Michel Waldschmidt
email: miw@math.jussieu.fr

On Ramachandra's Contributions to Transcendental Number Theory

, which will be our central subject: in section 1 we state his Main Theorem, in section 2 we apply it to algebraically additive functions. Next we give new consequences of Ramachandra's results to density problems; for instance we discuss the following question: let E be an elliptic curve which is defined over the field of algebraic numbers, and let Γ be a finitely generated subgroup of algebraic points on E; is Γ dense in E(C) for the complex topology? The other contributions of Ramachandra to transcendental number theory are dealt with more concisely in section 4. Finally we propose a few open problems.

Ramachandra's Main Theorem

Hilbert's seventh problem on the transcendence of α β (for algebraic α and β) was solved in 1934 by Gel'fond and Schneider, using two different approaches: while Gel'fond's solution [START_REF] Gel | On Hilbert's seventh problem[END_REF] involved the differential equation (d/dz)e z = e z of the exponential function, Schneider's proof [START_REF] Schneider | Transzendenzuntersuchungen periodischer Funktionen, (I); Transendenz von Potenzen[END_REF] rested on the addition formula e z1+z2 = e z1 e z2 . Later, both methods were developed and applied to other functions, notably the elliptic functions. In particular Schneider in [START_REF] Schneider | Transzendenzuntersuchungen periodischer Funktionen, (II); Transzendenzeigenschaften elliptischer Funktionen[END_REF] proved an elliptic analog of the theorem on the transcendence of α β , using the differential equation which is satisfied by a Weierstrass elliptic function: ℘ ′ 2 = 4℘ 3 -g 2 ℘ -g 3 . Sometimes, one refers to Schneider's method when no derivative is needed, and to Gel'fond's method when differential equations are there; but, as pointed out by A. Baker, this terminology is somewhat deficient, since for instance Schneider's early results on elliptic functions [START_REF] Schneider | Transzendenzuntersuchungen periodischer Funktionen, (II); Transzendenzeigenschaften elliptischer Funktionen[END_REF] involve derivatives, and furthermore the first result on functions of several variables (which yields the transcendence of the values of the Beta function at rational points) has been proved by Schneider in [START_REF] Schneider | Zur Theorie der Abelschen Funktionen und Integrale[END_REF] using a variant of Gel'fond's method!

The first general criterion dealing with analytic or meromorphic functions of one variable and containing the solution to Hilbert's seventh problem appears in [START_REF] Schneider | Ein Satz über ganzwertige Funktionen als Prinzip für Transzendenzbeweise[END_REF]; in fact one can deduce the transcendence of α β from this criterion by both methods, either by using the two functions z and α z (Schneider's method), or else e z and e βz (Gel'fond's method). This criterion is somewhat complicated, and Schneider made successful attempts to simplify it [START_REF] Schneider | Einführung in die transzendenten Zahlen[END_REF]; however these last results deal only with Gel'fond's method, i.e. derivatives are needed. A further simplification for functions satisfying differential equations was provided by Lang later ( [START_REF]Algebraic values of meromorphic functions[END_REF] and [START_REF]Introduction to Transcendental Numbers[END_REF]); the socalled Schneider-Lang criterion was used by Bertrand and Masser to derive Baker's Theorem on linear independence of logarithms, as well as its elliptic analog [START_REF] Bertrand-Masser | Linear forms in elliptic integrals[END_REF]; also it was extended to functions of several variables by Bombieri, solving a Conjecture of Nagata [START_REF] Bombieri | Algebraic values of meromorphic maps[END_REF].

Thus the situation for functions satisfying differential equations (Gel'fond's method) is rather satisfactory; but it is not the same for Schneider's method. The difficulty of providing simple criteria without assuming differential equations is illustrated by examples due to Weierstrass, Stäckel and others (see [START_REF] Mahler | Lectures on Transcendental Numbers[END_REF]). The work of Ramachandra which we consider here deals with this question. Simple criteria are known, the first one being Pólya's Theorem: there is no entire function which is not a polynomial, which maps the natural integers into Z, and which has a growth order less than 2 z (see [START_REF] Gel | Transcendental Number Theory[END_REF] Chap. III, §2, for related results, [START_REF] Waldschmidt | Fonctions entières et nombres transcendants[END_REF] and [START_REF] Waldschmidt | Algebraic values of analytic functions[END_REF] for surveys, and [START_REF] Waldschmidt | Pólya's Theorem by Schneider's method[END_REF] for a proof which is inspired by Ramachandra's work).

The first part of [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF] contains an introduction, the statement of some results and of the Main Theorem, and the proof of it. The second part is devoted to corollaries of the Main Theorem. We reproduce here the Main Theorem.

We denote by Q the field of complex algebraic numbers (algebraic closure of Q in C). The size of an algebraic number α is defined by size α = den α + α , where den α is the denominator of α (the least natural integer d such that dα is an algebraic integer) and α is the house of α (maximum of the absolute values of the complex conjugates of α). We also need the following definition: an entire function f in C is of order ≤ ̺ if there exists C > 0 such that for R ≥ 1 log sup

|z|=R |f (z)| ≤ CR ̺ .
(1) Let d ≥ 2 be a natural number and ̺ a positive real number; for 1 ≤ i ≤ d, let g i and h i be two entire functions without common zeros, of order ≤ ̺, and let M (i) (R) denote the quantity

M (i) (R) = 1 + max |z|=R |h i (z) 1 + max |z|=R |g i (z) .
Assume further that the d meromorphic functions f i = h i /g i , (1 ≤ i ≤ d) are algebraically independent over C.

Ramanujan Mathematical Society

(2) Let (ζ µ ) µ≥1 be an infinite sequence of distinct complex numbers, and (n µ ) µ≥1 be a non-decreasing sequence of natural numbers with lim µ→∞ n µ = ∞. For Q ≥ 1, define

N (Q) = Card{µ; µ ≥ 1, n µ ≤ Q} and D(Q) = max nµ≤Q |ζ µ |,
and assume lim inf

Q→∞ log N (Q) log D(Q) > ̺.
(3) Let (µ r ) r≥1 be a sequence of integers such that the number

N 1 (Q) = Card{µ r ; r ≥ 1, n µr ≤ Q}
tends to infinity as Q tends to infinity. Suppose that whenever a polynomial in f 1 , . . . , f d vanishes at all points ζ µr with n µr ≤ Q, then it vanishes also at all points ζ µ with n µ ≤ Q.

(4) Suppose that the numbers f i (ζ µ ) (for 1 ≤ i ≤ d and µ ≥ 1) are all algebraic numbers; denote by ∂(Q) the degree of the field obtained by adjoining the algebraic numbers

f i (ζ µ ), ( 1
≤ i ≤ d, 1 ≤ µ ≤ Q)
to the field of rational numbers, and set

M (i) 1 (Q) = 1 + max nµ≤Q size f i (ζ µ ) , ( 1
≤ i ≤ d).
(5) Finally set

M (i) 2 (Q) = 1 + max nµ≤Q 1 |g i (ζ µ )| , ( 1
≤ i ≤ d).
Main Theorem of [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF]]. Let q be a sufficiently large natural number, and L 1 , . . . , L d natural numbers related to q asymptotically by

L 1 • • • L d ∼ ∂(q) ∂(q) + 1 N 1 (q).
Suppose that the hypotheses (1) -( 4) above are satisfied. Then there exists a natural number Q, greater than q, such that for every positive quantity R, there holds

1 ≤ 8D(Q) R N (Q-1) s i=1 M (i) 1 (Q) 8∂(Q) M (i) 2 (Q)M (i) (R) Li .

Ramanujan Mathematical Society

Remark: It would be interesting to write down a proof of this result by means of Laurent's interpolation determinants (see [START_REF] Laurent | Sur quelques résultats récents de transcendance[END_REF][START_REF] Laurent | Hauteurs de matrices d'interpolation[END_REF][START_REF] Laurent | Linear forms in two logarithms and interpolation determinants[END_REF][START_REF]Formes linéaires en deux logarithmes et déterminants d'interpolation[END_REF] and [START_REF] Waldschmidt | Linear independence of logarithms of algebraic numbers[END_REF]): instead of using Dirichlet's box principle (lemma of Thue-Siegel) for constructing an auxiliary function, one considers the matrix of the related system of equations, and one estimates (from below using Liouville's inequality, from above thanks to Schwarz's lemma) a non-vanishing determinant. It is to be expected that such an argument will produce a slightly different explicit inequality, but it is unlikely that these differences will have any effect on the corollaries.

Further works in this direction have been developed; see in particular [START_REF] Waldschmidt | Propriétés arithmétiques des valeurs de fonctions méromorphes algébriquement indépendantes[END_REF][START_REF] Mignotte-Waldschmidt | Approximation des valeurs de fonctions transcendantes Koninkl[END_REF][START_REF] Waldschmidt | Pólya's Theorem by Schneider's method[END_REF][START_REF] Gramain-Mignotte | Fonctions entières arithmétiques; Approximations diophantiennes et nombres transcendants[END_REF][START_REF]Valeurs algébriques de fonctions analytiques[END_REF][START_REF] Philippon | Nouveaux aspects de la transcendance, Journées Arithmétiques[END_REF].

