1.

Transcendence, exponential polynomials and commutative linear algebraic groups

We start with two examples of transcendence results; their proofs involve exponential polynomials and they occur as corollaries of a general result on commutative algebraic groups: the linear subgroup Theorem.

In this context, there is duality, which can be explained by means of the Fourier-Borel transform of exponential polynomials. This duality is revisited by S. Fischer from the view point of commutative linear algebraic groups, using Hopf algebras.

Transcendence results

Here is our first transcendence result ( [B] Theorem 2.1).

Theorem 1.1. (Baker). Let β 0 , . . . , β n be algebraic numbers and ↵ 1 , . . . , ↵ n be non-zero algebraic numbers. For 1  i  n, denote by log ↵ i any complex logarithm of ↵ i . Assume

β 0 + β 1 log ↵ 1 + • • • + β n log ↵ n = 0.
Then it holds that 1. β 0 = 0. 2. If (β 1 , . . . , β n ) 6 = (0, . . . , 0), then the numbers log ↵ 1 , . . . , log ↵ n are Q-linearly dependent.

3.If (log ↵ 1 , . . . , log ↵ n ) 6 = (0, . . . , 0), then the numbers β 1 , . . . , β n are Q-linearly dependent.

As is well known this result includes Hermite's result (1873) on the transcendence of e, Lindemann's result (1882) on the transcendence of ⇡ and more generally Corollary 1.2. (Hermite-Lindemann). If β is a non-zero algebraic number, then e β is a transcendental number. Equivalently, if ↵ is a non-zero algebraic number and if log ↵ is any non-zero logarithm of ↵, then log ↵ is a transcendental number.

This includes the transcendence of numbers like e, ⇡, e √ 2 , log 2. Denote by Q the field of all complex algebraic numbers, which is the algebraic closure of Q in C and by L = exp -1 (Q ⇥ ) the Q-vector space of logarithms of algebraic numbers:

L = {λ ∈ C ; e λ ∈ Q ⇥ } = {log ↵ ; ↵ ∈ Q ⇥ }.
Hermite-Lindemann's Theorem asserts that L does not contain any nonzero algebraic number:

L ∩ Q = {0}.
Another corollary of Baker's Theorem 1.1 is the answer to Hilbert's seventh problem, given by A.O. Gel'fond and Th. Schneider in 1934: Corollary 1.3. (Gel'fond-Schneider). If β is an irrational algebraic number, ↵ a non-zero algebraic number and log ↵ a non-zero logarithm of ↵, then the number

↵ β = exp(β log ↵)
is transcendental.

Gel'fond-Schneider's Theorem (Corollary 1.3) asserts that the quotient of two non-zero elements in L is either rational or else transcendental; Baker's Theorem 1.1 implies more generally that Q-linearly independent elements of L are Q-linearly independent.

Theorem 1.1 also yields the transcendence of numbers like e

√ 2 2 √ 3 , Z 1 0 dx 1 + x 3 = 1 3 ✓ log 2 + ⇡ √ 3
◆ and more generally (under suitable assumptions -see [B] Theorems 2.2, 2.3 and 2.4) of numbers of the form

e β 0 ↵ β 1 1 • • • ↵ βm m and β 0 + β 1 log ↵ 1 + • • • + β m log ↵ m
when the numbers ↵ i and β j are algebraic.

It is to be remarked that Baker's Theorem does not include all known transcendence results related to the exponential function: here is an example ( [W] p. 386).

Theorem 1.4. (Sharp six exponentials Theorem). Let x 1 , x 2 be two complex numbers which are Q-linearly independent and y 1 , y 2 , y 3 three complex numbers which are also Q-linearly independent. Further let β ij (i = 1, 2, j = 1, 2, 3) be six algebraic numbers. Assume

e x i y j -β ij ∈ Q for i = 1, 2, j = 1, 2, 3. Then x i y j = β ij for i = 1, 2, j = 1, 2, 3.
The special case β ij = 0 for all i, j is known as the six exponentials Theorem (due to Lang and Ramachandra in the 60's -see references in [W], § 1.3): if x 1 , x 2 are Q-linearly independent and y 1 , y 2 , y 3 are also Q-linearly independent, then at least one of the six numbers

e x i y j (i = 1, 2, j = 1, 2, 3) is transcendental.
The four exponentials Conjecture ( [W] Conjecture 1.13) asserts that two values for y should suffice: if x 1 , x 2 are Q-linearly independent and y 1 , y 2 are also Q-linearly independent, then at least one of the four numbers e x i y j (i = 1, 2, j = 1, 2) is transcendental.

A sharper result is expected, which we call here the sharp four exponentials Conjecture: under the same assumptions as in the four exponentials Conjecture, if β ij (i = 1, 2, j = 1, 2) are four algebraic numbers such that

e x i y j -β ij ∈ Q for i = 1, 2, j = 1, 2, then one should have x i y j = β ij for i = 1, 2, j = 1, 2.
Conjecture 1.5. (Sharp five exponentials Conjecture). If x 1 , x 2 are Q-linearly independent, if y 1 , y 2 are Q-linearly independent and if ↵, β 11 , β 12 , β 21 , β 22 , γ are six algebraic numbers such that e x 1 y 1 -β 11 , e x 1 y 2 -β 12 , e x 2 y 1 -β 21 , e x 2 y 2 -β 22 , e (γx 2 /x 1 )-↵ are algebraic, then x i y j = β ij for i = 1, 2, j = 1, 2 and furthermore γx 2 = ↵x 1 .

The case β ij = 0 of Conjecture 1.5 is an easy consequence of the sharp six exponentials Theorem 1.4: this is the five exponentials Theorem ( [W] p. 385): If x 1 , x 2 are Q-linearly independent, if y 1 , y 2 are Q-linearly independent and if γ is a non-zero algebraic, then at least one of the five numbers e x 1 y 1 , e x 1 y 2 , e x 2 y 1 , e x 2 y 2 , e γx 2 /x 1 is transcendental. Moreover, in the special case where the three numbers y 1 , y 2 and γ/x 1 are Q-linearly independent, the sharp five exponentials Conjecture 1.5 follows from the sharp six exponentials Theorem 1.4 by setting

y 3 = γ/x 1 , β 13 = γ, β 23 = ↵, so that
x 1 y 3 -β 13 = 0 and x 2 y 3 -β 23 = (γx 2 /x 1 ) -↵.

In the case where the three numbers y 1 , y 2 and γ/x 1 are linearly dependent over Q, the conjecture is open. A consequence of the sharp five exponentials Conjecture 1.5 is the transcendence of the number e ⇡ 2 : take

x 1 = y 1 = γ = 1, x 2 = y 2 = i⇡, ↵ = 0, β 11 = 1, β ij = 0 for (i, j) 6 = (1, 1).
So far, we only know (W.D. Brownawell and the author) that at least one of the two statements holds: • e ⇡ 2 is transcendental.

• The two numbers e and ⇡ are algebraically independent.

In the same way, setting

x 1 = y 1 = γ = 1, x 2 = y 2 = λ, ↵ = 0, β 11 = 1, β ij = 0 for (i, j) 6 = (1, 1),
we deduce from Conjecture 1.5 the transcendence of e λ 2 when λ is a nonzero logarithm of an algebraic number. Writing ↵ = e λ or λ = log ↵, we have

e λ 2 = ↵ log ↵ .
Only the following weaker statement is known: at least one of the two numbers

e λ 2 = ↵ log ↵ , e λ 3 = ↵ (log ↵) 2
is transcendental, which was proved initially by W.D. Brownawell and the author as a consequence of a result of algebraic independence; however it is also a consequence of the sharp six exponentials Theorem 1.4 with

x 1 = y 1 = 1, x 2 = y 2 = λ, y 3 = λ 2 , β 11 = 1, β ij = 0 for (i, j) 6 = (1, 1).
The sharpest known result on this subject is the strong six exponentials Theorem due to D. Roy ([W] Corollary 11.16). Denote by e L the Q-vector space spanned by 1 and L: hence e L is nothing else than the set of complex numbers of the form

β 0 + n X i=1 β i log ↵ i ,
with n ≥ 0, β j algebraic numbers, ↵ i non-zero algebraic numbers and all values of their logarithm are considered. The strong six exponentials Theorem states that if x 1 , x 2 are Q-linearly independent and if y 1 , y 2 , y 3 are Q-linearly independent, then at least one of the six numbers

x i y j (i = 1, 2, j = 1, 2, 3) is not in e L.
The strong four exponentials Conjecture ( [W] Conjecture 11.17) claims that the same should hold with only two values y 1 , y 2 in place of three.

Exponential polynomials

The proofs of both Theorems 1.1 and 1.4 involve exponential polynomials. Here are basic facts on them.

For the proof of Baker's Theorem 1.1, assume

β 0 + β 1 log ↵ 1 + • • • + β n-1 log ↵ n-1 = log ↵ n .
In Gel'fond-Baker's Method (B 1 ), we consider the following n + 1 functions z 0 , e z 1 , . . . , e z n-1 , e

β 0 z 0 +β 1 z 1 +•••+β n-1 z n-1 of n variables z 0 , . . . , z n-1 . At the points Z(1, log ↵ 1 , . . . , log ↵ n-1 ) ∈ C n ,
all these functions take algebraic values. Moreover we also get algebraic numbers by taking derivatives with respect to the operators ∂/∂z i , (0  i  n -1).

Notice that there are n variables, n + 1 functions, 1 point (together with its multiples) and n derivations (together with their compositions).

In Generalized Schneider's Method (B 2 ), we consider the n + 1 functions: z 0 , z 1 , . . . , z n-1 and

e z 0 ↵ z 1 1 • • • ↵ z n-1 n-1 = exp{z 0 + z 1 log ↵ 1 + • • • + z n-1 log ↵ n-1 } at the points: {0} ⇥ Z n-1 + Z(β 0 , , . . . , β n-1 ) ∈ C n .
Only one derivation yields algebraic numbers, namely ∂/∂z 0 . In this alternative approach there are again n variables and n + 1 functions, but a single derivation, while the points form a group of Zrank n.

1.3

Data for the proof of Theorem 1.4

Here are the main data for the proof of Theorem 1.4. Assume x 1 , . . . , x a are Q-linearly independent, y 1 , . . . , y b are Q-linearly independent, β ij are algebraic numbers and λ ij are elements in L such that

λ ij = x i y j -β ij for i = 1, . . . , a, j = 1, . . . , b with ab > a + b.
For Theorem 1.4 it would be sufficient to restrict to a = 2, b = 3, but it will be useful to introduce these two parameters a and b so that the situation becomes symmetric. As we shall see, we should assume ab > a + b, which means either a ≥ 2 and b ≥ 3 or else a ≥ 3 and b ≥ 2.

Consider the functions:

z i , e (x i /x 1 )(z a+1 +z 1 )-z i (1  i  a)
at the points of the subgroup in C spanned by

(β 1j , . . . , β aj , λ 1j ) ∈ C a+1 (1  j  b).
These values are algebraic and the same holds for the values at the same points of the derivatives of these functions with respect to the differential operators ∂/∂z i (2  i  a) and ∂/∂z a+1 -∂/∂z 1 .

Hence we are dealing with 2a functions in a + 1 variables, b points (linearly independent) and a derivations.

