} ⋯ Y_{n−1}^{t_{n−1}} for the space ℂ[Y_{0}, …, Y_{n−1}], for instance △(Y_{0}; τ, T_{0}^{♯})Y_{1}^{t_{1}} ⋯ Y_{n−1}^{t_{n−1}}.

5.2.i Removing n^{n} under Kummer’s Condition

In [Mat I], E.M. Matveev succeeds to remove the factor n^{n} in (5.8) under Kummer’s condition (5.17) for p = 2. The idea is the following. He restricts the set of exponents (τ, \underline{t}) of his auxiliary polynomial to a subset satisfying

(5.19)  |t_{1} log α_{1} + ⋯ + t_{n} log α_{n}| ≤ \frac{U}{MES_{1}} ,

where M ≥ 1 is a new parameter. This allows him to take a larger radius R for the disk where he will apply a Schwarz Lemma (or an approximate Schwarz Lemma): in place of R = Er, he may take (almost without extra cost at this place) R = EMr. In the conclusion, in place of (log E)^{−n−1} appears now (log(ME))^{−n−1}. On the other hand, for application of Thue-Siegel’s Lemma in the construction of P, it is necessary to estimate from below the number of such elements (τ, \underline{t}). Matveev in [Mat I] uses arguments from geometry of numbers. Yu Kunrui [Y2 II] worked out simpler arguments, using only Dirichlet’s box principle, which suffice (the final numerical estimate may be not as sharp, but the argument works as well for the p-adic case; see also [W5] § 9.3 for another adaptation of this argument). Essentially, requiring (5.19) divides the number of tuples (τ, \underline{t}) by M. So in the final result the new parameter U is the old one multiplied by M(log(ME))^{−n−1}. Assuming E = e and taking M = c^{n}, one gets rid of the unwanted term n^{n} in (5.8).

Appendix. From 2 to n Logarithms

In this appendix we develop a remark which originates in Gel’fond’s work and has been also used by Bombieri, Bilu and Bugeaud (see [W5], Theorem 1.9 and Corollary 10.18). The goal is to deduce, from a nontrivial irrationality measure for the quotient of two logarithms of algebraic numbers, a nontrivial measure of linear independence for n logarithms. The idea is to write the coefficients b_{i} as b̃q_{i} + r_{i} with integers b̃, q_{1}, …, q_{n}, r_{1}, …, r_{n}, so that the linear form

b_{1}X_{1} + ⋯ + b_{n}X_{n}

has the same value at the point (λ_{1}, …, λ_{n}) than the binary form

b̃Y_{1} + Y_{2}

at the point

(q_{1}λ_{1} + ⋯ + q_{n}λ_{n}, r_{1}λ_{1} + ⋯ + r_{n}λ_{n}).

We need to select b̃ so that the “remainders” r_{i} ∈ ℤ have comparatively small absolute values.

We expand this argument in the following lemma.

Lemma 5.20 Let K be a number field of degree D, λ_{1}, …, λ_{n} elements of L such that the algebraic numbers α_{i} = e^{λ_{i}} are in K. Define V_{1}, …, V_{n} and V by

V_{i} = max \{h(α_{i}), \frac{|λ_{i}|}{D},\frac{1}{D} \} (i = 1, …, n) and V = max\{V_{1}, …, V_{n}\}.

Further, let b_{1}, …, b_{n} be rational integers and let B be a positive integer satisfying B ≥ max{|b_{1}|, …, |b_{n}|}.

Then there exist λ̃_{1}, λ̃_{2} in L and b̃ in ℤ such that α̃_{1} = e_{λ_{1}} and α̃_{2} = e_{λ_{2}} are in K,

b_{1}λ_{1} + ⋯ + b_{n}λ_{n} = b̃ λ̃_{1} + λ̃_{2},

1 ≤ b̃ ≤ B, and such that the numbers

Ṽ_{i} = max \{h(α̃_{i}), \frac{|λ̃_{i}|}{D},\frac{i}{D} \} (i = 1, 2)

satisfy

Ṽ_{1}Ṽ_{2} ≤ 2n^{2}B^{1−κ_{n}}V^{2} with κ_{n} = \frac{1}{2n − 1}·

For the proof of Lemma 5.20, we shall use Minkowski’s Linear Forms Theorem (see for instance [S], Chap. II, S 1, Theorem 2C)^{19}.