The paper [START_REF] Bombieri-Lang | Analytic subgroups of group varieties[END_REF] can be considered as the first extension of Ramachandra's Theorem to higher dimension; more recent results connected with functions of several variables are given in [START_REF] Waldschmidt | On the transcendence methods of Gel'fond and Schneider in several variables[END_REF].

2 Pseudo-algebraic points of algebraically additive functions a) Statement of Ramachandra's upper bound When f is a meromorphic function in the complex plane and y is a complex number, we shall say that y is a pseudo-algebraic point of f if either y is a pole of f or else f (y) is an algebraic number (this is the definition in [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF] p. 84).

Let f 1 , . . . , f d be meromorphic functions; we define δ(f 1 , . . . , f d ) (which is either a non-negative integer or else ∞) as the dimension of the space of pseudo-algebraic point of f i .

This notation is convenient to state a few classical transcendence results: the Theorem of Hermite-Lindemann is δ(z, e z ) = 0, the Theorem of Gel'fond-Schneider can be stated either as -(Gel'fond's method): for any irrational algebraic number β, δ(e z , e βz ) = 0; -(Schneider's method): for any non-zero complex number t, δ(z, e tz ) ≤ 1.

Schneider's results on elliptic functions (see for instance [START_REF] Schneider | Einführung in die transzendenten Zahlen[END_REF] Chapitre 2 §3 Théorèmes 15, 16 et 18) can also be stated as follows:

if ℘ and ℘ * are Weierstrass elliptic functions with algebraic invariants g 2 , g 3 , g * 2 , g * 3 , if β and γ are non-zero algebraic numbers such that the two functions ℘(z) and ℘ * (γz) are algebraically independent, if ζ is the Weierstrass zeta function associated to ℘, and if a, b are algebraic numbers with (a, b) = (0, 0),

then δ e βz , ℘(z) = 0, δ ℘(z), az + bζ(z) = 0 and δ ℘(z), ℘ * (γz) = 0. ( 1
)
These results of Schneider on elliptic functions depend heavily on the fact that these functions satisfy differential equations with algebraic coefficients. The main point in Ramachandra's work is that similar results are achieved for functions which do not satisfy differential equations with algebraic coefficients. In place of derivatives, the addition theorem which is satisfied by these functions plays a crucial role. According to [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF] p. 85, a meromorphic function f is said to possess an algebraic addition theorem if there exists a non-zero polynomial P ∈ C[T 1 , T 2 , T 3 ] such that the meromorphic function of three variables P f (z 1 + z 2 ), f (z 1 ), f (z 2 ) is the zero function.

Further, if there is such a polynomial P with algebraic coefficients, then f will be called algebraically additive.

If f 1 , . . . , f d are algebraically additive functions, then the set of common pseudoalgebraic points of f 1 , . . . , f d can be shown to be a Q-vector space, and δ(f 1 , . . . , f d ) is nothing else than the dimension of this vector space. The fundamental result in part II of [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF] (Theorem 1 p. 74) is the following upper bound for this dimension:

Ramachandra's δ-Theorem. -Let f 1 , . . . , f d , with d ≥ 2, be algebraically independent meromorphic functions; assume that for 1 ≤ i ≤ d, the function f i is algebraically additive and is of order

≤ ̺ i . Define κ = 1 if f 1 , . . . , f d have a common non-zero period, 0 otherwise. Then δ(f 1 , . . . , f d ) ≤ ̺ 1 + • • • + ̺ d -κ d -1 .
Lang's criterion for Schneider's method in [START_REF]Introduction to Transcendental Numbers[END_REF] (Chapter 2, Theorem 2) is the following special case:

For d = 2 the inequality δ(f 1 , f 2 ) ≤ 2̺ * holds with ̺ * = max{̺ 1 , ̺ 2 }. (2)
We quote from [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF] p. 87: "It may be possible to improve the bound for the dimension given by Theorem 2 probably to 1 in all cases; but even a slight improvement such as ≤ ̺ * + (̺ * -κ)/d appears to be very difficult". (We have substituted κ and d to θ and s respectively to cope with our own notations). The number ̺ * stands for max{̺ 1 , . . . , ̺ d }, while ̺ * stands for min{̺ 1 , . . . , ̺ d }.

It is quite remarquable that no substantial improvement of Ramachandra's δ-Theorem has been obtained after more than a quarter of a century! We now describe one situation where the assumption of Ramachandra's δ -Theorem are satisfied, and nevertheless the estimate δ(f 1 , . . . , f d ) ≤ 1 does not hold. Take a Weierstrass elliptic function ℘ with algebraic invariants g 2 and g 3 ; let t be a non-zero complex number; consider the two functions f 1 (z) = z and f 2 (z) = ℘(tz). From Ramachandra's δ-theorem follows δ z, ℘(tz) ≤ 2. To start with, assume equality holds: let α and β be two Q-linearly independent common pseudo-algebraic points of f 1 and f 2 ; then u = tα and v = tβ are Q-linearly independent common pseudo-algebraic points of ℘, and the quotient γ = u/v = α/β is algebraic irrational; hence δ ℘(z), ℘(γz) ≥ 1; from Schneider's above mentioned theorem [START_REF] Alaoglu-Erdős | On highly composite and similar numbers[END_REF] with ℘ * = ℘, we deduce that the two functions ℘(z) and ℘(γz) are algebraically dependent; now γ is irrational, hence we are in the so-called "CM case": when (ω 1 , ω 2 ) is a fundamental pair of periods, τ = ω 2 /ω 1 is an imaginary quadratic number and the associated elliptic curve has a non trivial ring of endomorphisms.

Conversely, if ℘ has complex multiplications, for any t ∈ C × the set of common pseudo-algebraic points of the two functions z and ℘(tz) is a Q(τ )-vector space, and therefore δ z, ℘(tz) is even; one example where this vector space has positive dimension is when t is a rational multiple of a period of ℘; for instance

δ z, ℘(ω 1 z) = 2.
We repair Ramachandra's Conjecture as follows:

Ramachandra's δ-Conjecture. -For any non-zero complex number t and any Weierstrass elliptic function ℘ (resp. ℘ * ) with algebraic invariants g 2 , g 3 (resp. g * 2 , g * 3 ), assuming the two functions ℘(tz) and ℘ * (z) are algebraically independent,

δ e z , e tz ≤ 1, δ e z , ℘(tz) ≤ 1, δ ℘(tz), ℘ * (z) ≤ 1.
The first inequality is equivalent to the so-called four exponentials Conjecture, which was apparently known to Siegel (see [START_REF] Alaoglu-Erdős | On highly composite and similar numbers[END_REF]), which was also considered by A. Selberg in the early 40's (personal communication, Hong-Kong, July 1993) and later was proposed by Lang in [START_REF]Introduction to Transcendental Numbers[END_REF]; an equivalent question is the first of Schneider's eight problems in [START_REF] Schneider | Einführung in die transzendenten Zahlen[END_REF]: Four exponentials Conjecture. -Let x 1 , x 2 be two Q-linearly independent complex numbers, and y 1 , y 2 be also two Q-linearly independent complex numbers; then one at least of the four numbers e x1y1 , e x1y2 , e x2y1 , e x2y2 is transcendental.

A partial result can be proved (which is called the five exponentials Theorem in [START_REF] Waldschmidt | On the transcendence methods of Gel'fond and Schneider in several variables[END_REF]; the fifth number is e x1/x2 ). A result which is stronger than both the six and the five exponentials Theorems, but which does not include the four exponentials Conjecture, is due to D. Roy [START_REF] Roy | Matrices whose coefficients are linear forms in logarithms[END_REF]). Denote by L the set of linear combinations of logarithms of algebraic numbers, which is the Q-vector space spanned by {1} ∪ {ℓ ∈ C; e ℓ ∈ Q× }.

Theorem (D. Roy). -If x 1 , x 2 , x 3 are Q-linearly independent complex numbers and y 1 , y 2 are Q-linearly independent complex numbers, then one at least of the six numbers x i y j is not in L.

b) Sketch of proof of Ramachandra's δ-Theorem as a consequence of Ramachandra's Main Theorem

The deduction of the δ-Theorem from the Main Theorem is by no means trivial; Ramachandra had to use Weyl's criterion of equidistribution in order to check the hypotheses concerning the poles of the elliptic functions. Another approach is due to Serre (see lemma 3.4 p. 46 of [START_REF] Waldschmidt | Propriétés arithmétiques des valeurs de fonctions méromorphes algébriquement indépendantes[END_REF]). Here, in this sketch of proof, we hardly quote problems arising from the poles.