1.4

Commutative linear algebraic groups

Theorems 1.1 and 1.4 are special cases of the linear subgroup Theorem. Consider a commutative linear algebraic group, say

G = G d 0 a ⇥G d 1 m
(where G a denotes the additive group and G m the multiplicative group), over the field Q of algebraic numbers. Its dimension is

d = d 0 + d 1 . Let W ⇢ T e (G) be a C-subspace which is rational over Q. Denote by `0 its dimension. Let Y ⇢ T e (G) be a finitely generated subgroup such that Γ = exp(Y ) is contained in G(Q) = Q d 0 ⇥ (Q ⇥ ) d 1 . Let `1 be the Z-rank of Γ. Finally let V ⇢ T e (G) be a C-subspace containing both W and Y .
Let n be the dimension of V.

The conclusion of the linear subgroup Theorem below ([W] Theorem 11.5) is non-trivial only when n(`1 + d 1 ) < `1d 1 + `0d 1 + `1d 0 .

(1.6)

For each connected algebraic subgroup G ⇤ of G, defined over Q, we define

Y ⇤ = Y ∩ T e (G ⇤ ), V ⇤ = V ∩ T e (G ⇤ ), W ⇤ = W ∩ T e (G ⇤ ) and d ⇤ = dim(G ⇤ ), `⇤ 1 = rank Z (Y ⇤ ), n ⇤ = dim C (V ⇤ ), `⇤ 0 = dim C (W ⇤ ).
We may write

G ⇤ = G ⇤ 0 ⇥ G ⇤ 1 where G ⇤ 0 is an algebraic subgroup of G 0 and G ⇤ 1 is an algebraic subgroup of G 1 . Define d ⇤ 0 = dim(G ⇤ 0 ), d ⇤ 1 = dim(G ⇤ 1 ), so that d ⇤ = d ⇤ 0 + d ⇤ 1 . If we set G 0 0 = G 0 G ⇤ 0 , G 0 1 = G 1 G ⇤ 1 , G 0 = G G ⇤ = G 0 0 ⇥ G 0 1 , Y 0 = Y Y ⇤ , V 0 = V V ⇤ , W 0 = W W ⇤ ,
and

d 0 0 = dim(G 0 0 ), d 0 1 = dim(G 0 1 ), d 0 = dim(G 0 ), `0 1 = rank Z (Y 0 ), n 0 = dim C (V 0 ), `0 0 = dim C (W 0 ), then d 0 = d ⇤ 0 + d 0 0 , d 1 = d ⇤ 1 + d 0 1 , d = d ⇤ + d 0 , `1 = `⇤ 1 + `0 1 , n = n ⇤ + n 0 , `0 = `⇤ 0 + `0 0 .
Theorem 1.7. (Linear subgroup Theorem).

(1) Assume d > n. Then there exists a connected algebraic subgroup G ⇤ of G such that

d 0 > `0 0 and `0 1 + d 0 1 d 0 -`0 0  d 1 d -n • (1') Assume `1 > 0.
Then there is a G ⇤ for which

(d ⇤ 1 , `⇤ 1 ) 6 = (0, 0) and d ⇤ -`⇤ 0 d ⇤ 1 + `⇤ 1  n - `0 `1 • (2) Assume d > n and `1 > 0. Assume further that for any G ⇤ for which Y ⇤ 6 = {0}, we have n ⇤ -`⇤ 0 `⇤ 1 ≥ n - `0 `1 •
Assume also that there is no G ⇤ for which the three conditions `0 1 = 0, n 0 = `0 0 and d 0 > 0 simultaneously hold. Then

d 1 > 0 and `1(d -n)  d 1 (n -`0).
(2') Assume d > n and `1 > 0. Assume further that for any G ⇤ for which d 0 > n 0 , we have

d 1 d -n  d 0 1 d 0 -n 0 • Assume also that there is no G ⇤ for which the three conditions d ⇤ 1 = 0, d ⇤ = n ⇤ and d ⇤ > 0 simultaneously hold. Then n > `0 and `1(d -n)  d 1 (n -`0).
(3) Assume `1 > 0. Then the family of G ⇤ for which `⇤ 1 6 = 0 and (n ⇤ -`⇤ 0 )/`⇤ 1 is minimal is not empty. Let G ⇤ be such an element for which d ⇤ is minimal. Then either d ⇤ = n ⇤ or else

d ⇤ 1 > 0 and n - `0 `1 ≥ n ⇤ -`⇤ 0 `⇤ 1 ≥ d ⇤ -n ⇤ d ⇤ 1 • (3') Assume d > n.
Then the family of G ⇤ for which d 0 > n 0 and d 0 1 /(d 0 -n 0 ) is minimal is not empty. Let G ⇤ be such an element for which d 0 is minimal. Then either `0 1 = 0 or else

n 0 > `0 0 and d 1 d -n ≥ d 0 1 d 0 -n 0 ≥ `0 1 n 0 -`0 0 •

Fourier-Borel duality

Unifying the notation of §1.2 involving exponential polynomials, we let d 0 + d 1 be the number of functions, d 0 of which are linear and d 1 are exponential, `0 the number of derivations, `1 the number of points and n the number of variables.

d0 d1 `0 `1 n Baker B1 1 n n 1 n Baker B2 n 1 1 n n Sharp six exponentials a a a b a + 1
The inequality (1.6)

n(`1 + d 1 ) < `1d 1 + `0d 1 + `1d 0 .
is satisfied in the case of Baker's Theorem 1.1 since

n(`1 + d 1 ) = n 2 + n, `1d 1 + `0d 1 + `1d 0 = n 2 + n + 1.
For Theorem 1.4 the condition a + b < ab is required:

n(`1 + d 1 ) = a 2 + ab + a + b, `1d 1 + `0d 1 + `1d 0 = a 2 + 2ab.
There is a duality in each two cases: it consists in permuting

(d 0 , d 1 , `0, `1) ←→ (`0, `1, d 0 , d 1 )
For Baker's Theorem 1.1 it permutes methods B 1 and B 2 . As pointed out to me by S. Fischler, for Theorem 1.4 it is not a mere permutation of a and b. Indeed the proof in § 1.3 involved the parameters

d 0 = d 1 = `0 = a, `1 = b, n = a + 1,
henceforth the dual proof will involve the parameters

`0 = `1 = d 0 = a, d 1 = b, n = a + 1.
In the dual proof there are d 0 + d 1 = a + b functions, namely z 1z a+1 , z 2 , . . . , z a together with

e β 1j z 1 +•••+β aj za+λ 1j z a+1 (1  j  b),
the derivative operators are ∂/∂z i (1  i  a) and the points are (0, . . . , 0, 1) together with

(x i /x 1 , -δ i2 , . . . , -δ ia , x i /x 1 ) (2  i  a)
where δ ij is Kronecker's symbol. This duality rests on the analytic formula

✓ d dz ◆ s z t e xz z=y = ✓ d dz ◆ t z s e yz z=x .
(1.8)

This formula (1.8) is related to the Fourier-Borel transform as follows. For s a non-negative integer and y a complex number, consider the analytic functional

L sy : f 7 -→ ✓ d dz ◆ s f (y).
Its Fourier-Borel transform is the analytic function L sy (f ⇣ ) of ⇣ ∈ C which is the transform of the function f ⇣ : z 7 → e z⇣ :

f ⇣ (z) = e z⇣ , L sy (f ⇣ ) = ⇣ s e y⇣ .
This yields (1.8) for t = 0. The general case follows from

L sy (z t f ⇣ ) = ✓ d d⇣ ◆ t L sy (f ⇣ ).
Formula (1.8) extends to the higher dimensional case (that is when n > 1). For

v = (v 1 , . . . , v n ) ∈ C n , set D v = v 1 ∂ ∂z 1 + • • • + v n ∂ ∂z n • Let w 1 , . . . , w `0 , u 1 , . . . , u d 0 , and in C n , t ∈ N d 0 and s ∈ N `0 . For z ∈ C n , write (uz) t = (u 1 z) t 1 • • • (u d 0 z) t d 0 and D s w = D s 1 w 1 • • • D s `0 w `0 .
Then D s w (uz) t e z z= = D t u (wz) s e z z= .

(1.9)

Example. In the proof of the sharp six exponentials Theorem 1.4 given in § 1.3 where d 0 = d 1 = `0 = a, `1 = b and n = a + 1,

u i = (δ i1 , . . . , δ ia , 0) (1  i  a),
w 1 = (1, 0, . . . , 0, -1) and

w i = (0, δ i2 , . . . , δ ia , 0) (2  i  a),
is a linear combination of 1 , . . . , a with 1 = (0, . . . , 0, 1) and

i = (x i /x 1 , -δ i2 , . . . , -δ ia , x i /x 1 ) (2  i  a)
while is a linear combination of 1 , . . . , b with j = (β 1j , . . . , β aj , λ 1j ) (1  j  b).

Remark. The Fourier-Borel duality is not the same as the duality introduced by D. Roy in [Ro] which relates (1) and (1'), ( 2) and (2'), ( 3) and (3').

Bicommutative Hopf algebras

We consider commutative and cocommutative Hopf algebras (also called bicommutative Hopf algebras) over a field of characteristic zero.

As the first example, the algebra of polynomials in one variable H = [X] is endowed with a Hopf algebra structure with the coproduct ∆, the co-unit ✏ and the antipode S defined as the algebra morphisms for which ∆(X) = X ⌦ 1 + 1 ⌦ X, ✏(X) = 0 and S(X) = -X. [G a ] is a bicommutative Hopf algebra of finite type.

If we identify

[X] ⌦ [X] with [T 1 , T 2 ] by mapping X ⌦ 1 to T 1 and 1 ⌦ X to T 2 , then ∆P (X) = P (T 1 + T 2 ), ✏P (X) = P (0), SP (X) = P (-X). Since G a (K) = Hom ( [X], K) and [G a ] = [X], it follows that
Our next example is the algebra of Laurent polynomials H = [Y, Y -1 ] which becomes a Hopf algebra with the coproduct ∆ satisfying ∆(Y ) = Y ⌦Y , the co-unit ✏ for which ✏(Y ) = 1 and the antipode S with

S(Y ) = Y -1 . The algebra isomorphism between H ⌦ H and [T 1 , T -1 1 , T 2 , T -1 2 ] defined by Y ⌦ 1 7 → T 1 , 1 ⌦ Y 7 → T 2 gives ∆P (Y ) = P (T 1 T 2 ), ✏P (Y ) = P (1), SP (Y ) = P (Y -1 ). Since G m (K) = Hom ( [Y, Y -1 ], K), we have [G m ] = [Y, Y -1
] and again [G m ] is a bicommutative Hopf algebra of finite type.

Combining these two examples, one gets a whole family of Hopf algebras. Indeed let d 0 ≥ 0 and d 1 ≥ 0 be two integers with

d = d 0 + d 1 > 0. The Hopf algebra [X] ⌦d 0 ⌦ [Y, Y -1 ] ⌦d 1 is isomorphic to H = [X 1 , . . . , X d 0 , Y 1 , Y -1 1 , . . . , Y d 1 , Y -1 d 1 ],
hence is isomorphic to [G] with

G = G d 0 a ⇥ G d 1 m .
According to [A] Chap. 4 (p. 163), the category of -linear algebraic groups is anti-equivalent to the category of commutative -Hopf algebras of finite type. Hence the category of commutative linear algebraic groups over is anti-equivalent to the category of bicommutative Hopf algebras of finite type over .