19If one applies Dirichlet’s box principle in place of Minkowski’s Theorem, one deduces a weaker estimate for Ṽ_{1}Ṽ_{2}, where 2n^{2}B^{1−κ_{n}}V^{2} is replaced by 4n^{2}B^{1−κ_{n}}V^{2}.

Lemma 5.21 Let (x_{ij})_{1≤i,j≤n} be a n × n matrix with determinant ±1. Let A_{1}, …, A_{n} be positive real numbers with product A_{1} ⋯ A_{n} = 1. Then there exists (q_{1}, …, q_{n}) ∈ ℤ^{n} \ {0} such that

|q_{1}x_{i1} + ⋯ + q_{n}x_{in}| < A_{i} (1 ≤ i < n)

and

|q_{1}x_{n1} + ⋯ + q_{n}x_{nn}| ≤ A_{n}.

Proof of Lemma 5.20. In the trivial case b_{1} = ⋯ = b_{n} = 0 we set b̃ = 1, λ̃_{1} = λ̃_{2} = 0 and the conclusion is satisfied. So we may assume (b_{1}, …, b_{n}) ≠ 0. By symmetry, we may assume b_{1} ≥ |b_{i}| for 1 ≤ i ≤ n.

Further, if n = 1, we set

b̃ = b_{1}, λ̃_{1} = λ_{1}, λ̃_{2} = 0,

so that Ṽ_{1} = V_{1}, Ṽ_{2} = 2/D and again the conclusion is satisfied. Hence we shall assume n ≥ 2.

Define

Q = B^{(n−1)κ_{n}}.

Notice that the exponent is

(n − 1)κ_{n} = \frac{1}{2}(1 − κ_{n}).

Using Lemma 5.21, we deduce that there exist rational integers q_{1}, …, q_{n}, not all of which are zero, satisfying

|\frac{b_{i}}{b_{1}} q_{1} − q_{i}| < Q^{−1/(n−1)} for 2 ≤ i ≤ n and |q_{1}| ≤ Q.

Since

|q_{i}| < |\frac{b_{i}}{b_{1}} · q_{1}| + 1 ≤ |q_{1}| + 1,

we have |q_{i}| ≤ |q_{1}|. In particular q_{1} ≠ 0. Replacing, if necessary, all q_{i} by −q_{i}, we may assume q_{1} > 0.

We define

b̃ = [b_{1}/q_{1}].

Clearly the inequalities 1 ≤ b̃ ≤ B are satisfied. Further, set

r_{i} = b_{i} − b̃q_{i} (1 ≤ i ≤ n),

λ̃_{1} = ∑_{i=1}^{n} q_{i}λ_{i} and λ̃_{2} = ∑_{i=1}^{n} r_{i}λ_{i},

so that

b̃λ̃_{1} + λ̃_{2} = ∑_{i=1}^{n}(b̃q_{i} + r_{i})λ_{i} = ∑_{i=1}^{n} b_{i}λ_{i}.

It remains to estimate Ṽ_{1} and Ṽ_{2}. Define R = max_{1≤i≤n} |r_{i}|. We have

Ṽ_{1} ≤ nq_{1}V, Ṽ_{2} ≤ nRV,

hence

Ṽ_{1}Ṽ_{2} ≤ n^{2}q_{1}RV^{2}.

Finally from the inequalities

|b_{i} − b̃q_{i}| ≤ \frac{1}{q_{1}} |b_{i}q_{1} − b_{1}q_{i}| + |q_{i}| |\frac{b_{1}}{q_{1}} − b̃| ≤ q_{1}^{−1}B^{1−κ_{n}} + q_{1}

and

q_{1}^{2} ≤ Q^{2} ≤ B^{1−κ_{n}}

we deduce

q_{1}R ≤ B^{1−κ_{n}} + q_{1}^{2} ≤ 2B^{1−κ_{n}}.
□

We shall combine Lemma 5.20 with the following sharp estimate for two logarithms (Corollary 1 in [LaMiN]).

Theorem 5.22 . Let λ_{1}, λ_{2} be two elements in L and b_{1}, b_{2} two nonzero rational integers. Define

α_{1} = e^{λ_{1}}, α_{2} = e^{λ_{2}} and D = [ℚ(α_{1}, α_{2}) : ℚ].