Inside the group of common pseudo-algebraic points of f 1 , . . . , f d , select a finitely generated subgroup Y = Zy 1 + • • • + Zy ℓ of rank ℓ. In case κ = 1, choose for y 1 a common period to f 1 , . . . , f d . Let h 1 , h 2 , . . . be a numbering of Z ℓ , with h µ = (h 1µ , . . . , h ℓµ ), (µ ≥ 1), such that the sequence

n µ = max |h 1µ |, . . . , |h ℓµ | , (µ ≥ 1)
is non-decreasing, and define the sequence ζ 1 , ζ 2 , . . . by

ζ µ = h 1µ y 1 + • • • + h ℓµ ζ ℓ , (µ ≥ 1), so that ζ µ , µ ≥ 1 = h 1 y 1 + • • • + h ℓ ζ ℓ , h = (h 1 , . . . , h ℓ ) ∈ Z ℓ = Zy 1 + • • • + Zy ℓ .
In fact the poles of any f i should be removed from this sequence; even more, any point which is too close to a pole should also be omitted, in order to estimate M (i) 2 (Q); but, as mentioned above, we give here only a sketch of the proof. Suitable positive constants c 0 , c 1 , . . . are then selected, which do not depend on the large integer q, so that the Main Theorem can be used with the following inequalities:

∂(q) ≤ c 0 , M (i) (R) ≤ exp(c 1 R ̺i ), ( 1
≤ i ≤ d), c 2 Q ℓ ≤ N (Q) ≤ c 3 Q ℓ , D(Q) ≤ c 4 Q, max{M (i) 1 (Q), M (i) 2 (Q)} ≤ exp(c 5 Q ̺i ), ( 1
≤ i ≤ d).
In the case where κ = 1, we define {µ 1 , µ 2 , . . .} as he sequence of integers µ such that h 1µ = 0, and we use the bound

N 1 (Q) ≤ c 6 Q ℓ-κ ;
in the case κ = 0, we define µ r = r, (r ≥ 1), and again we have the same upper bound for N 1 (Q). Now choose

L i = c 7 q λ-̺i , (1 ≤ i ≤ d), with λ = (ℓ -κ + ̺ 1 + • • • + ̺ d )/d,
and choose also R = c 10 Q, where c 10 is sufficiently large, so that the desired conclusion follows from the Main Theorem.

Remark: We need to take for L i natural integers; we have introduced an integral part, but we need to check that L i does not vanish; hence the sketch of proof is valid only when λ > ̺ * , where ̺ * = max{̺ 1 , . . . , ̺ d }.

This condition can be written

̺ * ≤ ̺ 1 + . . . + ̺ d + ℓ -κ d ;
since the goal is to prove the upper bound ℓ ≤ (̺

1 + • • • + ̺ d -κ)/(d -1)
, we may consider that this condition on ̺ * is satisfied as soon as

̺ * ≤ ̺ 1 + . . . + ̺ d -κ d -1 ;
this assumption occurs explicitly in Theorem 1 p. 74 of [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF]; however it is possible to remove it, by means of an induction argument on d; see p. 78 of [START_REF] Waldschmidt | Propriétés arithmétiques des valeurs de fonctions méromorphes algébriquement indépendantes[END_REF] and p. 52 of [START_REF] Waldschmidt | Nombres transcendants[END_REF].

The statement of Theorem 1 in [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF] also involves a notion of weighted sequences which has been used in [START_REF] Waldschmidt | Propriétés arithmétiques des valeurs de fonctions méromorphes algébriquement indépendantes[END_REF].

c) Corollaries

We first provide a collection of upper bounds for δ(f 1 , . . . , f d ), where f i are either linear, or exponential, or elliptic functions; all these estimates follow from Ramachandra's δ-Theorem.

We start with Gel'fond-Schneider's Theorem (already quoted above): for any non-zero complex number t, δ(z, e tz ) ≤ 1.

Another example which do not involve elliptic functions is the six exponentials Theorem (see below): if x 1 , . . . , x d are Q-linearly independent complex numbers with d ≥ 2, then δ e x1z , . . . , e x d z ≤ d/(d -1). As a matter of fact it suffices to select either d = 2 or else d = 3 to cover all cases (see below).

The next examples all involve elliptic functions. Notations are as follows: ℘, ℘ * , ℘ 1 , . . . are Weierstrass elliptic functions, all of whose invariants g 2 , g 3 ,g * 2 , g * 3 ,. . . are algebraic. The numbers t, t * , t 1 , . . . are non-zero complex numbers, while ω, ω * , ω 1 . . . are respectively non-zero periods of ℘, ℘ * , ℘ 1 , . . . Then

δ z, ℘(z) ≤ 2, δ e z , ℘(tz) ≤ 3, δ e 2πiz , ℘(ωz) ≤ 2, δ ℘(tz), ℘ * (z) ≤ 4, δ ℘(ωz), ℘ * (ω * z) ≤ 3, δ e z , ℘(tz), ℘ * (t * z) ≤ 2, δ ℘ 1 (t 1 z), ℘ 2 (t 2 z), ℘ 3 (z) ≤ 3, δ ℘ 1 (ω 1 z), ℘ 2 (ω 2 z), ℘ 3 (ω 3 z) ≤ 2, δ ℘ 1 (t 1 z), ℘ 2 (t 2 z), ℘ 3 (t 3 z), ℘ 4 (z) ≤ 2.
We tacitly assumed that the functions we consider are algebraically independent; by the way, it was a non-trivial problem to provide explicit conditions which guarantee the algebraic independence of the functions; an important contribution to this question is Lemma 7 in [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF] p. 83; this problem has been solved later in [START_REF] Brownawell-Kubota | The algebraic independence of Weierstrass functions and some related numbers[END_REF].

We now consider more closely a few of these results.

Example 1. δ z, e tz . -Hilbert's seventh problem by Schneider's method Corollary 1. -Let α be a non-zero complex algebraic number, and β an irrational algebraic number; choose any determination log α of the logarithm of α with log α = 0 in case α = 1. Then α β = exp(β log α) is a transcendental number.

Proof: This statement (Theorem of Gel'fond-Schneider) follows from Ramachandra δ-Theorem by taking

d = 2, f 1 (z) = z, f 2 (z) = α z = exp(z log α), ̺ 1 = 0, ̺ 2 = 1, κ = 0; since y 1 = 1 is a common pseudo-algebraic point of f 1 and f 2 , we deduce δ(f 1 , f 2 ) = 1
, and hence β is not a pseudo-algebraic point of f 2 , which means that α β is a transcendental number.

Notice that Lang's above mentioned criterion (2) for Schneider's method in [START_REF]Introduction to Transcendental Numbers[END_REF] does not cover the transcendence of α β : the point is that the orders of the two functions are 0 and 1 respectively, and it is not sufficient to consider the maximum of both numbers.

Example 2. δ e z , e tz . -The six exponentials Theorem

The story starts with Ramanujan's study of highly composite numbers; see [START_REF] Alaoglu-Erdős | On highly composite and similar numbers[END_REF][START_REF]Nombres transcendants[END_REF][START_REF]Algebraic values of meromorphic functions[END_REF][START_REF]Introduction to Transcendental Numbers[END_REF][START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF][START_REF] Ramachandra | Lectures on transcendental numbers[END_REF]; see also [START_REF] Baker | Transcendental Number Theory[END_REF][START_REF] Waldschmidt | Nombres transcendants[END_REF] and [START_REF] Waldschmidt | Linear independence of logarithms of algebraic numbers[END_REF] for further references.

Corollary 2. -Let x 1 , . . . , x d be Q-linearly independent complex numbers, and y 1 , . . . , y ℓ be also Q-linearly independent complex numbers; assume that the dℓ numbers

e xiyj , ( 1
≤ i ≤ d, 1 ≤ j ≤ ℓ)
are all algebraic; then dℓ ≤ d + ℓ.

Proof: Take A nice consequence of part b) of this statement is Corollary p. 87 of [Ramachandra, 1968] which we reproduce here:

f i (z) = e xiz and ̺ i = 1 for 1 ≤ i ≤ d.
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

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.
Ramachandra deduces from his results some new transcendental complex numbers, by means of the clean trick (p. 68): if x and y are real numbers, then the complex number x + iy is transcendental if and only if one at least of the two numbers x, y is transcendental. Further consequences of this idea have been worked out by G. Diaz in [START_REF] Diaz | La conjecture des quatre exponentielles et les conjectures de D. Bertrand sur la fonction modulaire[END_REF] and [START_REF] Diaz | Utilisation de la conjugaison complexe dans l'tude de la transcendance de valeurs de la fonction exponentielle[END_REF]. See also [START_REF] Waldschmidt | Variations on the Six Exponentials Theorem; with an appendix by H. Shiga: Periods of a Kummer surface[END_REF].

Example 4. δ ℘(tz), ℘ * (z)
We consider now two elliptic functions (compare with [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF] p. 68).

Corollary 4. -Let ℘ and ℘ * be two Weierstrass elliptic functions with algebraic invariants g 2 , g 3 and g * 2 , g * 3 respectively; let u 1 , . . . , u ℓ be Q-linearly independent complex numbers, each of which is a pseudo-algebraic point of ℘; similarly, let u * 1 , . . . , u * ℓ be pseudo-algebraic points of ℘ * which are linearly independent over Q. Assume that the two functions ℘(u 1 z) and ℘ * (u * 1 z) are algebraically independent. a) Assume ℓ ≥ 5. Then the matrix

u 1 . . . u ℓ u * 1 . . . u * ℓ has rank 2. b) Assume u 1 is a period of ℘ and u * 1 is a period of ℘ * ; then the rank of the matrix is 2 also when ℓ = 4. c) If Ramachandra's δ-Conjecture δ ℘(tz), ℘ * (z) ≤ 1 holds, the same conclusion is true if only ℓ ≥ 2.
An impressive collection of further consequences to Ramachandra's Main Theorem is displayed in [START_REF] Brownawell-Kubota | The algebraic independence of Weierstrass functions and some related numbers[END_REF] section III C. Several of the previous corollaries deal with Ramanujan Mathematical Society elliptic integrals of the first of second kind; further consequences concern elliptic integrals of the third kind as well. More generally, a natural situation where all hypotheses are satisfied is connected with analytic subgroups of commutative algebraic groups (see for instance [START_REF] Waldschmidt | Propriétés arithmétiques des valeurs de fonctions méromorphes algébriquement indépendantes[END_REF] §3 section 5:"Application du théorème de Ramachandra à l'étude de sous-groupes à un paramètre de certaines variétés de groupes"; see also [START_REF] Bombieri-Lang | Analytic subgroups of group varieties[END_REF] and [START_REF] Waldschmidt | On the transcendence methods of Gel'fond and Schneider in several variables[END_REF]); however it seems to the author that this is not exactly the right place to develop this aspect of the theory.