The commutative and connected linear algebraic groups over an algebraically closed fields are the groups

d 0 a ⇥ G d 1 m and the Hopf algebras [G d 0 a ⇥ G d 1 m
] are the bicommutative Hopf algebras of finite type over without zero divisors. In [G d 0 a ⇥ G d 1 m ] the -vector space of primitive elements has dimension d , while the rank of group-like elements is d 1 .

We exhausted the list of examples of bicommutative Hopf algebras without zero divisors and of finite type. However this is not the end of the story: let W be a -vector space of dimension `0. Then the symmetric algebra Sym(W ) on W has a natural structure of bicommutative Hopf algebra of finite type [H3]. If ∂ 1 , . . . , ∂ `0 is a basis of W over , then Sym (W )

is isomorphic to [∂ 1 , . . . , ∂ `0 ], hence to [G `0 a ].
If Γ is a free Z-module of finite type and rank `1, then the group algebra Γ is a bicommutative Hopf algebra of finite type isomorphic to [G `1 m ].

Therefore the category of bicommutative Hopf algebras without zero divisors and of finite type over is equivalent to the category of pairs (W, Γ) where W is a -vector space of finite dimension and Γ a free Zmodule of finite type. In H ' Sym(W ) ⌦ Γ, the space of primitive elements has dimension `0 = dim W , while the group-like elements have rank `1 = rankΓ.

We now take = Q. S. Fischler [F1] further introduces two more categories.

Let C 1 be the category whose objects are the triples (G, W, Γ) with

G = G d 0 a ⇥G d 1 m a commutative linear algebraic group over Q, W ⇢ T e (G)
a subspace which is rational over Q and Γ ⇢ G(Q) a torsion free finitely generated subgroup; moreover G is minimal for these properties: no algebraic subgroup

G ⇤ other than G itself satisfies W ⇢ T e (G ⇤ ) and Γ ⇢ G ⇤ (Q).
We denote by `0 the dimension of W and by `1 the rank of Γ.

The morphisms f :

(G 1 , W 1 , Γ 1 ) → (G 2 , W 2 , Γ 2 ) are given by a mor- phism f : 1 → G 2 of algebraic groups such that f (Γ 1 ) ⇢ Γ 2 such that the linear tangent map to f df : T e (G 1 ) -→ T e (G 2 ) satisfies df (W 1 ) ⇢ W 2 .
The definition of the category C 2 requires the following additional data. Let H be an bicommutative Hopf algebra of finite type over Q and without zero divisors. Denote by d 0 the dimension of the Q-vector space spanned by the primitive elements and by d 1 the rank of the group-like elements. Let H 0 be also a Hopf algebra, which is again bicommutative, without zero divisors and of finite type over Q. The dimension of the Q-vector space spanned by the primitive elements in H 0 is denoted by `0 while `1 is the rank of the group-like elements in H 0 . Let h•i : H ⇥ H 0 -→ Q be a bilinear map such that hx, yy 0 i = h∆x, y ⌦ y 0 i and hxx 0 , yi = hx ⌦ x 0 , ∆yi.

(2.1)

We used the notation h↵ ⌦ β, γ ⌦ δi = h↵, γihβ, δi.

The objects of the category C 2 are the triples (H, H 0 , h•i) given by two bicommutative Hopf algebras, without zero divisors and of finite type over Q, and a bilinear product satisfying (2.1). The morphisms are the pairs (f, g)

: (H 1 , H 0 1 , h•i 1 ) → (H 2 , H 0 2 , h•i 2 ) where f : H 1 → H 2 and g : H 0 2 → H 0 1 are Hopf algebra morphisms such that 1 , g(x 0 2 )i 1 = hf (x 1 ), x 0 2 i 2 .
One composes two morphisms (f 1 , g 1 ) :

(H 1 , H 0 1 , h•i 1 ) → (H 2 , H 0 2 , h•i 2 ) and (f 2 , g 2 ) : (H 2 , H 0 2 , h•i 2 ) → (H 3 , H 0 3 , h•i 3 ) as (f 2 • f 1 , g 1 • g 2 ) : (H 1 , H 0 1 , h•i 1 ) → (H 3 , H 0 3 , h•i 3 ).
Stphane Fischler [F1] proves:

Theorem 2.2. (S. Fischler). Both categories C 1 and C 2 are equivalent. This equivalence preserves the parameters d 0 , d 1 , `0, `1.

The category C 2 has a natural contravariant involution which consists in permuting H and H 0 . The corresponding involution in the category C 1 is the Fourier-Borel duality (1.9) we discussed above, which exchanges (d 0 , d 1 ) and (`0, `1) in Theorem 1.7.

The main goal in [F1] is to establish new interpolation lemmas. Theorem 2.2 enables Fischler to obtain them by duality, starting from known zero estimates.

Roughly speaking, a zero estimate (see for instance [W] § 2.1) is a lower bound for the degree of a polynomial vanishing at a given finite set of points (multiplicities may be considered). An interpolation lemma provides a lower bound for an integer D with the following property: given a finite set of points (maybe with multiplicities), there is a polynomial of degree at most D given values at these points. In terms of matrices, the zero estimates states that a matrix, whose entries are the values of monomials at the given points, has maximal rank, if only there are enough monomials (hence the matrix is sufficiently rectangular), while the interpolation lemma states that such a matrix has maximal rank once there are enough points (again this means that the matrix is sufficiently rectangular, but in the other direction).

This method using a duality to deduce interpolation lemmas from zero estimates works only for linear commutative algebraic groups. Zero estimates are known more generally for commutative algebraic groups (hence for abelian and semi-abelian varieties), but duality does not extend to the non-linear case. Fischler [F2] uses other arguments to obtain interpolation lemmas for non-linear commutative algebraic groups.

Hopf algebras and multiple zeta values

Let S denote the set of sequences s = (s 1 , . . . , s k ) ∈ N k with k ≥ 0,

s 1 ≥ 2, s i ≥ 1 (2  i  k).
The weight |s| of s is

s 1 + • • • + s k , while k is the depth of s. For s ∈ S set ⇣(s) = X n 1 >•••>n k ≥1 n -s 1 1 • • • n -s k k .
When s is the empty sequence (of weight and depth 0), we require ⇣(s) = 1.

Goncharov's Conjecture

Denote by Z the Q-vector subspace of C spanned by the numbers

(2i⇡) -|s| ⇣(s) (s ∈ S).
As is well known (and as we shall see), for s and s 0 in S, the product ⇣(s)⇣(s 0 ) is in two ways a linear combination with positive coefficients of numbers ⇣(s 00 ). Hence Z is a Q-sub-algebra of C with a double filtration by weight and depth.

For a graded Lie algebra C • denote by UC • its universal envelopping algebra and by

UC ∨ • = M n≥0 (UC) ∨ n
its graded dual, which is a commutative Hopf algebra.

Conjecture 3.1. (Goncharov [G]). There exists a graded Lie algebra C • and an isomorphism Z ' ∨

• of bifiltered algebras, by the weight on the left and by the depth on the right.

Hopf algebras also occur in this theory in a non-conjectural way. They are used to describe the above mentioned quadratic relations expressing the product of two multiple zeta values as a linear combination of zeta values.

The concatenation Hopf algebra

Let X = {x 0 , x 1 } be an alphabet with two letters. The free monoid (of words) on X is

X ⇤ = {x ✏ 1 • • • x ✏ k ; ✏ i ∈ {0, 1}, (1  i  k), k ≥ 0}
whose product is concatenation, and its unity is the empty word e.

Let H denote the free algebra QhXi on X. An element P ∈ H is written

P = X w∈X ⇤ hP |wiw with coefficients hP |wi ∈ Q.
The concatenation Hopf algebra is (H, •, e, ∆, ✏, S) where the coproduct is ∆P = P (x 0 ⌦ 1 + 1 ⌦ x 0 , x 1 ⌦ 1 + 1 ⌦ x 1 ), co-unit ✏(P ) = hP | ei and the antipode

S(x 1 • • • x n ) = (-1) n x n • • • x 1
for n ≥ 1 and x 1 , . . . , x n in X.

It is a cocommutative, not commutative Hopf algebra.

3.3

The shuffle Hopf algebra

The shuffle product x : H ⇥ H → H is defined inductively by the conditions uxe = exu = u and xuxyv = x(uxyv) + y(xuxv) for x and y in X, u and v in X ⇤ . It endows H with a structure of commutative algebra H x .

According to [Re] Theorem 3.1, for P ∈ H,

∆P = X u,v∈X ⇤ (P |uxv)u ⌦ v. (3.2)
The shuffle Hopf algebra is the commutative (not cocommutative) Hopf algebra (H, x, e, Φ, ✏, S), with Φ :

H → H ⌦ H defined by hΦ(w) | u ⌦ vi = huv | wi. Hence Φ(w) = X u,v∈X ⇤ uv=w u ⌦ v.
From (3.2) it follows that the shuffle Hopf algebra is the graded dual of the concatenation Hopf algebra (see [Re] Chap. 1).

We need to consider subalgebras of H.

For s ≥ 1 define y s = x s-1 0 x 1 . The subalgebra H 1 of H spanned by {y 1 , y 2 , . . .} is free, and so is the subalgebra H 0 of H 1 spanned by {y 2 , y 3 , . . .}. Also H 1 is the Q-vector space Qe + Hx 1 spanned by {e} ∪ X ⇤ x 1 , while H 0 is the Q-vector space Qe + x 0 Hx 1 spanned by {e} ∪ x 0 X ⇤ x 1 .