Assume α_{1} and α_{2} are multiplicatively independent. Let V_{1}, V_{2} and W be positive real numbers satisfying

V_{i} = max \{h(α_{i}), \frac{|λ_{i}|}{D},\frac{1}{D} \} (i = 1, 2)

and

W ≥ 1, W ≥ \frac{21}{D}, e^{W} ≥ \frac{|b_{2}|}{DV_{1}} + \frac{|b_{1}|}{DV_{2}} ·

Then

|b_{1}λ_{1} + b_{2}λ_{2}| ≥ exp{−31D^{4}V_{1}V_{2}W^{2}}.

Corollary 5.23 Under the assumptions of Theorem 5.1, assuming α_{1}, …, α_{n} are multiplicatively independent, if we set V = max{V_{1}, …, V_{n}}, E = e,

B = max{e, e^{21/D}, |b_{1}|, …, |b_{n}|}

and κ_{n} = 1/(2n − 1), then we have

|Λ| > exp{−62n^{2}D^{4}B^{1−κ_{n}}(log B)^{2}V^{2}}.

Further similar estimates are easy to produce.

Remark. Such an argument is used in [GL] in order to reduce a linear form in logarithms from 5 to 2 terms and to show that all integer solutions to the Diophantine equation in 3 variables

x^{5}+(z−1)^{2}x^{4}y−(2z^{3}+4z+4)x^{3}y^{2}+(z^{4}+z^{3}+2z^{2}+4z−3)x^{2}y^{3}+(z^{3}+z^{2}+5z+3)xy^{4}+y^{5} = ±1

are the trivial ones given by

±(x, y) = \{ \begin{matrix}(1, 0), (0, 1), & z ≠ −1, 0, \\ (1, 0), (0, 1), (±1, 1), (−2, 1), & z ∈ \{−1, 0\}. \end{matrix}

6 Matveev’s Theorem with Interpolation Determinants

We refer to Nesterenko’s lectures for an introduction to Matveev’s proof of the following Theorem:

Theorem 6.1 There exists an absolute positive constant C with the following property. Let n be a positive integer, α_{1}, …, α_{n} nonzero algebraic numbers and log α_{1}, …, log α_{n} logarithms of α_{1}, …, α_{n} respectively. Assume that the numbers log α_{1}, …, log α_{n} are ℚlinearly independent. Let b_{1}, …, b_{n} be rational integers, not all of which are zero. Denote by D the degree of the number field ℚ(α_{1}, …, α_{n}) over ℚ. Further, let W, V_{1}, …, V_{n} be positive real numbers. Assume

V_{j} ≥ max \{h(α_{j}), \frac{e| log α_{j}|}{D}\} (1 ≤ j ≤ n) and W ≥ 1 + log D.

Further, assume b_{n} ≠ 0 and

e^{W} ≥ \frac{1}{D} max_{1≤j≤n−1} \{\frac{|b_{n}|}{V_{j}} + \frac{|b_{j}|}{V_{n}}\} .

Then the number

Λ = b_{1} log α_{1} + ⋯ + b_{n} log α_{n}

has absolute value bounded from below by

|Λ| > exp{−C^{n}D^{n+2}(1 + log D)W V_{1} ⋯ V_{n}}.

Our goal is to explain how to prove Matveev’s Theorem 6.1 by means of interpolation determinants and to avoid the construction of an auxiliary function involving Thue-Siegel-Bombieri-Vaaler’s Lemma in [Mat II]. We aim at obtaining the conclusion with an unspecified (but effectively computable) value for the absolute constant C.

6.1 First Extrapolation

Matveev uses several auxiliary polynomials which are of the form

P (\underline{Y}) = ∑_{(τ,\underline{t})∈L} c_{τ,\underline{t}} △ (Y_{0}; τ, T_{0}^{♯})Y_{1}^{t_{1}} ⋯ Y_{n}^{t_{n}} ∈ ℂ[Y_{0}, …, Y_{n}],

where L is a suitable set of (τ, \underline{t}) ∈ ℕ × ℕ^{n} with 0 ≤ τ < T_{0} and 0 ≤ t_{j} < T_{j} (1 ≤ j ≤ n). He starts the construction by means of Bombieri-Vaaler’s version of Thue-Siegel’s Lemma and solves a system of equations (3.4). Next a first extrapolation enables him to get more relations like (3.7), say

(6.2) ∂\underline{σ}P (s, α_{1}^{s}, …, α_{n}^{s}) = 0 for ∥\underline{σ}∥ < S_{0}^{(j)} and 0 ≤ s < S_{1}^{(j)} (0 ≤ j ≤ J).