Application to density statements

Ramachandra's results on algebraic values of algebraically additive functions can be used to prove some density results. We give here a sample of results dealing with

(R × ) 2 , C × , R × × E(R), E(R) × E * (R
) and E(C). Further topological groups are considered in [START_REF] Waldschmidt | Densité de points rationnels sur un groupe algébrique[END_REF]. a1) Consequences of the six exponentials Theorem: real case Let γ 1 , . . . , γ ℓ be multiplicatively independent elements in (R × + ) 2 ; write γ j = (α j , β j ), (j = 1, . . . , ℓ).

By means of a well-known result due to Kronecker † , one can show that the subgroup Γ which is generated by γ 1 , . . . , γ ℓ is dense in (R × + ) 2 if and only if for each s ∈ Z ℓ \{0}, the matrix with three rows and ℓ columns det are Q-linearly independent. We assume now that this condition is satisfied, and also that the 2ℓ numbers α j and β j are algebraic. a) According to the six exponentials Theorem, if ℓ ≥ 4, then Γ is dense in (R × + ) 2 . For instance if ℓ = 4 and if the eight numbers α j , β j are multiplicatively independent, the group Γ whose rank is 4 is dense in (R × + ) 2 . b) If we take for granted the four exponentials Conjecture (see section 2 a), Γ is dense in (R × + ) 2 as soon as ℓ ≥ 3. For instance when ℓ = 3 and when the six numbers α j , β j are multiplicatively independent, then we expect Γ to be dense in (R × + ) 2 . † According to his own taste, the reader will find a reference either in N. Bourbaki, Eléments de Mathématique, Topologie Générale, Herman 1974, Chap. VII, § 1, N • 1, Prop. 2; or else in G.H. Hardy and A.M. Wright, An Introduction to the Theory of Numbers, Oxford Sci. Publ., 1938, Chap. XXIII.

  log α 1 • • • log α ℓ log β 1 • • • log β ℓ s 1 • • • s ℓ  

Ramanujan Mathematical Society

Example: the field Q( √ 2).

From the six exponentials Theorem follows that the subgroup of (R × ) 2 , of rank 4, which is generated by the images of

2 √ 2 -1, -3 √ 2 -1, 4 √ 2 -1, 6 √ 2 -1,
under the canonical embedding of the real quadratic field Q( √ 2), is dense in (R × ) 2 . If we knew the four exponentials Conjecture we could omit 6 √ 2 -1 and get a dense subgroup of rank 3. a2) Consequences of the six exponentials Theorem: complex case Consider ℓ multiplicatively independent complex numbers γ 1 , . . . , γ ℓ . The subgroup Γ of C × generated by γ 1 , . . . , γ ℓ is dense if and only if for all s ∈ Z ℓ+1 \ {0}, the matrix with three rows and ℓ + 1 columns

  0 log |γ 1 | • • • log |γ ℓ | 2πi log(γ 1 /γ 1 ) • • • log(γ ℓ /γ ℓ ) s 0 s 1 • • • s ℓ  
has rank 3 (this condition clearly does not depend on the choice of the logarithms log(γ j /γ j )). A first necessary condition is that at least two of the numbers |γ 1 |, . . ., |γ ℓ | are multiplicatively independent; a second necessary condition is that the numbers γ 1 /|γ 1 |, . . . , γ ℓ /|γ ℓ | are not all roots of unity. These two conditions mean that the projection of Γ on each of the two factors R × + and R/Z which arises from z → (|z|, z/|z|), has a dense image. We assume that these conditions are satisfied, and furthermore that the ℓ complex numbers γ j are algebraic. According to the four exponentials Conjecture, Γ should be dense in C × without any further assumption. On the other hand, if we use the six exponentials Theorem, assuming that three at least of the numbers |γ 1 |, . . . , |γ ℓ | are multiplicatively independent, we deduce that Γ is dense in C × .

Example: the field Q(i).

The subgroup of rank 3 in C × which is generated by

2 + i, 2 + 3i, 4 + i, is dense C × .
℘ ′ 2 = 4℘ 3 -g 2 ℘ -g 3 ;
the set of real points on the corresponding elliptic curve E, namely

E(R) = (x : y : t) ∈ P 2 (R); y 2 t = 4x 3 -g 2 xt 2 -g 3 t 3 ,
has one or two connected components, according as the polynomial 4X 3 -g 2 X -g 3 has one or three real roots; we denote by E(R) 0 the connected component of E(R) which contains the origin (0 : 1 : 0); hence E(R) 0 is a subgroup of E(R) of index 1 or 2. For simplicity of notation, when ω is a pole of ℘, then ℘(ω) : ℘ ′ (ω) : 1 means (0 : 1 : 0). With this convention the map exp E : u → ℘(u) : ℘ ′ (u) : 1 is a surjective homomorphism from R onto E(R) 0 whose kernel is of the form Zω, where ω is a fundamental real period of ℘. A point γ = exp E (u) = ℘(u) : ℘ ′ (u) : 1 ∈ E(R) 0 is a torsion point if and only if the two number u and ω are linearly dependent over Q. More generally, when u 1 , . . . , u ℓ are real numbers, the rank over Z of the subgroup Γ generated by the ℓ points γ j = ℘(u j ) : ℘ ′ (u j ) : 1 , (1 ≤ j ≤ ℓ) in E(R) 0 is related to the rank of the subgroup Y of R generated by the ℓ + 1 real numbers u 1 , . . . , u ℓ , ω by rank Z Γ = rank Z Y -1.

Recall (Kronecker's Theorem again) that a subgroup of rank ≥ 1 in R/Z is dense. We deduce:

Let E be an elliptic curve which is defined over R and let γ be a point of infinite order on E(R) 0 ; then the subgroup Zγ is dense for the real topology in E(R) 0 .

The next density result deals with the product of the multiplicative group of non-zero real numbers with E(R); it will be proved as a consequence of Corollary 3.

Corollary 5. -Let α 1 , . . . , α ℓ be multiplicatively independent positive real algebraic numbers; let γ 1 , . . . , γ ℓ be points on E( Q) ∩ E(R) 0 , which are not all torsion points. Denote by Γ the subgroup of R × + × E(R) 0 which is spanned by the ℓ points (α j , γ j ), (1

≤ j ≤ ℓ). a) Assume that ℓ ≥ 4. Then Γ is dense in R × + × E(R) 0 . b)
According to Ramachandra's δ-Conjecture δ e z , ℘(tz) ≤ 1, the same conclusion should hold as soon as ℓ ≥ 2.

Proof: The exponential map

R 2 -→ R × + × E(R) 0 (x 1 , x 2 ) -→ e x1 ; ℘(x 2 ) : ℘ ′ (x 2 ) : 1
is a topological surjective homomorphism with kernel Z(0, ω) for some ω ∈ R × ; define y 0 = (0, ω) and y j = (log α j , u j ), (1 ≤ j ≤ ℓ), where u j ∈ R is such that γ j = ℘(u j ) : ℘ ′ (u j ) : 1 . Now the goal is to prove that the subgroup Y = Zy 0 + Zy

1 + • • • + Zy ℓ is dense in R 2 .
Let ϕ : R 2 → R be a linear form which satisfies ϕ(Y ) ⊂ Z. According to Kronecker's above mentioned Theorem (see footnote ( 1)), we only need to prove ϕ = 0. Write ϕ(y j ) = s j with s j ∈ Z, (0 ≤ j ≤ ℓ); then the matrix

  0 log α 1 • • • log α ℓ ω u 1 • • • u ℓ s 0 s 1 • • • s ℓ  
has rank < 3; it follows that the rank of the matrix

s 0 log α 1 • • • s 0 log α ℓ s 0 u 1 -s 1 ω • • • s 0 u ℓ -s ℓ ω is < 2. Since u 1 , .
. . , u ℓ are not all torsion points and log α 1 , . . . , log α ℓ are linearly independent over Q, it follows from Corollary 3 that s 0 = 0; using once more the linear independence of the log α's, we deduce s 1 = • • • = s ℓ = 0, and ϕ = 0.

Remark: Ramachandra's δ-Conjecture δ e z , ℘(tz) ≤ 1 implies the following:

Let log α 1 and log α 2 be two Q-linearly independent logarithms of algebraic numbers. Let E be an elliptic curve which is defined over the field Q of algebraic numbers, ω be a non-zero period of exp E and u ∈ C be an elliptic logarithm of a point of infinite order in E( Q). Then the three numbers

log α 1 log α 2 , u ω , 1
are linearly independent over Q.