The shuffle x makes H 0 and H 1 subalgebras of H x :

H 0 x ⇢ H 1 x ⇢ H x . Define a mapping ⇣ : x 0 X ⇤ x 1 → C as follows. Each element w in x 0 X ⇤ x 1 can be written in a unique way y s 1 • • • y s k with s = (s 1 , . . . , s k ) ∈ S.
The number of letters x 1 in w is the depth k of, while the total number of letters of w is the weight s

1 + • • • + s k of s. Define ⇣(y s 1 • • • y s k ) = ⇣(s 1 , . . . , s k ).
By Q-linearity one extends ⇣ to a map from H 0 to C with ⇣(e) = 1. Using the representation of ⇣ as Chen iterated integrals, namely (

[K], § XIX.11), for w ∈ x 0 X ⇤ x 1 , ⇣(x ✏ 1 • • • x ✏p) = Z 1>t 1 >•••>tp>0 ω ✏ 1 (t 1 ) • • • ω ✏p (t p ) with ✏ i ∈ {0, 1} (1  i  p), ✏ 0 = 0, ✏ p = 1, ω 0 (t) = dt t and ω 1 (t) = dt 1 -t ,
one checks that ⇣ is a commutative algebra morphism of H 0 x into C. The structure of the commutative algebra H x is given by Radford Theorem [Re] Chap. 6. Consider the lexicographic order on X ⇤ with x 0 < x 1 . A Lyndon word is a word w ∈ X ⇤ such that, for each decomposition w = uv with u 6 = e and v 6 = e, the inequality w < v holds. Examples of Lyndon words are x 0 , x 1 , x 0 x k 1 (k ≥ 0), x 0x 1 (`≥ 0), x 2 0 x 2 1 . Denote by L the set of Lyndon words. Then the three shuffle algebras are (commutative) polynomial algebras

H x = K[L] x , H 1 x = K ⇥ L \ {x 0 } ⇤ x and H 0 x = K ⇥ L \ {x 0 , x 1 } ⇤ x . Therefore H x = H 1 x [x 0 ] x = H 0 x [x 0 , x 1 ] x and H 1 x = H 0 x [x 1 ] x .
(3.3)

Harmonic algebra

is another product being shuffle-like law on H, called harmonic product by M. Hoffman ([H1], [H2]) and stuffle by other authors [BBBL], denoted with a star, which also gives rise to subalgebras

H 0 ? ⇢ H 1 ? ⇢ H ? .
It is defined as follows. First on X ⇤ , the map ? : X ⇤ ⇥ X ⇤ → H is defined by induction, starting with

x n 0 ? w = w ? x n 0 = wx n 0 for any w ∈ X ⇤ and any n ≥ 0 (for n = 0 it means e ? w = w ? e = w for all w ∈ X ⇤ ) and then y s u ? y t v = y s (u ? y t v) + y t (y s u ? v) + y s+t (u ? v)

for u and v in X ⇤ , s and t positive integers. The harmonic product is an efficient way of writing the quadratic relations among multiple zeta values arising from the expression of ⇣(s) as series: ⇣ is a commutative algebra morphism of 0 ? into C. Hoffman [H1] gives the structure of the quasi-harmonic algebra H ? as well as of its subalgebras H 1 ? and H 0 ? : they are again polynomial algebras on Lyndon words:

H ? = K[L] ? , H 0 ? = K ⇥ L \ {x 0 , x 1 } ⇤ ? and H 1 ? = K ⇥ L \ {x 0 , x 1 } ⇤ ? .
⇣(uxv) = ⇣(u) ⇣(v) and ⇣(u ? v) = ⇣(u) ⇣(v) for u and v in H 0

show that, for any u and v in H 0 , uxv -u ? v belong to the kernel of ⇣. The equations ⇣(uxv -u ? v) = 0 for u and v in H 0 (3.5) are called the standard linear relations among multiple zeta values.

Other elements belong to the kernel of Hoffman's relations (see for instance [Z]) are ⇣(x 1 xv -x 1 ? v) = 0 for v in H 0 .

(3.6)

Notice that x 1 xv -x 1 ? v ∈ H 0 for v ∈ H 0 . The simplest example

x 1 xy 2 -x 1 ? y 2 ∈ ker ⇣ yields the relation ⇣(2, 1) = ⇣(3) known by Euler. was conjectured in [MJOP] that the elements uxv -u ? v and x 1 xv -x 1 ? v, when u and v range over the set H 0 , span the Q vector space ker ⇣. This conjecture is not yet disproved, but there is a little doubt about it for the following reason. From (3.3) and (3.4) it follows that there are two uniquely determined algebra morphisms Ẑx : H 1

x -→ R[T ] and Ẑ? :

H 1 ? -→ R[T ]
which extend ⇣ and map x 1 to T . According to [C], the next result is due to Boutet de Monvel and Zagier (see also [I-K]).

Proposition 3.7. There is a R-linear % : R[T ] → R[T ] which makes the following diagram commutative:

H 1 Ẑ?
------→ R[T ] ? ? ? ? ? y %

H 1 Ẑx ------→ R[T ]
An explicit formula for % is given by means of the generating series Recall that H x = H 0 [x 0 , x 1 ] x . Denote by reg x the Q-linear map H → H 0 which maps w ∈ H to its constant term in its expansion as a

  to compare the right hand side of (3.8) with the formula giving the expansion of the logarithm of Euler Gamma function:Γ(1 + t) = expÄ the image of e tT ). In [I-K] Ihara and Kaneko propose a regularization of the divergent multiple zeta values as follows.

Hence

H ? = H 1 ? [x 0 ] ? = H 0 ? [x 0 , x 1 ] ? and H 1 ? = H 0 ? [x 1 ] ? .

(3.4)

The quasi-shuffle Hopf algebra is the commutative algebra H 1 ? with the coproduct ∆ defined by the conditions ∆(y i ) = y i ⌦ e + e ⌦ y i for i ≥ 1, the co-unit ✏(P ) = hP | ei and the antipode

This quasi-shuffle Hopf algebra is isomorphic to the Hopf algebra of noncommutative symmetric series, whose graded dual is the Hopf algebra of quasi-symmetric series (see [H2] and [H3]) .

Regularized double shuffle relations

As we have seen the map ⇣ is a commutative algebra morphism of H 0

x into C and also of H 0 ? into C. Hence the kernel of ⇣ in H 0 is an ideal for the two algebra structures x and ?. A fundamental question (cf. Goncharov's Conjecture 3.1) is to describe this kernel.

The relations polynomial in x 0 , x 1 in the shuffle algebra H 0 [x 0 , x 1 ] x . Then reg x is an algebra morphism H x → H 0 x . Clearly for w ∈ H 0 we have reg x (w) = w.

Theorem 3.9. (Ihara, Kaneko). Let w be any word in

Special cases are:

Moreover there is an explicit expression for w as a polynomial in x 0 and x 1 in the algebra 0 [x 0 , x 1 ] x :

The regularized double shuffle relations of Ihara and Kaneko in [I-K] produce a number of linear relations among multiple zeta values: Theorem 3.10. (Ihara, Kaneko). For w ∈ H 1 and w 0 ∈ H 0 , reg x (wxw 0 -w ? w 0 ) ∈ ker ⇣.

(3.11) Special cases of (3.11) -for which no regularization is required -are the standard relations (3.5) which correspond to w ∈ H 0 and Hoffman's relations (3.6) which correspond to w = x 1 .

An example of u and v in H 1 for which uxv -

3.6

The main diophantine Conjecture

The main diophantine Conjecture arose after the works of several mathematicians, including D. Zagier, A.B. Goncharov, M. Kontsevich, M. Hoffman, M. Petitot and Hoang Ngoc Minh, K. Ihara and M. Kaneko, J. calle, P. Cartier (see [C]).

Conjecture 3.12. The kernel of ⇣ is spanned by the elements reg x (wxw 0 -w ? w 0 ) where w ranges over H 1 and w 0 over H 0 . Conjecture 3.12 means that the ideal of algebraic relations among multiple zeta values is generated by the double shuffle relations of Ihara and Kaneko in Theorem 3.10.

More precisely, we introduce independent variables Z u , where u ranges over the set

where Z e = 1. In particular, for u 1 and u

Denote by R the ring of polynomials with coefficients in Q in the variables Z u , where u ranges over the set of words in x 0 X ⇤ x 1 which start with x 0 and end with x 1 . Further, denote by I the ideal of R consisting of all polynomials which vanish under the specialization map R → R which is the Q-algebra morphism defined by

The Q-sub-algebra in C of multiple zeta values (up to the normalization with powers of 2⇡i, this is the algebra Z of Goncharov's Conjecture 3.1) is isomorphic to the quotient R/I.

Let be the ideal of R generated by the polynomials Z u Z v -Z uxv and Z r with r = reg x (wxw 0 -w ? w 0 ), where u, v, w 0 range over H 0 and w over H 1 . Theorem 3.10 can be written ⇢ I and Conjecture 3.12 means = I.

The ideal of R associated to the above mentioned conjecture of [MJOP] (see § 3.5) is the ideal, contained in , generated by the polynomials

where u and v range over the set of elements in x 0 X ⇤ x 1 .

The structure of the quotient of R/ is being studied by Jean calle [E].


Keywords: Systèmes repartis, gestion des données, système à large échelle, surveillance, visualisation

Un service de surveillance pour BlobSeer

Résumé : Ce rapport présente un service de surveillance spécialement conçu pour BlobSeer, une plate-forme de gestion de données réparties à large échelle. Nous décrivons comment cette application distribuée à large échelle est suivie et visualisée en utilisant MonALISA, ainsi que certains détails techniques. Un tutoriel pour les développeurs est joint en annexe.

BlobSeer [START_REF] Nicolae | Distributed management of massive data. an efficient fine grain data access scheme[END_REF], a large-scale data-sharing system, aims to efficiently manage the storage of large and unstructured binary data blocks which are referred to as BLOBs. This report introduces the basic concept of monitoring BlobSeer and how it could be done with the help of the MonALISA [START_REF]The MonALISA Website[END_REF] monitoring framework.

1 Background Knowledge

BlobSeer

BlobSeer is a data-sharing system that manages the storage of large and unstructured data blocks called binary large objects, referred to as BLOBs further in this report. The blobs are striped into small chunks that have the same size, called pages.

BlobSeer addresses the problem of efficiently storing massive blobs in largescale distributed environments. It provides an efficient fine-grained access to the pages belonging to each blob, as well as the possibility to modify them, in a distributed, multi-user environment.

One of the entities involved in the architecture of BlobSeer is the client, which initiates all blob operations: CREATE, READ, WRITE and APPEND. There can be many concurrent clients accessing the same blob or different blobs in the same time. The support for concurrent operations is enhanced by storing the pages belonging to the same blob on multiple storage providers.

The system has to keep track of the pages distribution across providers. Therefore, it associates some metadata to each blob. For each blob, the metadata is organized as a distributed segment tree [START_REF] Zheng | Distributed segment tree: Support of range query and cover query over DHT[END_REF], where each node corresponds to a version and to a page range within that version. Each leaf covers just one page, recording the information about the data provider where the page is physically stored. The metadata trees are stored on the metadata providers, which are processes organized as a Distributed Hash Table .  BlobSeer provides versioning support, so as to prevent pages from being overwritten and to be able to handle highly concurrent WRITE and APPEND operations. For each of them, only a patch composed of the range of written pages is added to the system, and a new metadata tree is created. The new metadata tree corresponds to a new version and points to the newly added pages and to the pages from the previous versions that were not overlapped by the added page range.

The system comprises two more entities: the version manager that deals with the serialization of the concurrent WRITE/APPEND requests and with the assignment of version numbers for each new WRITE/APPEND operation; the provider manager, the one that keeps track of all the storage providers in the system.

As far as this paper is concerned, an APPEND operation is only a special case of WRITE. Therefore, we disregard this aspect in the rest of the paper. Everything stated about WRITEs is also true for APPENDs, unless explicitly specified.

A typical setting of the BlobSeer system involves the deployment of a few hundreds of provider nodes, each of them storing data in the order of GB, and even tens of GB in the case of the use of the disk storage for each node. This implies that sizes within the order of TB can be easily reached for the blobs INRIA stored in the system. Furthermore, the typical size for a page within a blob can be smaller that 1 MB, whence the need to deal with hundreds of thousands of pages belonging to just one blob.

MonALISA

BlobSeer is a storage system that deals with massive data, which are striped into a huge number of pages scattered across numerous storage providers. A monitoring tool tuned for presenting the state of a system like BlobSeer has to cope with two major challenges. On one side, it has to accommodate the immense number of pages that the system comprises once it stores several blobs. On the other side, the monitoring system has to be able to deal with a huge amount of monitoring information generated when an application accesses the nodes that make up the storage service. It is the case when multiple clients simultaneously access various parts of the stored blobs, as they generate a piece of monitoring information for each page accessed on each provider. MonALISA is suitable for this task, as it is a system designed to run in grid environments and it proved to be a scalable and reliable system.