At this stage of his proof we consider the matrix M of the linear system given by equations (6.2). We write this matrix as follows. Consider the column vector with |L| rows and entries in ℚ[\underline{Y}]:

P(\underline{Y}) = (△(Y_{0}; τ, T_{0}^{♯})Y_{1}^{t_{1}} ⋯ Y_{n}^{t_{n}})_{(τ,\underline{t})∈L}.

For each \underline{σ} ∈ ℕ^{n}, define

G\underline{σ}(z) = ∂\underline{σ}P(z, α_{1}^{z}, …, α_{n}^{z})

and set

M = (G\underline{σ}(s))_{(\underline{σ},s)},

where (\underline{σ}, s) ranges over the set of elements in ℕ^{n}×ℕ such that ∥\underline{σ}∥ < S_{0}^{(j)} and 0 ≤ s < S_{1}^{(j)} for at least one j in the range 0 ≤ j ≤ J.

We consider two cases. The first one is when the rank of M is maximal, equal to |L|. In this case the proof is very short: we select a maximal nonvanishing determinant

Δ = det(G_{\underline{σ}_{μ}}(s_{μ}))_{1≤μ≤|L|}

by taking the first |L| columns in minimal lexicographic ordering like in § 2.2 and § 3.5, we bound its absolute value from below by means of Liouville’s inequality (Proposition 1.19), and from above thanks to analytic arguments. Corollary 3.22 is not quite sufficient for this purpose: it corresponds to a Schwarz’ Lemma like Lemma 1.5 for a function with many zeroes, while we need a statement corresponding to an approximated Schwarz’ Principle like Lemma 1.18 for a function with many small values (see Lemma 6.6). These estimates provide the required conclusion.

Now we assume M has rank < |L|. Denote by L_{1} − 1 this rank. We select a subset L_{1} of L with L_{1} elements such that, if we set

P_{1}(\underline{Y}) = (△(Y_{0}; τ, T_{0}^{♯})Y_{1}^{t_{1}} ⋯ Y_{n}^{t_{n}})_{(τ,\underline{t})∈L_{1}} and G(1) ; \underline{σ}(z) = ∂\underline{σ}P_{1}(z, α_{1}^{z}, …, α_{n}^{z}),

then the associated truncated matrix with L_{1} rows only

M_{1} = (G(1) ; \underline{σ}(s))_{(\underline{σ},s)}

has rank L_{1} − 1. Thanks to Lemma 5.18 we may assume that all (τ, \underline{t}) ∈ L with t_{n} = 0 belong to L_{1}.

Again we select the first L_{1} − 1 columns of M_{1} which are linearly independent like in § 2.2 and § 3.5, say

G_{\underline{σ}_{1}}^{(1)}(s_{1}), …, G_{\underline{σ}_{L_{1}−1}}^{(1)}(s_{L_{1}−1}),

and we consider the polynomial

P_{1}(\underline{Y}) = det(G_{\underline{σ}_{1}}^{(1)}(s_{1}), …, G_{\underline{σ}_{L_{1}−1}}^{(1)}(s_{L_{1}−1}), P_{1}(\underline{Y})).

From the construction it follows that P_{1} is not zero. We extrapolate on the division points as follows: for ∥\underline{σ}∥ < S_{0}^{(J+1)} and 0 ≤ s < S_{1}^{(J+1)} we prove

∂\underline{σ}P_{1}(s/2, α_{1}^{s/2}, …, α_{n}^{s/2}) = 0.

Thanks to Lemma 5.12, for odd s each such equation decomposes into 2^{n} equations. We explain now how to use this fact.

6.2 Using Kummer’s Condition

Lemma 6.3. Let ℓ, L_{1}, …, L_{ℓ}, M, N be positive integers, 𝕃 a field, 𝕂 a subfield of 𝕃, t_{1}, …, t_{ℓ} elements in 𝕃 which are linearly independent over 𝕂, A_{1}, …, A_{ℓ}, B_{1}, …, B_{ℓ} matrices with entries in 𝕂, where A_{ν} has size L_{ν} × M and B_{ν} size L_{ν} × N (1 ≤ ν ≤ ℓ). Define

A = ( A_{1} ; ⋮ ; A_{ℓ} ) and C = ( \begin{matrix}A_{1} & t_{1}B_{1} \\ ⋮ & ⋮ \\ A_{ℓ} & t_{ℓ}B_{ℓ} \end{matrix} )

and assume

rank(C) = rank(A) < L_{1} + ⋯ + L_{ℓ}.