Indeed, this means, for each (λ, µ) ∈ Q 2 , det log α 1 log α 2 u + λω µω = 0.

c) Product of two elliptic curves

As a consequence of Corollary 4 we have a density result for the product of two elliptic curves over the real number field. Corollary 6. -Let E and E * be two Weierstrass elliptic curves with real algebraic invariants g 2 , g 3 and g * 2 , g * 3 respectively; assume for simplicity that there is no isogeny between them, which means that for t ∈ C × , the two functions ℘(tz) and ℘ * (z) are algebraically independent. Denote by ω (resp. ω * ) a non-zero real period of ℘ (resp. of ℘ * ). Let ℓ be a positive integer and let γ 1 , . . . , γ ℓ (resp. γ * 1 , . . . , γ * ℓ ) be elements in

E(R) 0 ∩ E( Q) (resp. in E * (R) 0 ∩ E * ( Q)) such that a) γ 1 , . . . , γ ℓ are not all torsion points on E( Q); b) γ * 1 , . . . , γ * ℓ are not all torsion points on E * ( Q); c) the subgroup Γ of E( Q) × E * ( Q) generated by the ℓ points (γ j , γ * j ), (1 ≤ j ≤ ℓ) has rank ℓ. Then 1) if ℓ ≥ 3, Γ is dense in E(R) 0 × E * (R) 0 . 2) if Ramachandra's δ-Conjecture δ ℘(ωz), ℘ * (ω * z) ≤ 1 is
true, the same conclusion holds already for ℓ ≥ 1.

Proof: We first translate the hypotheses concerning the points on the elliptic curves in terms of elliptic logarithms. Let ω (resp. ω * ) be a real fundamental period of ℘ (resp. of ℘ * ); for 1 ≤ j ≤ ℓ, let u j ∈ R satisfy ℘(u j ) : ℘ ′ (u j ) : 1 = γ j , and let u * j ∈ R satisfy ℘ * (u * j ) : ℘ * ′ (u * j ) : 1 = γ * j . Now u 1 , . . . , u ℓ (resp. u * 1 , . . . , u * ℓ ) are pseudo-algebraic points of ℘ (resp. of ℘ * ), such that a) two at least of the ℓ + 1 numbers u 1 , . . . , u ℓ , ω are linearly independent over Q, b) two at least of the ℓ + 1 numbers u * 1 , . . . , , u * ℓ , ω * are linearly independent over Q, c) the ℓ + 2 points

y 0 = ω 0 , y * 0 = 0 ω * , y j = u j u * j , (1 ≤ j ≤ ℓ)
are linearly independent over Q. We want to prove that the subgroup of R 2 of rank ℓ+2 which is generated by y 0 , y * 0 , y 1 , . . . , y ℓ is dense in R 2 . Indeed, let s 0 , s * 0 , s 1 , . . . , s ℓ be rational integers, not all of which are zero; we shall deduce from Corollary 4 (with a shift of notations ℓ → ℓ + 1) that the matrix

  ω 0 u 1 • • • u ℓ 0 ω * u * 1 • • • u * ℓ s 0 s * 0 s 1 • • • s ℓ  
has rank 3. If s 0 = 0 (resp if s * 0 = 0), this follows from the hypothesis b) (resp. a)) above. Assume now s 0 = 0 and s * 0 = 0; we want to prove that the matrix

-s * 0 ω s 0 u 1 -s 1 ω • • • s 0 u ℓ -s ℓ ω ω * u * 1 • • • u * ℓ
has rank 2; in order to use part b) of Corollary 4, we need to check that the ℓ + 1 elements on the first row are linearly independent over Q, and the same for the ℓ + 1 elements on the second row; if this were not the case, and if the matrix had rank < 2, we would get a non trivial linear dependence relation between the ℓ+2 elements

ω 0 , 0 ω * , u j u * j , (1 ≤ j ≤ ℓ),
which would contradict the assumption rank Z Γ = ℓ.

Remark: Ramachandra's δ-Conjecture δ ℘(ωz), ℘ * (ω * z) ≤ 1 implies the following:

Let ℘ (resp. ℘ * ) be a Weierstrass elliptic function with algebraic g 2 , g 3 (resp. g * 2 , g * 3 ); let ω (resp. ω * ) be a non-zero period of ℘ (resp. of ℘ * ), and u ∈ C (resp. u * ∈ C) be a pseudo-algebraic point of ℘ (resp. of ℘ * ), with u/ω and u * /ω * both irrational numbers; assume that the two complex functions ℘(ωz) and ℘ * (ω * z) are algebraically independent; then the three numbers

1, u ω , u * ω *
are linearly independent over Q.

In other words, according to this conjecture, for rational integers s 0 , s * 0 , s, the determinant det

  ω 0 u 0 ω * u * s 0 s * 0 s   can vanish only for s 0 = s * 0 = s = 0. d)

Complex points on an elliptic curve

Let E be an elliptic curve over C

E(C) = (x : y : t) ∈ P 2 (C); y 2 t = 4x 3 -g 2 xt 2 -g 3 t 3 ,
and let γ ∈ E(C); we ask whether the subgroup Zγ spanned by γ is dense in the topological group E(C). Denote as before by ℘ the Weierstrass elliptic function with invariants g 2 and g 3 :

℘ ′ 2 = 4℘ 3 -g 2 ℘ -g 3 ,
by Ω = Zω 1 + Zω 2 the lattice of periods of ℘, by Ω = Zω 1 + Zω 2 the complex conjugate lattice and by E = C/Ω the Weierstrass elliptic curve with invariants g 2 and g 3 ; select u = x 1 ω 1 + x 2 ω 2 ∈ C (with real x 1 and x 2 ) such that γ = (℘(u) :

℘ ′ (u) : 1). The three conditions (i) γ is not a torsion point; (ii) the three numbers u, ω 1 , ω 2 are Q-linearly independent; (iii) (x 1 , x 2 ) ∈ Q 2 are equivalent.
On the other hand the three following conditions are also equivalent:

(j) Zγ is dense in the topological group E(C); (jj) the three numbers 1, x 1 , x 2 are Q-linearly independent over Q;

(jjj) if ω ∈ Ω is any non-zero period of ℘ and n ≥ 1 any positive integer, then nγ does not belong to the 1-parameter subgroup (℘(tω) :

℘ ′ (tω) : 1); t ∈ R of E(C).
Of course conditions (j), (jj) et (jjj) imply conditions (i), (ii) et (iii); clearly the converse does not hold without any further assumption. Let us assume that g 2 and g 3 are algebraic, as well as ℘(u) and ℘ ′ (u).

The curve E is defined over R if and only if there exists θ ∈ C × such that θΩ = θΩ; the set

E(E) = θ ∈ C × ; rank Z θΩ ∩ θΩ = 2};
is empty if and only if the two curves E and E are not isogeneous. We start with the easiest case:

Corollary 7.
-Let E be an elliptic curve which is defined over the field Q of algebraic numbers and is not isogeneous to its complex conjugate. 1) Any subgroup of E( Q) of rank ≥ 3 is dense in E(C) for the complex topology.

2) If Ramachandra's δ-Conjecture δ ℘(ωz), ℘ * (ω * z) ≤ 1 is true, any element in E( Q) which is not a torsion point spans a dense subgroup of E(C).

Proof: The proof is the same as for Corollary 6; also, Corollary 7 will follow from Corollary 8 below.

Before we study the general case, we prove the following auxiliary lemma. and

Ω θ = θω -θω ′ ; (ω, ω ′ ) ∈ Ω × Ω ⊂ C. If Y θ ∩ Ω θ is a subgroup of finite index in Y θ , then Y is not dense in C.
Proof: a) From the hypotheses we deduce that there exists a positive integer m such that mθω 1 ∈ θΩ and mθω 2 ∈ θΩ. Define a, b, c, d is Z by

mθω 1 = aθω 1 + bθω 2 and mθω 2 = cθω 1 + dθω 2 .
We show the relations a + d = 0 and m 2 + ad -bc = 0.

Using complex conjugation, we get

m 2 θω 1 = amθω 1 + bmθω 2 = a(aθω 1 + bθω 2 ) + b(cθω 1 + dθω 2 ) Ramanujan Mathematical Society and m 2 θω 2 = cmθω 1 + dmθω 2 = c(aθω 1 + bθω 2 ) + d(cθω 1 + dθω 2 ), hence m 2 = a 2 + bc, (a + d)b = 0, m 2 = d 2 + bc, (a + d)c = 0.
The solution a = d and b = c = 0 is not possible because ω 2 /ω 1 ∈ R.

b) We show that there exist ω 0 and ω ′ 0 which generate a subgroup of finite index in Ω such that θω 0 ∈ R and θω ′ 0 ∈ iR. We want to find λ, µ in Z such that ω 0 = λω 1 + µω 2 satisfies θω 0 = θω 0 : we need to solve the system (a -m)λ + cµ = 0 bλ + (d -m)µ = 0 whose determinant ad -bc -m(a + d) + m 2 vanishes; hence there is a non trivial solution, which means that θΩ ∩ R is a Z-module of rank 1. Similarly, since adbc + m(a + d) + m 2 = 0, the system

(a + m)λ ′ + cµ ′ = 0 bλ ′ + (d + m)µ ′ = 0 has a non trivial solution (λ ′ , µ ′ ) ∈ Z 2 , and θΩ ∩ iR is generated by a non-zero element ω ′ 0 = λ ′ ω 1 + µ ′ ω 2 . c) For each v ∈ Ω θ ∩ iR
ϕ(x 1 ω 1 + x 2 ω 2 ) = µx 1 -λx 2 ,
where (λ, µ) satisfies (as before)

ω 0 = λω 1 + µω 2 ∈ Ω ∩ R. If Y θ ∩ Ω θ is a subgroup of finite index in Y θ , then we shall deduce ϕ(Y ) ⊂ Q (from which it follows that Y is not dense in C).
For the proof, we take y = x 1 ω 1 + x 2 ω 2 ∈ Y with (x 1 , x 2 ) ∈ R 2 . Let m be a positive integer such that m(θyθy) ∈ Ω θ ; it follows from c) that there exist k ≥ 1 and ω ∈ Ω with mk(θyθy) = θω -θω, and therefore mkθ(y -ω) ∈ R. Put n = mk and write ω = aω 1 + bω 2 :

(nx 1 -a)θω 1 + (nx 2 -b)θω 2 ∈ R;
however θω 1 and θω 2 are linearly independent over R and satisfy λθω 1 + µθω 2 ∈ R.