The MonALISA (Monitoring Agents in a Large Integrated Services Architecture) [START_REF]The MonALISA Website[END_REF] system is a JINI-based [START_REF] Waldo | The Jini architecture for network-centric computing[END_REF], scalable framework of distributed services, which provides the necessary tools for collecting and processing monitoring information.

Its architecture is based on four layers of services. It complies with the Grid Monitoring Architecture (GMA) [START_REF] Tierney | A grid monitoring architecture[END_REF] proposed by the Global Grid Forum (GGF) [START_REF]Global Grid Forum[END_REF], which includes three components: consumers, producers and a directory service.

The first layer corresponds to a network of Lookup Discovery Services that provide discovery and notification mechanisms for all the other services.

The second layer is composed of MonALISA services, the components that perform the data collection tasks. Each MonALISA service is part of a group and registers itself with a set of Lookup Services, together with several describing attributes.

The interaction between clients and services is made available through transparent Proxy services, which represent the third layer in the MonALISA architecture. Every MonALISA service discovers the Proxy Services by using the discovery mechanism implemented into the Lookup Services layer, and permanently keeps a TCP connection with each of them.

The top-level layer is represented by the MonALISA clients, which offer an intuitive graphical interface of the states of the monitored systems. It allows users to subscribe to and to visualize global parameters gathered from multiple MonALISA services. It also provides detailed tracking of parameters for any individual MonALISA service or component in the entire system. The Mon-ALISA repository is a "pseudo-client" for the MonALISA services, developed as a Web server. It is able to store the monitoring data and to present historical and real-time values, statistics and graphical charts for a specific group of MonALISA services. Each type of MonALISA client has to connect to the layer of Lookup services in order to request access to data gathered by one or more specified groups of MonALISA services. It is then transparently connected to the nearest and less loaded proxy service, which will forward the data that the client has subscribed to, from all the MonALISA services.

RT n°0368

#include "ApMon.h" ApMon * apm = new ApMon(ConfigFile);

... apm -> sendParameter("MyParameterGroup", "NodeName", "MyParameter", XDR_REAL64, (char * )&value); ... The MonALISA system is a well-suited choice for monitoring a distributed storage system, thanks to several features that it provides. First of all, it can monitor both a set of predefined parameters and various user-defined parameters. This is due to an application instrumentation library, called ApMon, that enables any application to send monitoring information to one or more MonALISA services.

BlobSeer is instrumented using the ApMon library, requiring each provider to report to the monitoring system each time a page is written or read, by sending a parameter and its value to a predefined MonALISA service, as described in Figure 1.

The version manager is monitored in the same way. An ApMon-based daemon parses its log file each time it is updated, in order to report the written page ranges and their associated versions. The state of the physical resources on each node is monitored through an ApMon thread that periodically sends the data to the monitoring service.

The MonALISA system also enables the user to create new data from the collected values, through the use of filters. In this way, new or aggregated values can be dynamically created within independent threads while the MonALISA service or repository receives the monitoring information. Since by default the repository can accommodate only time series in its database, all the specific data monitored from BlobSeer go through a filter that stores them into the corresponding database tables.

Another element is essential for defining a visualization tool tuned for a particular storage system. It is the possibility of having customized graphical charts, appropriate to the collected parameters. The MonALISA repository supports the integration of external graphical libraries, thus opening the way to the generation of any type of chart for any type of user-defined parameters.

2 General structure of monitoring

How the system works

As we talked in the background section, a BlobSeer system can have hundreds or thousands of data providers and lots of clients accessing the data concurrently. Monitoring the behavior of the system, as well as monitoing the clients, is a challenging task. In our approach, the version manager and each provider send monitoring data to the MonALISA system. This is done by additional monitoring infrastructures, for instance, listeners. The monitoring data is col-INRIA lected by the MonALISA services and then forwarded to a Repository. The repository uses a database to store those received data. To process the users' requests, the repository queries the database, and then it creates and displays the visualization charts.

Types of monitoring information

The current monitoring service involves two types of monitoring information, which are the general information and BlobSeer specific information respectively.

General parameters.

The general type are such parameters that can be handled by the MonALISA repository module through meaningful monitoring predicates. Such parameters may include CPU usage, network traffic, disk access, memory usage etc. The repository subscribes to the specified parameters and stores them into its internal database. With these stored data, charts can be created simply by providing the repository with a dedicated property configuration file. This type of parameters described above are basically time series parameters, which means the x-axis of the charts typically contains time stamps. Sometimes, it is not sufficient. Therefore, we need application specific parameters.

Specific parameters.

The second type, namely the BlobSeer specific parameters, are such parameters that are dedicated to BlobSeer and can not be collected or shown simply by predicates and property configuration files. We thus need to create customized parameters which are subscribed to by the repository. When the monitoring data arrive at the repository, they first go through some filters that enable the creation of new aggregated parameters and the execution of some specific actions depending on the types and values of the incoming parameters. We may also discard parameters which we are not interested in.

Approaches of monitoring

Untill now, we have tackled the problem of collecting monitoring data from MonALISA and storing them in the database. Now we discuss the problem of collecting monitoring data from the application, in this case, BlobSeer. This task can be done my means of two different approaches. The first one is adding listeners to the entities that we want to monitor. The listener sends the monitoring data to the MonALISA service using ApMon. The second approach is to parse the log of the interested entity so as to obtain the monitoring information. In our case, we employed both approaches.

After the monitoring data are collected, we are ready to visualize them. As we mentioned above, the general information can be directly visualized by providing the repository with certain configuration files which are usually located at REPOSITORY_HOME/tomcat/webapps/ROOT/WEB-INF/conf. These configuration files can be used to create several different types of charts such as realtime charts, history charts, statistics charts and spider charts etc. Using these time-series based charts, we could display most of the general monitoring data.

RT n°0368

For the second type of monitoring information, it's not so straightforward since the parameters are customized and we need to create the visualization charts from scratch. In our case, we are using the open-source chart library which is called JFreeChart to create the visualization figures.

Collecting the monitoring data

The regular monitoring data could be directly collected by the MonALISA service and repository's subscription function. Therefore here we only talk about how to collect application customized parameters.

Customized parameters

In the case of BlobSeer, the most important metrics are such parameters as how the blobs are accessed(read/write) and how the blob pages are distributed among the data providers etc. In the current implementation, we analyze two major aspects of BlobSeer parameters: blob I/O traffic and blob storage. The blob I/O traffic metric deals with how the blobs are read and written and we record every read and write operation in the database from two levels of perspectives: client level and provider level (see below). The blob storage metric deals with how each blob page is stored and it records some other basic metadata of each blob and blob page.

Due to the distributed working mechanism of BlobSeer, the read and write operations from the clients are split and distributed among data providers to achieve load balancing. Therefore, we record two levels of I/O traffic, namely client I/O and provider I/O. These two levels of I/O traffic are collected in different manners.

Provider level I/O

The data are collected by adding a listener to each provider and sending the monitored data using ApMon whenever a read/write operation occurs. The data sent to MonALISA service may include some basic metadata of that read/write operation, such as blob_id, page_size, time stamp etc.

Client level I/O.

For the I/O at the client level, we currently only deal with the WRITE operation. We use a parser to analyze the log file of the version manager and detect the client write operation by a special flag surrounded by the a pair of brackets. This process is done by the log parsing daemon which is running as long as BlobSeer is running. Whenever this daemon detects that a new blob or version is created, it inserts this new blob or version into the database. On the other hand, a more general way to handle it is to send it to the MonALISA service using ApMon and then insert into the database through the repository filters instead of communicating with the database directly. Actually this is what we have implemented. The MonALISA repository is mainly composed of an embedded database server and a web server. It also contains some other components like shell scripts and configuration files. But in this report, we focus on the former two major components.

Database

The database component is responsible for storing the data collected. When the repository subscribes to a specific predicate, it stores the corresponding data. This predicate could be a system-defined one or a customized one.

Web Server

The web server, which is based on a Tomcat web server, is responsible for dynamically creating and displaying the visualization charts. It uses JavaBeans to process the user requests and outputs them through a servlet. In next subsections we will discuss each of these components.

Configuring the database

The embedded PostgreSQL database is responsible for storing the collected monitoring data. Nevertheless, we could also use a standalone database instead of the embedded one for the purposes of having a newer version or a better performance. First we discuss the embedded PostgreSQL and later we will introduce how to connect to an external database.

The embedded database is located at REPOSITORY_HOME/pgsql_store. To start it, first go to the bin directory and type: pg_ctl start -D ../data. The -D switch specifies where the data folder of the database is located. We may also use a different location for the storage, say on a more stable partition of the hard disk. Use the command psql -d mon_data to get into console of the default database. Using the \dt command helps to show all the tables available in the current database. The monitor_ids table contains all of the regular monitoring parameter for every monitored node. Typically, every monitoring metric will have 2 or more resolutions which are indicated by the name of the table, such as 1y_1min and 1y_1sec (one value per min/sec during one year).

Common database commands

When using PostgreSQL, we sometimes need to list all of the available resources in the database. These command are usually called listing commands. They help you answer questions like "What tables are in this PostgreSQL database" or "What databases do I have within PostgeSQL". For example, \dt will show you all the tables. Other popular queries are related to permissions, indexes, views, and sequences. Following is a checklist.

RT n°0368

Jing Cai

Listing

Configuring the repository

To enable the repository to subscribe to the desired monitoring data, we need to specify proper monitoring predicates in the Store Configuration Section of App.properties file. The repository is ready to receive monitoring data after it is started (or restarted) by launching the start.sh script in its home folder.

Creating visualization charts

In our implementation, we use JFreeChart to visualize the data we collected. JFreeChart is a free but powerful chart library for producing professional quality charts for developers and it's extremely easy to get started with it. To use latest version of JFreeChart, first download the library from the official website and unpack it. To make the repository recognize the JFreeChart library, put the jf reechart.jar and jcommon.jar into REPOSITORY_HOME/tomcat/common/lib INRIA Bear in mind that not always the latest version of JFreeChart is the best. Since the some of the regular charts coming with the repository internally use an older version of JFreeChart and since JFreeChart is itself under development, its API might change. If you come across an exception thrown out by the repository's regular charts, this might be the reason. Now we are ready to create a new chart.

In the current implementation, a new chart typically consists of three components:

• the JavaBean which creates the data set and the chart,

• the servlet which encodes the chart and outputs it and • the JSP web page which finally displays the chart.

Chart Beans

Usually, each chart will have a JavaBean which actually creates it. The Bean typically consists of two major methods. One method which has a name like getDataset() does the job of communicating with the database and creating an appropriate data set for the chart. The other method which has a name like getChartViewer() creates a chart using the data set returned by the above method. In this method, different types of charts could be created and the characteristics of the chart, like color, axis, background image etc, could be tuned. The chart is then put in the http session after creation thus making it available for the image reading action. Finally, the getChartViewer method constructs an URL string and passes the image displaying job to the servlet. This URL string is what we will use in the JSP webpage for the <img> tag to display that chart. If the JavaBean needs any input parameters such as the blob_id, we need to write extra set() and get() functions. All of the chart beans are located in REPOSITORY_HOME/tomcat/common/lib/BSMonitoring.jar. Note that, we need to restart the repository to make the Java Beans/servlets take effect if they are modified or recompiled.