Then for at least one index ν in the range 1 ≤ ν ≤ ℓ we have

rank(B_{ν}) < L_{ν}.

Proof. For 1 ≤ ν ≤ ℓ, write

B_{ν} = (b_{λj}^{(ν)})_{{1≤λ≤Lν \atop 1≤j≤N}} = (\underline{b}_{1}^{(ν)}, …, \underline{b}_{N}^{(ν)}),

where \underline{b}_{1}^{(ν)}, …, \underline{b}_{N}^{(ν)} are the column vectors in 𝕂^{L_{ν}}. Let r be the rank of the matrix A. Since C has also rank r, each of the column vectors

( t_{1}\underline{b}_{j}^{(1)} ; ⋮ ; t_{ℓ}\underline{b}_{j}^{(ℓ)} ) of ( t_{1}B_{1} ; ⋮ ; t_{ℓ}B_{ℓ} )

(1 ≤ j ≤ N) belongs to the space spanned by the column vectors of A. We write the N relations

rank ( \begin{matrix}A_{1} & t_{1}\underline{b}_{j}^{(1)} \\ ⋮ & ⋮ \\ A_{ℓ} & t_{ℓ}\underline{b}_{j}^{(ℓ)} \end{matrix} ) = r (1 ≤ j ≤ N)

expanding by minors:

∑_{ν=1}^{ℓ} ∑_{λ=1}^{L_{ν}} α_{νλ}^{(ϱ)}t_{ν}b_{λj}^{(ν)} = 0 for 1 ≤ ϱ ≤ R and 1 ≤ j ≤ N,

where the coefficients α_{νλ}^{(ϱ)} are in 𝕂 (independent of j) and are not all zero (because A has rank r). Since t_{1}, …, t_{ℓ} are linearly independent over 𝕂, we deduce

(6.4)  ∑_{λ=1}^{L_{ν}} α_{νλ}^{(ϱ)}b_{λj}^{(ν)} = 0 for 1 ≤ ϱ ≤ R, 1 ≤ j ≤ N and 1 ≤ ν ≤ ℓ.

There is at least one ν in the range 1 ≤ ν ≤ ℓ such that the coefficients α_{νλ}^{(ϱ)} with 1 ≤ ϱ ≤ R and 1 ≤ j ≤ N are not all zero. For such a ν the relations (6.4) yield a nontrivial dependence relation between the row vectors of B_{ν}, hence B_{ν} has rank < L_{ν}. □

6.3 Second Extrapolation

At this stage of the proof, after the first extrapolation including division points, we get a new matrix

M_{2} = (G(2) ; \underline{σ}(s))_{(\underline{σ},s)}

built with

G(2) ; \underline{σ}(s) = ∂\underline{σ}P_{2}(s, α_{1}^{s}, …, α_{n}^{s}) and P_{2}(\underline{Y}) = (△(Y_{0}; τ, T_{0}^{♯})Y_{1}^{t_{1}} ⋯ Y_{n}^{t_{n}})_{(τ,\underline{t})∈L_{2}}.

Now the set L_{1} has been reduced to L_{2}, with

|t_{i}| < T_{i}/2 for (τ, \underline{t}) ∈ L_{2} and 1 ≤ i ≤ n,

while the set of (\underline{σ}, s) is bigger than for M_{1}, say

(6.5)  ∥\underline{σ}∥ < S_{0}^{(2,j)} and 0 ≤ s < S_{1}^{(2,j)} (0 ≤ j ≤ J_{2}).

The upper index (2, j) corresponds to the j-th step in the second extrapolation (hence we may set S_{0}^{(1,j)} = S_{0}^{(j)} and S_{1}^{(1,j)} = S_{1}^{(j)}).

Before starting our second induction let us have a look at Matveev’s arguments in [Mat II]: his second auxiliary polynomial P_{2} is a linear combination of the components of P_{2}(\underline{Y}); he shows by induction on j the relations

∂\underline{σ}P_{2}(s, α_{1}^{s}, …, α_{n}^{s}) = 0

for all (\underline{σ}, s) ∈ ℕ^{n} × ℕ in (6.5).