We deduce λ(nx 2 -b) = µ(nx 1 -a), which completes the proof.

We can now state and prove the following result:

Corollary 8.

-Let E = C/Ω be a Weierstrass elliptic curve with algebraic g 2 , g 3 . Define

E(E) = θ ∈ C × ; rank Z θΩ ∩ θΩ = 2};
for each θ ∈ E(E), define

Ω θ = θω -θω ′ ; (ω, ω ′ ) ∈ Ω × Ω ⊂ C. Let Γ = Zγ 1 + • • • + ZΓ ℓ be a finitely generated subgroup of rank ℓ in E( Q). Define Y ⊂ C by Y = exp -1 E (Γ). For each θ ∈ E(E), put Y θ = θy -θy; y ∈ Y .
Assume that for each θ ∈ E(E), the subgroup

Y θ ∩ Ω θ is not of finite index in Y θ . a) If ℓ ≥ 3, then Γ is dense in E(C). b) If Ramachandra's δ-Conjecture δ ℘(ωz), ℘ * (ω * z) ≤ 1 is true, then Γ is a dense subgroup of E(C).
It follows that for any elliptic curve E which is defined over Q, there exists an algebraic number field K such that E(K) is dense in the topological group E(C).

Proof: A necessary and sufficient condition for Γ to be dense in E(C) is that Y is dense in C. For 1 ≤ j ≤ ℓ, let u j ∈ C be an elliptic logarithm of γ j ; then Y is dense in C if and only if Zω 1 +Zω 2 +Zu 1 +• • •+Zu ℓ is dense in C; according to Kronecker's Theorem, this is equivalent to the following assertion: for each (s ′ 0 , s ′′ 0 , s 1 , . . . , s ℓ ) ∈ Z ℓ+2 \ {0}, the matrix

  ω 1 ω 2 u 1 • • • u ℓ ω 1 ω 2 u 1 • • • u ℓ s ′ 0 s ′′ 0 s 1 • • • s ℓ  
has rank 3. This condition is clearly satisfied if either s ′ 0 = 0 or s ′′ 0 = 0, because we assume Γ has rank ℓ. If s ′ 0 = 0 and s ′ 0 = 0, we define

ω = s ′ 0 ω 2 -s ′′ 0 ω 1 , v j = s ′ 0 u j -s j ω 1 , (1 ≤ j ≤ ℓ),
and we want to prove that the matrix

ω v 1 • • • v ℓ ω v 1 • • • v ℓ has rank 2.
If the two functions ℘(ωz) and ℘(ωz) are algebraically independent, we can apply parts b) and c) of Corollary 4 (with ℓ replaced by ℓ + 1 again). Otherwise, since the period lattices of ℘(ωz) and ℘(ωz) are respectively (1/ω)Ω and (1/ω)Ω, we can use our assumption on Y θ with θ = 1/ω: firstly θΩ ∩ θΩ is of finite index in θΩ, secondly the numbers

s ′ 0 (θu 1 -θu 1 ) -s 1 (θω 1 -θω 1 )
do not all vanish; hence the above matrix has rank 2.

4 Further contributions of Ramachandra to transcendental number theory a) On the numbers 2 π k , (k = 1, 2, 3, . . .) From the six exponentials Theorem follows that one at least of the three numbers 2 π , 2 π 2 and 2 π 3 is transcendental. The result can be made effective, and a transcendence measure for at least one of these three numbers can be derived. Using a Theorem of Szemeredi, Srinivasan obtained a result which he himself states as follows:

for almost all k, the number 2 π k has a transcendence measure of the type

2 π k -α ≥ exp -(log H ν ) 1+ǫ
(for any ǫ > 0 with respect to a sequence of heights H ν → ∞, with height of α bounded by H ν ).

Starting from such a statement, he investigated the number of algebraic numbers among the numbers 2 π k , (1 ≤ k ≤ N ); in [START_REF] Srinivasan | On algebraic approximations to 2 π k (k = 1[END_REF] and [START_REF] Srinivasan | On algebraic approximations to 2 π k (k = 1[END_REF] he got the upper bound O √ N (conjecturally, none of them is algebraic). The O constant was bounded by 2 in [START_REF] Ramachandra-Srinivasan | A note to a paper by Ramachandra on transcendental numbers[END_REF] and by √ 2 in [START_REF] Balasubramanian-Ramachandra | Transcendental numbers and a lemma in combinatorics[END_REF]. In fact, O √ N can be replaced by a more explicit expression of the type c √ N + smaller order terms.

b) A note on Baker's method Ramachandra has several contributions to Baker's theory on linear forms in logarithms and its applications; see in particular [START_REF] Ramachandra | A note on Baker's method[END_REF][START_REF] Ramachandra | Application of Baker's theory to two problems considered by Erdős and Selfridge[END_REF], and [START_REF] Ramachandra-Shorey | On gaps between numbers with a large prime factor[END_REF]. A survey of this subject is given in [START_REF] Baker | The theory of linear forms in logarithms[END_REF]. It is fair to quote here also the important work of Shorey: [50, 51] and [START_REF] Shorey | On linear forms in the logarithms of algebraic numbers[END_REF] (one of the main contributions of Ramachandra's to transcendental number theory is Shorey).

For his investigations concerning lower bounds for linear forms in logarithms of algebraic numbers, Ramachandra used Baker's method (which is a generalization of

Ramanujan Mathematical Society

Gel'fond's solution to Hilbert's seventh problem). It turns out that a method closely related to [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF] yields similar results; see [START_REF] Mignotte-Waldschmidt | Linear forms in two logarithms and Schneider's method Math[END_REF][START_REF] Waldschmidt | Linear independence of logarithms of algebraic numbers[END_REF] and [START_REF]Formes linéaires en deux logarithmes et déterminants d'interpolation[END_REF] for "usual" logarithms, and [START_REF] Kunrui | Linear forms in elliptic logarithms[END_REF] for elliptic logarithms.

The paper [START_REF] Ramachandra | A note on Baker's method[END_REF] provides the first lower bound for simultaneous linear forms in logarithms. This was done at a very early stage of Baker's method, and the estimate has been superseded now, but the interest lies in the idea of improving the bound by considering several simultaneous linear forms; also the suggestion that the simultaneous result should allow one to conjecture a stronger result for a single linear form turned out to be correct.

The subject has been developed more recently in [START_REF] Loxton | Some problems involving powers of integers[END_REF][START_REF] Philippon-Waldschmidt | Formes linéaires de logarithmes simultanées sur les groupes algébriques commutatifs[END_REF] and [START_REF] Hirata | Approximations simultanées sur les groupes algébriques commutatifs[END_REF]. Using the same argument as in [START_REF] Ramachandra | A note on Baker's method[END_REF], one might consider that these work give partial evidence towards the Lang-Waldschmidt Conjectures (see the Introduction to Chapters X and XI of [START_REF]Elliptic Curves Diophantine Analysis[END_REF]). c) An easy transcendence measure for e When θ is a complex transcendental number, a transcendence measure for θ is a lower bound for |P (θ)| when P is a non-zero polynomial with rational integer coefficients. Such a lower bound should depend on the degree of the polynomial P as well as on the height H(P ) of the same (we consider the so-called "usual height", namely the maximum absolute value of the coefficients of P ). The very first transcendence measure goes back to the nineteenth century: Borel gave a transcendence measure for e in 1899. Later Popken (1929), Mahler (1932), Fel'dman (1963), Galochkin (1972), Cijsouw (1974), Durand (1980), [START_REF] Ramachandra | An easy transcendence measure for e[END_REF], Khassa and Srinivasan (1991) as well as other authors gave further transcendence measure for the same number e. We quote here the result of [START_REF] Khassa-Srinivasan | A transcendence measure for e[END_REF] which rests on Ramachandra's method in [START_REF] Ramachandra | An easy transcendence measure for e[END_REF].

For every positive integer n there exists a constant H 0 which depends only on n such that for each positive integer m, each positive real number H ≥ H 0 and each non-zero polynomial P ∈ Z[X] of degree ≤ n, with at most m non-zero coefficients, and with height H(P ) ≤ H,

|P (e)| ≥ H

-m -cmn log(m + 1) log log H .