Chart Servlet

The chart servlet, which is called ChartViewer, simply gets the BufferedImage object from the current session, encodes it using PNG format and finally outputs it. It is a component that we may not need to touch unless we need to add more advanced features.

JSP webpage

The JSP webpage handles the user interface of the monitoring service and what we usually need to do is to write a JSP file which calls the chart bean.The chart bean will return an URL string which points to the servlet. This URL string will serve as the src attribute of the <img> tag and finally the servlet displays the desired chart. Note that, when creating the web page, we need to incorporate it in the master page which is used by the repository in order to achieve an uniformed user interface.

For more details, please reference the tutorial in the last section.

Conclusion and further work

In this report, we introduce the monitoring service which is specially designed for the BlobSeer, a large-scale distributed data management platform. We describe how the the large scale distributed application is monitored and visualized using MonALISA and also reveal some technical details. Further, we will take into account the security problem of the BlobSeer system. We are currently dealing with the user accounting aspect. The user accounting is a recently introduced module and is still under development. In this module, we aim to collect informations about the users who do "malicious" operations to BlobSeer system. The malicious operations can be: writing data to the providers without publishing, publishing without writing any data to providers or publishing a part of the written data.

A tutorial on BlobSeer monitoring

This section we give a step-by-step tutorial for the newbies who would like to do more developments based on the current implementation of the BlobSeer monitoring service. In this tutorial, we will discuss the major components of the monitoring service, how to use them, and furthermore, how to create new modules using the existing infrastructure.

Before starting, please note that, from now on, we are always assuming that the current directory is REPOSITORY_HOME. Also, more detailed information can be found in the README file of each component.

Installation and Basic Configurations

MonALISA Service Installation

MonALISA is a large scale distributed monitoring service which has been used in many real monitoring applications. To start, simply install it in your preferred location, say $HOME/monalisa. The installation script install.sh will prompt you INRIA for some information. One thing you need to notice is that you should enable the ApMon support since we will use ApMon to send the monitoring information later in this tutorial. You also have to give the correct path of java home on your system.

To start/stop/restart it, just go to the Service/CMD, and type ML_SER start/ stop/restart to proceed. In case you have given an incorrect java home path or want to change the name of your MonALISA service, simply modify the ml_env file in this directory.

Repository Installation

The second step is to set up the Repository. Similarly, install the repository using the install.sh shell first. Next, you may need to take some care of configuration files located in the /conf directory. Make sure you put the right value in the env.JAVA_HOME file which contains the path to the JAVA_HOME on your system. If you want to move the repository, modify the env.REPOSITORY_DIR accordingly. To start or stop the repository, simply launch the start.sh and stop.sh scripts. But if you want to start or shutdown the embedded database separately, use the start_pgsql.sh and stop_pgsql.sh.

Using a standalone database

We can use an external database in order to have a newer version or a better performance. The supported databases are PostgreSQL and mySQL. Here we will take PostgreSQL for an example. Here we specify the server name or IP address, database name, database port and database user account information. You have to use this account to manipulate your database(for commands such as create, select, insert or update table ).

Configuring the repository.

Installing PostgreSQL. Go the the PostgreSQL website and download a latest release then unpack it. Here are the general steps for the installation. First we configure and install it.

./configure gmake su gmake install

RT n°0368

Then, we need to add a database user called postgres.

adduser postgres

We specify where the data is stored. Allowing External Connections. The installation is not the last step. To allow our program (such as the daemon and filter) which is running outside the local machine to connect to the PostgreSQL database, we need to specify who and how the database is accessed. We shall add client authentication lines in the PostgreSQL_HOME/data/pg _hba.conf file like this:

local DATABASE USER METHOD [OPTION] host DATABASE USER CIDR-ADDRESS METHOD [OPTION] hostssl DATABASE USER CIDR-ADDRESS METHOD [OPTION] hostnossl DATABASE USER CIDR-ADDRESS METHOD [OPTION]

If we are merely doing an experiment, we can trust all connecting clients. host all all 0.0.0.0/0 trust Also, we need to add authenticated listening addresses in the PostgreSQL _HOME-/data/posgresql.conf in order to allow outside connections. To allow all connections, we simply give:

listen_addresses = ' * '
Finally, the listening port of the database can also be modified here.

Monitoring BlobSeer

The original version of BlobSeer in SVN does not support monitoring in general. To make the BlobSeer monitor-able, we need to make additional efforts. We used different approaches for monitoring different entities of BlobSeer.

INRIA

Monitoring the Data Provider

The data provider's read/write operations are monitored by a listener. A monitoring listener is now already implemented and we may not need to change the class interface since we could extend it simply by adding more data-sending methods as follows:

void monitoring_listener::send_monitored_data (const monitored_params_t &params)

These monitoring data are sent to the MonALISA service and thus can be subscribed to by any repository which is interested in them.

To enable this monitoring capability of the data provider, we can simply check out the /blobseer/contrib/monitoringBlobSeer/BSmonsupport folder, install the monitoring provider and use it instead of the original BlobSeer provider.

Monitoring the Version Manager

To monitor the version manager, we developed a parser to analyze the log file of version manager. The version manager will write a line in the log vmanager.stdout (which is usually located in /tmp/BlobSeer/vmanager/), such as: The tags {CAV} and {CCB} mean "Client Allocate Version" and "Client Create Blob" respectively. These are the messages generated when the client writes on the blob or the client creates a new blob. The parser will read this log file from the beginning to the end and take corresponding actions whenever it finds one of these two tags. If no new data is available in the current log file, it sleeps for an interval whose value can be set in the script.

To use this parser daemon, we simple launch the program and provide it with the path to the log file as well as the configuration file.

Displaying Customized Monitoring Information

Configuring the subscription

Displaying the user-customized monitoring information is not so straightforward as above. As we mentioned before, the informations related to the data providers' I/O operations are collected via a listener and sent to the MonAL-ISA service. On the other hand, the monitoring information from the version manager is sent by the log parser to the MonALISA service.

First we need to subscribe to our own customized parameters and configure the predicates as in the previous section. Next, since the monitoring data will go through the filters before being inserted into the database, we can develop multiple filters for different monitoring information or we can just extend one filter to take different actions depending on the received parameter.

String sParam = r.param_name[0]; if (sParam.equals("my_new_parameter_name")) {....do sth here and then insert into DB....} else if (sParam.equals("my_old_parameter_name")) {....do sth here and then insert into DB....}

Creating a JavaBean

Now we can do some tests to write some data into the database. If everything goes fine, we have already got the data collected in the database. Now, let's create a demo JavaBean to generate a chart. Let's call this chart bean myBar-Chart. Now we create a project in the Eclipse or Netbeans, or whichever IDE you like. We need to add the JFreeChart library into the Build Path of the IDE. The following jars need to be added.

INRIA

In the myBarChart class, we first create a method called getDataset(). The return value of this method depends on what type of chart we want to create. Here we defined some series and categories and then create a CategoryDataset. String series1 = "First"; String category5 = "Category 5"; Then we create data set and add data into it.

DefaultCategoryDataset dataset = new DefaultCategoryDataset(); dataset.addValue(1.0, series1, category5);

The second function is called getChartViewer(). In this function, we first create the chart and set some characteristics of the chart. Next, we save it as BufferedImage object in the http session. Finally, we get the server name, port and context path and then concatenate them with the servlet's path which is /servlet/chartViewer. This concatenated string is actually the URL which will call our servlet and display this generated chart. Note that, the JavaBeans should be put in the directory of

Figure 1 :

 1 Figure 1: Instrumenting a code with the ApMon Library.

INRIA 4

 4 Storing data into the repository 4.1 Repository structure

  First, we need to configure Database Configuration Section of the App.properties file in the ./JStoreClient/conf folder. For example: lia.Monitor.jdbcDriverString = org.postgresql.Driver lia.Monitor.ServerName = 131.254.11.254 lia.Monitor.DatabaseName = repository lia.Monitor.DatabasePort = 5432 lia.Monitor.UserName = jcai lia.Monitor.Pass = pg

  mkdir /usr/local/pgsql/data Finally, we initialize it. chown postgres /usr/local/pgsql/data su -postgres /usr/local/pgsql/bin/initdb -D /usr/local/pgsql/data To connect to this database, go to the bin directory and type: ./psql -d postgres To start or shutdown it, type:./pg_ctl start/stop -D /usr/local/pgsql/dataThe -D switch specifies the location of the data.

  Here we specify the server name or IP address, database name, database port and database user account information. You have to use this account to manipulate your database(execute commands such as create, select, insert or update table).

		Command Argument
	Tables	\dt	name
	Indices	\di	name
	Sequences	\ds	name
	Views	\dv	name
	Permissions	\dp or \z	name
	System Tables	\dS	name
	Large Objects	\dl	name
	Types	\dT	name
	Functions	\df	name
	Operations	\do	name
	Aggregates	\da	name
	Comments	\dd	name
	Databases	\l	name
	4.2.2 Using a standalone database
		254
	lia.Monitor.DatabaseName = repository
	lia.Monitor.DatabasePort = 5432
	lia.Monitor.UserName = jcai
	lia.Monitor.Pass = pg

To use a standalone database, the only thing we need to do is to modify the Database Configuration Section of the App.properties file which is usually located in REPOSITORY_HOME/JStoreClient/conf/. lia.Monitor.jdbcDriverString = org.postgresql.Driver lia.Monitor.ServerName = 131.254.11.

  It is currently located in the same JAR archive as the chart Java Beans: REPOSITORY_HOME/tomcat/common/lib/BSMonitoring.jar. Note that, if a new servlet is created, we need to activate it in

	<servlet-mapping>
	<servlet-name>ChartViewer</servlet-name>
	<url-pattern>/servlet/ChartViewer</url-pattern>
	</servlet-mapping>
	REPOSITORY_HOME/tomcat/webapps/ROOT/WEB-INF/web.xml
	like this:
	<servlet>
	<servlet-name>ChartViewer</servlet-name>
	<servlet-class>myServlet.ChartViewer</servlet-class>
	<load-on-startup>1</load-on-startup>
	</servlet>
	RT n°0368

  Now, we are going to use the repository to display some regular monitoring information. First we need to set proper predicates in the App.properties, which is located in the ./JStoreClient/conf folder. If we want to collect the users' cpu usage of paralapeche node which is monitored by the BlobMonitor MonALISA service, we can simply use a predicate like this: Now, we create a history chart showing the user cpu usage over time. We need first create an empty file with a name extension of .properties in the ./conf/web-site_config_files directory. Let's say, it's called usr_cpu.properties. Inside this file, we may put some specifications about how the chart is displayed. Some example configurations are available in that folder. To make this chart take effect, we need add a link in the menu bar on the left side of the repository's web interface. To do this, simply add a line in the menu.js file which located in ./tomcat/webapps/ROOT/js as follows:The first and second parameters may be different depending on how you organize the links. Now, open a browser and type in the url of the repository, click on the new link you have just created. If everything works out, you will see a new time-series chart about the user cpu usage.