As far as we are concerned we repeat the argument of § 5.1. However it is necessary to be slightly careful with the construction of the next matrix: for an efficient application of the analytic argument we need to introduce a condition like (3.24). This difficulty did not arise in the first extrapolation (§ 6.1), because the condition

|L| > 2\binom{S_{0} + n − 1}{n}S_{1}

was in force: Matveev needed it to solve his system of linear equations and produce his first auxiliary polynomial. For the second extrapolation this condition is not there and we proceed as follows.

Let L_{2} − 1 be the rank of M_{2}. Define μ_{0} = L_{2}/2^{n+2}. We select the first μ_{0} columns which are linearly independent. Next we consider the matrix M̃_{2} which consists of these μ_{0} columns together with all other columns of M_{2} for which

∥\underline{σ}∥ < S̃_{0}^{(2,j)} and 0 ≤ s < S_{1}^{(2,j)} (0 ≤ j ≤ J_{2}),

where

\{ \begin{matrix}S̃_{0}^{(2,0)} = \frac{1}{2}(S_{0}^{(2,0)} + S_{0}^{(2,1)}) \\ S̃_{0}^{(2,j)} = S_{0}^{(2,j)} & for 1 ≤ j ≤ J_{2}. \end{matrix}

When μ_{0} is large, there is no point to distinguish between M_{2} and M̃_{2}, but otherwise the picture is the following:

S_{0}^{(2,0)} ; S̃_{0}^{(2,0)} ; S_{0}^{(2,1)}_{0  μ_{0}  S_{1}^{(2,0)}  S_{1}^{(2,1)}}

Denote by L̃_{2} the rank of M̃_{2}. We already selected μ_{0} columns, we select L̃_{2} − μ_{0} further columns in the usual way in order to get a maximal number of independent columns.

If L̃_{2} −1 = |L_{2}|, we easily conclude the proof by usual means: M̃_{2} is a nonsingular square matrix with nonzero determinant, say Δ_{2}. We bound the absolute value of Δ_{2} from below by arithmetic means and from above by analytic means.

If L̃_{2} < μ_{0} = L_{2}/2^{n+2}, then the second extrapolation is complete: our goal is precisely to produce a matrix with such a low rank.

Assume now L_{2}/2^{n+2} ≤ L̃_{2} ≤ |L_{2}|. We select L̃_{2} rows of M̃_{2}, corresponding to a subset L̃_{2} of L_{2} with L̃_{2} elements. Our new auxiliary polynomial is the determinant P̃_{2}(\underline{Y}) of the square matrix

(G̃_{\underline{σ}_{1}}^{(2)}(s_{1}), …, G̃_{\underline{σ}_{_{L_{2}}−1}}^{(2)}(s_{_{L_{2}}−1}), P̃_{2}(\underline{Y})).

We extrapolate on division points and complete the second induction as we did for the first one.

6.4 An Approximate Schwarz Lemma for Interpolation Determinants

The results of § 3.4 deal with zeroes of interpolation determinants and involve Schwarz’ Lemma. They are sufficient for the proof of qualitative statements like Baker’s Theorem 3.1.

In § 4, we were able to prove a quantitative result by means of Lemma 4.11, which is also a Schwarz’ Lemma; however only the zero at the origin is used, and there is no extrapolation; this explains why the required approximate Schwarz’ Lemma in the proof of Lemma 4.14 is just the interpolation formula of Lemma 1.17 (or just Taylor’s expansion as in the remark in § 4.4).

Such an argument does not seem to suffice when we want to extrapolate. Indeed the determinant

(∂\underline{σ}P(s, α_{1}^{s}, …, α_{n−1}^{z}, (α_{1}^{b_{1}} ⋯ α_{n−1}^{b_{n−1}})^{−z/b_{n}})(s_{μ}))

does not satisfy the assumption (3.17) of Proposition 3.16.

Another attempt is to follow the method of § 7.3 in [W5]: in this case we return to the results of § 3.4 but we replace in Proposition 3.16 D^{\underline{κ}^{μ}}φ_{λ}(\underline{ξ}_{μ}) by D^{\underline{κ}^{μ}}φ_{λ}(\underline{ξ}_{μ}) + ϵ_{λμ} with sufficiently small complex numbers ϵ_{λμ}. Unfortunately this does not seem to work either: the main hypothesis (3.17) of Proposition 3.16 is not satisfied for a matrix where some entries are replaced by constants.

Finally the only way so far is to combine Lemmas 1.18 and 3.15 by proving an approximate Schwarz Lemma for interpolation determinants.

Details will appear elsewhere.

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