An interesting feature of this statement is that it takes into account the number of non-zero coefficients of the polynomial in place of the degree; such an idea appears for instance in some works dealing with complexity theory; in Diophantine approximation it occurs in papers connected with Lehmer's Conjecture; it seems it never occurred before in connection with transcendence measures.

Open problems

We have already seen a few unsolved questions, notably the four exponentials Conjecture and Ramachandra's δ-Conjecture. Here are further unsolved questions.

a) Algebraic independence

Assume x 1 , . . . , x d are Q-linearly independent complex numbers, and y 1 , y 2 , y 3 are Q-linearly independent complex numbers; the six exponentials Theorem shows that at least d -1 of the numbers

e xiyj , ( 1
≤ i ≤ d, j = 1, 2 , 3)
are transcendental. A natural question is to ask whether at least d -1 of these numbers are algebraically independent. This amounts to ask if the transcendence degree of the field generated by these 3d numbers is at least d -1. This problem was raised in [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF]. The same argument was reproduced in [START_REF] Waldschmidt | Indépendance algébrique des valeurs de la fonction exponentielle[END_REF] as follows: according to the four exponentials Conjecture, at least d -1 of the numbers

e xiyj , ( 1
≤ i ≤ d, j = 1, 2)
are expected to be transcendental; is-it true that at least d -1 of these numbers are algebraically independent? Surprisingly enough, the answer to these questions is no! For instance take y 1 = 1, y 2 = β, y 3 = β 2 , where β is cubic (resp. One way of repairing the conjecture (see [START_REF] Waldschmidt | Nombres transcendants[END_REF] conjecture 7.5.5 and exercice 7.5.b) is to assume that y 1 , y 2 are linearly independent over the field of algebraic numbers. A better view of looking at this kind of problem from a conjectural point of view is to consider Schanuel's Conjecture [START_REF]Introduction to Transcendental Numbers[END_REF] Chapter 3 p. 30.

The first results of algebraic independence in this direction are due to Gel'fond [START_REF] Gel | Transcendental Number Theory[END_REF]: if ℓd ≥ 2(ℓ + d), then the transcendence degree t of the field generated by the dℓ numbers e xiyj , (1

≤ i ≤ d, 1 ≤ j ≤ ℓ)
is at least 2. Gel'fond's statement involved a so-called "technical hypothesis" (measure of linear independence for the x i 's, and also for the y j 's), which was removed later by Tijdeman (see [START_REF] Waldschmidt | Nombres transcendants[END_REF] Chapitre 7). In the early 70's, W.D.Brownawell and A.A.Smelev succeeded to prove t ≥ 3 under suitable assumptions; in 1974, Chudnovsky obtained 2 t ≥ ℓd/(ℓ+d); references are given in [START_REF] Chudnovsky | Contributions to the theory of transcendental numbers[END_REF]. P.Philippon reached the estimate t + 1 ≥ ℓd/(ℓ + d) in [START_REF] Philippon | Critères pour l'indépendance algébrique[END_REF]; this was improved by G. Diaz in [START_REF] Diaz | Grands degrés de transcendance pour des familles d'exponentielles en plusieurs variables[END_REF] as t ≥ ℓd/(ℓ + d) provided that ℓd > ℓ + d (without the proviso, the four exponentials Conjecture would follow!). An interesting fact is that all these results on "large transcendence degree" always involve a "technical hypothesis"; it is an open problem to remove it.

The above mentioned theorems of algebraic independence deal with the usual exponential function; here again, extensions can be given to commutative algebraic groups; we only quote [START_REF] Brownawell-Kubota | The algebraic independence of Weierstrass functions and some related numbers[END_REF] and [START_REF] Waldschmidt | Algebraic independence of values of exponential and elliptic functions[END_REF] which include extensions to results of algebraic independence of Ramachandra's transcendence results concerning the exponential and elliptic functions. b) Schneider's second problem in [START_REF] Schneider | Einführung in die transzendenten Zahlen[END_REF] Ramachandra's method might be the right way towards a solution of the second of Schneider's eight problems in [START_REF] Schneider | Einführung in die transzendenten Zahlen[END_REF]: to prove Schneider's Theorem on the transcendence of j(τ ) for τ an algebraic number in the upper half plane by means of the modular function (and not by mean of elliptic functions). The best known results in this direction are in [START_REF] Wakabayashi | Algebraic values of functions on the unit disk[END_REF]. c) Linear independence of elliptic logarithms in the non CM case by Schneider's method

We already quoted the paper [START_REF] Kunrui | Linear forms in elliptic logarithms[END_REF] where lower bounds for linear forms in elliptic logarithms are provided, by means of a method which is closely related to [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF]; as a matter of fact, an assumption is needed: namely one assumes that the elliptic curves has non trivial endomorphisms (CM case). It is not known how to extend the method to the non-CM case.

d) Effective results

Ramachandra was concerned (see top of p. 67 in [START_REF] Ramachandra | Contributions to the theory of transcendental numbers (I)[END_REF]) by the fact that his simplification of Schneider's method might be at the cost of making the proof ineffective in questions of transcendence measures. A quarter of a century later, we know that effectivity is not lost by avoiding derivatives. The earliest work in this direction is [START_REF] Srinivasan | On algebraic approximations to 2 π k (k = 1[END_REF]; further developments have already been quoted ( [START_REF] Mignotte-Waldschmidt | Linear forms in two logarithms and Schneider's method Math[END_REF] and [START_REF] Waldschmidt | Linear independence of logarithms of algebraic numbers[END_REF] for instance). However it is clear that a lot of work is still to be performed in this direction, and plenty of results are waiting to be unraveled by future generations of mathematicians.

Recent references

We give only a few references to papers or books which have been published during the last 10 years: [START_REF] Diaz | La conjecture des quatre exponentielles et les conjectures de D. Bertrand sur la fonction modulaire[END_REF][START_REF] Nesterenko-Philippon | Introduction to algebraic independence theory[END_REF][START_REF] Waldschmidt | Integer valued functions on products[END_REF][START_REF] Fel'dman-Nesterenko | Number theory. IV. Transcendental Numbers[END_REF][START_REF] Waldschmidt | Diophantine Approximation on Linear Algebraic Groups[END_REF][START_REF] Roy | An arithmetic criterion for the values of the exponential function[END_REF][START_REF] Diaz | Utilisation de la conjugaison complexe dans l'tude de la transcendance de valeurs de la fonction exponentielle[END_REF] and [START_REF] Waldschmidt | Variations on the Six Exponentials Theorem; with an appendix by H. Shiga: Periods of a Kummer surface[END_REF]. They contain further references to related works.

Example 3 .

 3 δ e z , ℘(tz) Corollary 3. -Let λ 1 , . . . , λ ℓ be complex numbers which are linearly independent over Q such that the ℓ numbers e λj ,(j = 1, . . . , ℓ) are algebraic. Let ℘ be a Weierstrass elliptic function with algebraic invariants g 2 , g 3 , and let v 1 , . . . , v ℓ be pseudo-algebraic point of ℘, not all of which are zero. a) If ℓ ≥ 4, then the matrix with 2 rows and ℓ columns λ 1 . . . λ ℓ u 1 . . . u ℓ Ramanujan Mathematical Society has rank 2. b) Assume λ 1 is a rational multiple of 2πi, and u 1 is a period of ℘; then the same conclusion holds for ℓ = 3. c) If Ramachandra's δ-Conjecture δ e z , ℘(tz) ≤ 1 holds, the same conclusion is valid already for ℓ ≥ 2.

has rank 3 .

 3 An obvious necessary condition is that for all (a, b) ∈ Z 2 \ {0}, at least two of the ℓ numbers a log α 1 + b log β 1 , . . . , a log α ℓ + b log β ℓ

Lemma 1 .

 1 -Let Ω = Zω 1 + Zω 2 be a lattice in C; let θ ∈ C × be such that θΩ ∩ θΩ is a subgroup of finite index in θΩ. Let Y be a finitely generated subgroup of C. Define two subgroups of C by Y θ = θyθy; y ∈ Y

  we show that there exist a positive integer k and an element ω ∈ Ω such that kv = θω -θω. Since v ∈ Ω θ there exist ω ∈ Ω and ω ′ ∈ Ω such that v = θωθω ′ . It follows from the previous result in b) that there exist integers h, a, b, c, d with h ≥ 1 such that hω = aω 0 + bω ′ 0 and hω ′ = cω 0 + dω ′ 0 . We deduce hv = (a -c)θω 0 + (b + d)θω ′ 0 . From the hypothesis v = -v we conclude a = c, and the result follows with k = 2h and ω = (b + d)ω ′ 0 . d) Define a linear form ϕ : C → R of R-vector spaces by

  y 1 = 1, y 2 = β, where β is quadratic) if one wishes to use the six exponentials Theorem (resp. the four exponentials Conjecture), and choosed = 3k, {x 1 , . . . , x d } = {log α j , β log α j , β 2 log α j ; (1 ≤ j ≤ k)} (resp. d = 2k, {x 1 , . . . , x d } = {log α j , β log α j ; (1 ≤ j ≤ k)}),where α 1 , . . . , α k are multiplicatively independent algebraic numbers; the transcendence degree is at most 2k = 2d/3 (resp. k = d/2).