	Jing Cai
	lia.Monitor.JiniClient.Store.predicates=BlobMonitor/Monalisa/
	paralapeche/-1/-1/cpu_usr.
	7.3.2 Creating a time-series chart using the properties
	export CLASSPATH = $PATH_TO_APMON/lib/apmon.jar:.
	java vmonitor/VManagerMonitor version_manager_log config/monitor.conf
	7.3 Displaying Regular Monitoring Information
	7.3.1 Configuring the predicates
	RT n°0368

d.add(10111,1011,'Node CPU Usage','display?page=usr_cpu');

  REPOSITORY_HOME/tomcat/webapps/ROOT/WEB-INF/classes/myBean or inside a JAR archive in REPOSITORY_HOME/tomcat/common/lib/ The servlet doesn't need to be modified. It is now located in REPOSITORY_HOME/tomcat/common/lib/BSMonitoring.jar Note that, if a new servlet is created, we need to activate it in REPOSITORY_HOME/tomcat/ webapps/ROOT/WEB-INF/web.xml

	like this:
	<servlet>
	<servlet-name>ChartViewer</servlet-name>
	<servlet-class>myServlet.ChartViewer</servlet-class>
	<load-on-startup>1</load-on-startup>
	RT n°0368

Jing Cai

</servlet> <servlet-mapping> <servlet-name>ChartViewer</servlet-name> <url-pattern>/servlet/ChartViewer</url-pattern> </servlet-mapping>

Creating the JSP webpage

Now, we create a corresponding web page to display this chart. All the webpages should be put in REPOSITORY_HOME/tomcat/webapps/ROOT/. In this JSP file we should first import the necessary libraries and then what we basically do is to call that JavaBean we have just created, like this:

String chartViewer=myBarChart.getChartViewer(request , response); sOut+="<img src="+chartViewer+" border=0 usemap='#imageMap'>";

Note that the java bean should be included it in the JSP page.

<jsp:useBean id="myBarChart" scope="session" class="myBean.myBarChart"/>

We also have to incorporate our web page into the master page. If everything works well, the new chart is completed and can be displayed.


Michel Waldschmidt

Professor T N Shorey

The Role of Complex Conjugation in Transcendental Number Theory

In his two well known 1968 papers "Contributions to the theory of transcendental numbers", K. Ramachandra proved several results showing that, in certain explicit sets {x1, . . . , xn} of complex numbers, one element at least is transcendental. In specific cases the number n of elements in the set was 2 and the two numbers x1, x2 were both real. He then noticed that the conclusion is equivalent to saying that the complex number x1 + ix2 is transcendental.

In his 2004 paper published in the Journal de Théorie des Nombres de Bordeaux, G. Diaz investigates how complex conjugation can be used for the transcendence study of the values of the exponential function. For instance, if log α1 and log α2 are two non-zero logarithms of algebraic numbers, one of them being either real of purely imaginary, and not the other, then the product (log α1)(log α2) is transcendental.

We will survey Diaz's results and produce further similar ones. * Many thanks to N. Saradha for the perfect organization of this Conference DION2005 and for taking care in such an efficient way of the publication of the proceedings.

Theorems of Hermite-Lindemann and Gel'fond-Schneider

Denote by Q the field of algebraic numbers and by L the Q-vector space of logarithms of algebraic numbers:

L = {λ ∈ C ; e λ ∈ Q × } = exp -1 (Q × ) = {log α ; α ∈ Q × }.
The Theorem of Hermite and Lindemann ( [START_REF] Waldschmidt | Diophantine approximation on linear algebraic groups[END_REF], Theorem 1.2) can be stated in several equivalent ways:

Theorem 1.1 (Hermite-Lindemann) (i) Let α be a non-zero algebraic number and let log α be any non-zero logarithm of α. Then log α is transcendental.

(ii) There is no non-zero logarithm of algebraic number:

L ∩ Q = {0}.
(iii) Let β be a non-zero algebraic number. Then e β is transcendental.

Another classical result is the Theorem of Gel'fond and Schneider ([13, Theorem 1.4]): Theorem 1.2 (Gel'fond-Schneider) Let λ and β be complex numbers. Assume λ = 0 and β ∈ Q. Then one at least of the three numbers β, e λ , e βλ is transcendental.

The proof of the Theorem 1.1 of Hermite and Lindemann rests on properties of the two functions z and e z : they are algebraically independent, they have finite order of growth, they satisfy differential equations with rational coefficients, and for any β ∈ C they take values in the field Q(β, e β ) for all values of z of the form sβ with s ∈ Z.

In a similar way, the proof by Gel'fond of Theorem 1.2 relies on the fact that, for β ∈ C \ Q, the two functions e z and e βz are algebraically independent, they have finite order of growth, they satisfy differential equations in the field Q(β), and for any λ ∈ C they take values in the field Q(e λ , e βλ ) for all values of z of the form sλ with s ∈ Z.

On the other hand, Schneider does not use differential equation: in his proof he uses the fact that for λ ∈ C × , the two functions z and e λz are algebraically independent (if λ = 0), they have finite order of growth and they take values in the field Q(β, e λ , e βλ ) for all values of z of the form

s 1 + s 2 β with (s 1 , s 2 ) ∈ Z × Z.
In connection with Theorem 1.1 of Hermite-Lindemann, G. Diaz [START_REF] Diaz | Utilisation de la conjugaison complexe dans l'étude de la transcendance de valeurs de la fonction exponentielle[END_REF] considered the following question:

• Which are the pairs (t, β), where t is a non-zero real number and β a non-zero algebraic number, such that e tβ is algebraic?

A first example is β ∈ R and t = (log 2)/β. A second example is β ∈ iR and t = iπ/β. In [START_REF] Diaz | Utilisation de la conjugaison complexe dans l'étude de la transcendance de valeurs de la fonction exponentielle[END_REF], G. Diaz shows that there are no further examples.

Theorem 1.3 (G. Diaz) Let β ∈ Q and t ∈ R × . Assume β ∈ R ∪ iR.
Then e tβ is transcendental. Equivalently: for λ ∈ L with λ ∈ R ∪ iR,

Rλ ∩ Q = {0}.
Proof Set α = e tβ . The complex conjugate α of α is e tβ = α β/β . Since β ∈ R ∪ iR, the algebraic number β/β is not real (its modulus is 1 and it is not ±1), hence it is not rational. Gel'fond-Schneider's Theorem 1.2 implies that α and α β/β cannot be both algebraic. Since α β/β = α, it follows that α and α are both transcendental.

Another consequence that G. Diaz deduces from Gel'fond-Schneider's Theorem 1.2 is the next one.

Corollary 1.4 (G. Diaz) There exists

β 0 ∈ R ∪ iR such that {β ∈ Q ; e β ∈ Q} = Qβ 0 .
Remark 1.5 Hermite-Lindemann's Theorem 1.1 tells us that in fact β 0 = 0.

Proof From Gel'fond-Schneider's Theorem 1.2 one deduces that the Qvector-space {β ∈ Q ; e β ∈ Q} has dimension ≤ 1 and is contained in R ∪ iR.

An interesting output of this argument is that it yields a proof of special cases of Hermite-Lindemann's Theorem 1.1 by means of Schneider's method, hence without involving derivatives. The details are provided in [START_REF] Diaz | Utilisation de la conjugaison complexe dans l'étude de la transcendance de valeurs de la fonction exponentielle[END_REF].

A challenge is to produce a transcendence proof using the function e z . The so-called elementary method in the last chapter of the book by Gelfond and Linnik [START_REF] Gel | Elementary methods in the analytic theory of numbers[END_REF] involves real analytic functions.

Another related challenge is to prove new results by means of the function e z 2 : so far it seems that all results which follow from Gel'ffond-Schneider's method involving this function can also be derived without using it.

The Six Exponentials Theorem and the Four Exponentials Conjecture

We first recall the statement of the six exponentials Theorem due to C.L. Siegel, S. Lang and K. Ramachandra ([13, Theorem 1.12]). is transcendental.

Example Take x 1 = 1, x 2 = π, y 1 = log 2, y 2 = π log 2, y 3 = π 2 log 2; the six exponentials are respectively 2, 2 π , 2 π 2 , 2 π , 2 π 2 , 2 π 3 ;
hence one at least of the three numbers

2 π , 2 π 2 , 2 π 3 is transcendental.
We refer to the papers [START_REF] Shorey | On a theorem of Ramachandra[END_REF] and [START_REF] Shorey | On the sum 3 k=1 2 π kα k , α k algebraic numbers[END_REF] by T.N. Shorey for effective versions of this result, namely lower bounds for

|2 π -α 1 | + |2 π 2 -α 2 | + |2 π 3 -α 3 |
for algebraic α 1 , α 2 , α 3 . The estimates depend on the heights and degrees of these algebraic numbers. Another subsequent effective version of Theorem 2.1 is given in [START_REF] Mignotte | Approximation simultanée de valeurs de la fonction exponentielle[END_REF] We now describe a trick which originates in the seminal paper [START_REF] Ramachandra | Contributions to the theory of transcendental numbers. I, II[END_REF] by K. Ramachandra. As often in such circumstances, the starting point is a pretty obvious remark.

Remark 2.5 Let x and y be two real numbers. The following properties are equivalent:

(i) one at least of the two numbers x, y is transcendental.

(ii) the complex number x + iy is transcendental.

We reproduce here Corollary p. 87 of [START_REF] Ramachandra | Contributions to the theory of transcendental numbers. I, II[END_REF]:

x = 1 240 + ∞ n=1 n 3 a n 1 -a n ∞ n=1 (1 -a n ) -8 , y = 6 (b 1/2 -b -1/2 ) 4 - 1 (b 1/2 -b -1/2 ) 2 - ∞ n=1 n 3 a n (b n + b -n ) 1 -a n ∞ n=1 (1-a n ) -8 ,
is transcendental. Therefore the complex number x + iy is transcendental.

This idea is developed by G. Diaz in ([1, Th. 1]), where he proves the following special case of the four exponentials Conjecture.

Corollary 2.7 (of the six exponentials Theorem) Let x 1 , x 2 be two elements in R ∪ iR which are Q-linearly independent. Let y 1 , y 2 be two complex numbers. Assume that the three numbers y 1 , y 2 , y 2 are Q-linearly independent. Then one at least of the four numbers e x1y1 , e x1y2 , e x2y1 , e x2y2 is transcendental.

Proof Set y 3 = y 2 . Then e xjy3 = e ±xjy2 for j = 1, 2 and Q is stable under complex conjugation. Hence one may use the six exponentials Theorem 2.1.

An alternate form (see [

The Strong Six Exponentials Theorem and the Strong Four Exponentials Conjecture

We consider now linear combinations of logarithms of algebraic numbers. Denote by L the Q-vector space spanned by 1 and L: hence L is the set of linear combinations with algebraic coefficients of logarithms of algebraic numbers:

L = {β 0 + β 1 λ 1 + • • • + β n λ n ; n ≥ 0, β i ∈ Q, λ i ∈ L}.
Here is the strong six exponentials Theorem of D. Roy [START_REF] Roy | Matrices whose coefficients are linear forms in logarithms[END_REF]; see also [START_REF] Waldschmidt | Diophantine approximation on linear algebraic groups[END_REF]Corollary 11.16]. Theorem 3.1 (Strong six exponentials Theorem) If x 1 , x 2 are Q-linearly independent complex numbers and y 1 , y 2 , y 3 are Q-linearly independent complex numbers, then one at least of the six numbers

x 1 y 1 , x 1 y 2 , x 1 y 3 , x 2 y 1 , x 2 y 2 , x 2 y 3 is not in L.
Also in [START_REF] Roy | Matrices whose coefficients are linear forms in logarithms[END_REF] Here is an alternate form of the strong four exponentials Conjecture 3.2.

Conjecture 3.4 (Strong four exponentials Conjecture again)

Let Λ 1 , Λ 2 , Λ 3 be non-zero elements in L. Assume the numbers Λ 2 /Λ 1 and Λ 3 /Λ 1 are both transcendental. Then the number

Λ 2 Λ 3 /Λ 1 is not in L.
For the equivalence between both versions of the strong four exponentials Conjecture, notice that the matrix

Λ 1 Λ 2 Λ 3 Λ 2 Λ 3 /Λ 1
has rank 1 and see remark 2.8.

Following [START_REF] Diaz | Utilisation de la conjugaison complexe dans l'étude de la transcendance de valeurs de la fonction exponentielle[END_REF][START_REF] Diaz | Produits et quotients de combinaisons linéaires de logarithmes de nombres algébriques: conjectures et résultats partiels[END_REF], we deduce a few consequences of the strong four exponentials Conjecture 3.2 and 3.4.

Product of Logarithms of Algebraic Numbers

No proof has been given so far that the number e π 2 is transcendental.

One can ask, more generally: for λ ∈ L \ {0}, is it true that λλ ∈ L? Again, more generally: for λ 1 and λ 2 in L\{0}, is it true that λ 1 λ 2 ∈ L? And finally, for λ 1 and λ 2 in L \ {0}, is it true that λ 1 λ 2 ∈ L?

Clearly a positive answer to the last question (hence to the previous ones) follows from the strong four exponentials Conjecture 3.2.

One of the few transcendence results concerning the product of logarithms of algebraic numbers reads as follows (it is a special case of Theorem 3 of [START_REF] Diaz | Utilisation de la conjugaison complexe dans l'étude de la transcendance de valeurs de la fonction exponentielle[END_REF] where λ is replaced by a transcendental element of L). Proof Apply Corollary 4.2 with Λ 1 = 1, Λ 2 = λ and Λ 3 = λ. The fact that Λ 3 /Λ 1 is transcendental follows from Hermite-Lindemann Theorem, while the fact that Λ 1 , Λ 2 , Λ 2 are linearly independent over Q is a consequence of Baker's Theorem (see for instance [START_REF] Waldschmidt | Diophantine approximation on linear algebraic groups[END_REF] This would provide a nice generalization to Theorem 1.1 of Hermite and Lindemann.

Further Results

From the strong six exponentials Theorem 3.1, one deduces the following result ([15] Theorem 1.2), which can be viewed as a first step towards the transcendence of non-zero numbers of the form Λ 1 Λ 2 -Λ 3 Λ 4 for Λ i ∈ L: Theorem 6.1 Let M be a 2 × 3 matrix with entries in L:

M = Λ 11 Λ 12 Λ 13 Λ 21 Λ 22 Λ 23 •
Assume that the five rows of the matrix

M I 3 =       Λ 11 Λ 12 Λ 13 Λ 21 Λ 22 Λ 23 1 0 0 0 1 0 0 0 1      
are linearly independent over Q and that the five columns of the matrix

I 2 , M = 1 0 Λ 11 Λ 12 Λ 13 0 1 Λ 21 Λ 22 Λ 23
are linearly independent over Q. Then one at least of the three numbers

∆ 1 = Λ 12 Λ 13 Λ 22 Λ 23 , ∆ 2 = Λ 13 Λ 11 Λ 23 Λ 21 , ∆ 3 = Λ 11 Λ 12 Λ 21 Λ 22 is not in L.
So far the conclusion of the above-mentioned statements (theorems or conjectures) were that some matrices had rank at least 2. Under suitable assumptions, it is possible to reach higher rank. Here is an example from [START_REF] Waldschmidt | Transcendence results related to the six exponentials theorem[END_REF] (Theorem 2.11). is not in L.

As pointed out in [START_REF] Waldschmidt | Further variations on the six exponentials Theorem[END_REF], the statement of Corollary 2.12, p. 347 of [START_REF] Waldschmidt | Transcendence results related to the six exponentials theorem[END_REF], should read as follows. is not in L.

Theorem 2 . 1 (

 21 The six exponentials Theorem) If x 1 , x 2 are Q-linearly independent complex numbers and y 1 , y 2 , y 3 are Q-linearly independent complex numbers, then one at least of the six numbers e x1y1 , e x1y2 , e x1y3 , e x2y1 , e x2y2 , e x2y3

Theorem 2 . 6 (

 26 K. Ramachandra) If a and b are real positive algebraic numbers different from 1 for which log a/ log b is irrational and a < b < a -1 , then one at least of the two numbers

Theorem 5 . 1 (

 51 G. Diaz) Let λ and λ be in L \ {0}. Assume λ ∈ R ∪ iR and λ ∈ R ∪ iR. Then λλ ∈ L.

  Theorem 1.6). According to the strong four exponentials Conjecture 3.2, one expects the assumptions λ ∈ R ∪ iR and λ ∈ R ∪ iR in Theorem 5.1 to be unnecessary. In particular, for λ ∈ L \ {0}, the number |λ| 2 = λλ should be transcendental. In other terms (Conjecture C(|u|) in § 5.1 of [1]): Conjecture 5.2 (G. Diaz) Let u ∈ C × . Assume |u| is algebraic. Then e u is transcendental.

Theorem 6 . 2

 62 Let M = (Λ ij ) 1≤i≤m;1≤j≤ be a m × matrix with entries in L. Denote by I m the identity m × m matrix and assume that the m + column vectors of the matrix (I m , M) are linearly independent over Q. Let Λ 1 , . . . , Λ m be elements of L. Assume that the numbers 1, Λ 1 , . . . , Λ m are Q-linearly independent. Assume further > m 2 . Then one at least of the numbers Λ 1 Λ 1j + • • • + Λ m Λ mj (j = 1, . . . , )

Corollary 6 . 3

 63 Let M be a 2 × 5 matrix with entries in L:M = Λ 11 Λ 12 Λ 13 Λ 14 Λ 15 Λ 21 Λ 22 Λ 23 Λ 24 Λ 25 .Assume that the three numbers 1, Λ 11 and Λ 21 are linearly independent over the field of algebraic numbers and that the seven columns of the matrix The Role of Complex Conjugation in Transcendental Number Theory 11 (I 2 , M) are linearly independent over Q. Then one at least of the four numbers Λ 11 Λ 2j -Λ 21 Λ 1j (j = 2, 3, 4, 5)

. Remark 2.2

  It is unknown whether one of the two numbers 2 π , 2 π 2 is transcendental. A consequence of Schanuel's conjecture 2.3 below ([13], Conjecture 1.14) is that each of the three numbers 2 π , 2 π 2 , 2 π 3 is transcendental and more precisely that the five numbers π, log 2, 2 π , 2 π 2 , 2 π 3 Let x 1 , . . . , x n be Q-linearly independent complex numbers. Then the transcendence degree over Q of the field Q(x 1 , . . . , x n , e x1 , . . . , e xn ) is at least n. The four exponentials Conjecture is the next statement which has been suggested by C.L. Siegel, Th. Schneider, S. Lang and K. Ramachandra ([13, Conjecture 1.13]). It is also a consequence of Schanuel's Conjecture 2.3.

	Conjecture 2.3 (Schanuel) Conjecture 2.
	are algebraically independent.

4 (The four exponentials Conjecture)

  If x 1 , x 2 are Qlinearly independent complex numbers and y 1 , y 2 are Q-linearly independent complex numbers, then one at least of the four numbers e x1y1 , e x1y2 , e x2y1 , e x2y2

	is transcendental.

  13, § 1.4]) of the six exponentials Theorem 2.1 (resp. the four exponentials Conjecture 2.4) is the fact that a 2 × 3 (resp. 2 × 2) matrix with entries in L λ 11 λ 12 λ 13 λ 21 λ 22 λ 23 resp.

							
	M =	   	. . .	. . .	. . .	. . .	   	.

λ 11 λ 12 λ 21 λ 22 , the rows of which are linearly independent over Q and the columns of which are also linearly independent over Q, has maximal rank 2. Remark 2.8 A d × matrix M has rank ≤ 1 if and only if there exist x 1 , . . . , x d and y 1 , . . . , y such that x 1 y 1 x 1 y 2 . . . x 1 y x 2 y 1 x 2 y 2 . . . x 2 y x d y 1 x d y 2 . . . x d y

(Strong four exponentials Conjecture)

  D. Roy proposes the strong four exponentials Conjecture ([START_REF] Waldschmidt | Diophantine approximation on linear algebraic groups[END_REF] Conjecture 11.17]. Again, it is a consequence of Schanuel's Conjecture 2.3. If x 1 , x 2 are Q-linearly independent complex numbers and y 1 , y 2 are Q-linearly independent complex numbers, then one at least of the four numbersx 1 y 1 , x 1 y 2 , x 2 y 1 , x 2 y 2 is not in L.An alternate form of the strong six exponentials Theorem 3.1 (resp. the strong four exponentials Conjecture 3.2) is the fact that a 2×3 (resp. 2×2) matrix with entries in L Λ 11 Λ 12 Λ 13 Λ 21 Λ 22 Λ 23 resp. Λ 11 Λ 12 Λ 21 Λ 22 ,

	Conjecture 3.2

the rows of which are linearly independent over Q and the columns of which are also linearly independent over Q, has maximal rank 2. Remark 3.3 Under suitable conditions ([13, Theorem 12.20]), one can show that a d × matrix with entries in L has rank ≥ d /(d + ). Such statements are consequences of the Linear Subgroup Theorem ([13, Theorem 11.5]). Quantitative refinements exist ([13, Chap. 13]).

Proposition 3.5 Assume the strong four exponentials Conjecture.

Recent Results

In is not in L.

Proof Set y 3 = y 2 . Then e xjy3 = e ±xjy2 for j = 1, 2 and L is stable under complex conjugation.

An equivalent formulation of Diaz's Theorem 4.1 is the next one (part 1 of Corollaire 1 of [START_REF] Diaz | Produits et quotients de combinaisons linéaires de logarithmes de nombres algébriques: conjectures et résultats partiels[END_REF]).

Corollary 4.2 (of Diaz's Theorem)

Let Λ 1 , Λ 2 , Λ 3 be three elements in L. Assume that the three numbers Λ 1 , Λ 2 , Λ 2 are linearly independent over Q. Further assume

Proof One deduces Corollary 4.2 from Theorem 4.1 by setting x 1 = 1,

One deduces examples where one can actually prove that numbers like 1/Λ, Λ 1 Λ 2 , Λ 2 /Λ 1 (with Λ, Λ 1 , Λ 2 in L) are not in L.