  Product of the multiplicative group with an elliptic curve Let ℘ be a Weierstrass elliptic function with real algebraic invariants g 2 , g 3 :

	are Q-linearly independent, which means that for each (λ, µ) ∈ Q 2 , the determinant  log 5 log 2 
		det	 log	3 + 4i 5	+ 2λπi 2µπi 
	does not vanish. This is not yet known.
	b)
	If the four exponential Conjecture holds, then for instance the subgroup
	of rank 2
			(2 + i) s 2 t ; (s, t) ∈ Z 2 ,
	generated by 2 and 2 + i is dense in C × ; a proof of this result would follow from a
	special case of the four exponentials Conjecture: it would suffice to show that the
	three numbers	log 2, log 5,	log 2 2πi	• log	3 + 4i 5
		Ramanujan Mathematical Society

Ramanujan Mathematical Society


G Derenty-Camenen

F Tavernier

O Chadebec

O Pinaud

L-L Rouve

S Zozor

R Scorretti

Low frequency numerical dosimetry of electric field by using multipolar sources defined on an arbitrary surface

Keywords: Finite Element method, inverse problem, multipoles, numerical dosimetry

Computation of low frequency induced electric fields in the human body requires the knowledge of the source field. The source field can be reconstructed by measuring the magnetic field over a regular grid, or by identifying a model, among which multipoles. Spherical multipoles are easy to compute, but cannot be used close to the source. In this work it is proposed to use a new multipolar model which holds outside an arbitrary geometry -not limited to a sphere.

I. INTRODUCTION

C OMPUTATION of induced electric field inside the human body is crucial to assess the respect of regulation [START_REF]Directive 2013/35/UE of the European Parliament and of the Council of 26 June 2013 on the minimum health and safety requirements regarding the expo[END_REF]. In the case of exposure to low frequency magnetic fields, it is natural to use scalar potential formulations, where the source term is given through a magnetic vector potential a 0 . Whatever the numerical method, the formulation writes:

div[-σ(grad v)] = div[σ(jωa 0 )] ( 1
)
where σ is the conductivity of tissues [START_REF] Gabriel | The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues[END_REF], the magnetic flux density b = curl a 0 is known through a 0 (the term from eddy currents is neglected) and the electric field writes e = -jωa 0grad v. The formulation is solved on the human body only, with boundary condition n • j = 0.

The determination of the source term a 0 can be done by using several approaches. Direct computation is possible only if a model of the source is available, which is seldom the case. Most often, the magnetic field is measured on a regular grid containing the human body. Then a vector potential can be computed by using so called uncurl methods [START_REF] Laakso | Modelling of induced electric fields based on incompletely known magnetic fields[END_REF], [START_REF] Conchin Gubernati | Comparison of Numerical Techniques for the Evaluation of Human Exposure From Measurement Data[END_REF]. By using this approach, a large number of measurement points are required, and the body cannot be displaced outside the volume defined by the measurement grid. In a recent work [START_REF] Tavernier | Realtime numerical dosimetry of low-frequency electromagnetic fields by using multipoles[END_REF], it has been proposed to model the source by using spherical multipoles. This allows to recompute quickly the source term a 0 for any position of the human body, but it has the major drawback that the body cannot lay inside a sphere centered on the source. This hinders using this method in proximity of the source -that is, precisely where the field is the more intense!

In this work we propose a model of (non-spherical) multipolar source which is defined on an arbitrary surface, and therefore aims at overcoming the limitations of [START_REF] Tavernier | Realtime numerical dosimetry of low-frequency electromagnetic fields by using multipoles[END_REF].

II. MATERIALS AND METHODS

A. Multipolar source

Let S be an arbitrary surface which encloses all the sources. It is possible to find an equivalent double-layer potential τ 0 which generates exactly the same field as the source in the space comprised between S and the Brillouin sphere [START_REF] Stratton | Electromagnetic theory, mcgrow-hill book company[END_REF], which is the smallest sphere which encloses the source.

We generate a basis for this double-layer potential by using the procedure described in [START_REF] Derenty-Camenen | Multipolar bases of equivalent magnetic sources on arbitrary surfaces[END_REF]. Hence τ 0 writes:

τ 0 (x) = ∞ k=1 k m=-k b km τ km (x) (2)
where τ km (x) are the terms of the basis of order k and degree m, b km are the coefficients of the development and x the points of the surface S. By construction, the following gauge condition is enforced:

S τ km (x)dx = 0 (3)
Outside the Brillouin sphere, the field generated by each distribution τ km decreases at least as fast as the one of a multipole of the same order. In order to make the problem computationally tractable, we truncate the basis up to a given order N , so that we obtain an approximate distribution τ :

τ (x) = N k=1 k m=-k b km τ km (x) ≃ τ 0 (x) (4)
In principle when τ 0 is known, the coefficients b km can be found by projecting τ 0 on the basis. The parameters can be identified via a model of the source. In practice, an inverse problem has to be solved from available experimental data.

The vector potential a is computed from τ at any point as [START_REF] Durand | Magnétostatique[END_REF]:

a 0 (x) = S (grad S τ (y)) × n G(x, y)dy. ( 5
)
where n is the outward normal to the surface S, and G(x, y) is the Green's function, which in 3D writes:

G(x, y) = 1 4π 1 ∥x -y∥ . (6)

B. Computational phantom

The computational phantom used in this work is Alvar [START_REF] Laakso | Alvar[END_REF]. Alvar is a voxel phantom based on freely available online anatomic atlas, BodyParts3D (http://lifesciencedb.jp/bp3d). It composed of 86 different tissues. It has a native resolution of 0.5 mm, but in this preliminary work it has been undersampled by a ratio 1:8 so as to achieve a resolution of 4 mm.

C. Dosimetric computation

Equation 1 has been solved with the Finite Element method by using a home-made software written in Matlab. The linear system is composed of 1'080'453 unknowns, and it is solved by using the AGMG algebraic multigrid solver [START_REF] Notay | AGMG software and documentation[END_REF].

III. RESULTS

In this preliminary test, a fictive source is enclosed in a box of dimensions 0.3 × 0.3 × 1 m located left to the human body. The chosen distribution of double-layer potential τ (x) corresponds to a dipole along the z axis. In the full paper, the considered source will be a wireless energy transfer system. The surface of the box is meshed with 624 triangles. On this mesh, τ (x) is represented by linear nodal shape functions.

The computed electric field is depicted at the surface of the human body in Figure 1. It is observed that the induced electric field is higher on the side close to the source, represented by a dashed box. For each tissue we computed the average, maximum and 99% percentile values of the electric field. These values are depicted in Figure 2 for some representative tissues (full data will be available in the extended article).

IV. CONCLUSION

It is demonstrated that by using non-spherical multipolar sources it is possible to simulate the exposure to a source of magnetic field located any position, including in proximity of the human body. Details on the validation and on the accuracy of the computation with a more realistic source will be presented in the extended article. Moreover, the multipolar source will be identified by solving an inverse problem.


Maxime Fays

Olivier Chadebec

Brahim Ramdane

Isogeometric mesh refinement through Non-conforming C 0 Isogeometric Mortaring

Keywords: Isogeometric Analysis, Mortar, Refinement

An original mesh refinement of Isogeometric model through C 0 isogeometric mortaring is proposed in this paper to address patch refinement propagation occuring when handling multipatch geometries, a particularly relevent side-effect in electromagnetic problems. The efficency of the proposed method is demonstrated on an academic problem.

I. INTRODUCTION

I SOGEOMETRIC analysis (IGA) is a numerical method created in 2005 [START_REF] Hughes | Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinemen[END_REF], to overcome the Finite Element Method (FEM) geometric deterioration arising during the meshing process. When designing a Computer Aided Design (CAD) model, geometries are largely described with generalized B-Splines called NURBS (rational combination of piece-wise polynomials), with exact precision, rather than FEM-suitable mesh geometries directly. In FEM, this exact representation is projected onto triangles/quadrangles, with a loss of information (curvature, normals, ...), and shape functions are defined on the mesh, completely separating the initial geometry and the functional space. In contrast, IGA uses the polynomials that generated the geometry as shape functions. In addition, as the NURBS are piece-wise polynomials, the support of each individual polynomial can be used to define a parametric mesh on the model, on which numerical integration can be performed with standard integration routines (quadrature, collocation, ... ). The geometry is then carried during the numerical resolution, and preserved. On simple geometries, a single NURBS is enough for a full model description, and refining the parametric mesh is easy, but on more complex geometries, a multipatch approach is needed, consisting in the description of the model with multiple NURBS. This approach is often used even for simply connected domains, leading to continuity issues at patch interfaces. For Electromagnetic applications, ensuring the C 0 continuity of the quantity of interest at patch interfaces is enough in general, and it can be done in the strong sense, by modifying the piecewise polynomials that have support at the interface, but force the parametric mesh to be conformal (i.e, the parametric mesh must match) at the interface. This side effect is very problematic: To generate a n-dimensional NURBS, a tensor product of NURBS of degree n -1 is commonly used. This approach is greatly appreciated in IGA as vectorization along the the n -1 dimensions is possible, greatly improving computation times. But forcing the parametric mesh to match on the interfaces propagates the mesh refinement from one patch to another, leading to an unwanted increasing number of degrees of freedom for the overall problem. This is especially true for electromagnetic problems, where the regions of interest can be local such as air gaps in electric machines. Solutions are emerging to avoid the refinement propagation problem and preserve a strong C 0 continuity: T-Spline, Hierarchical splines, ... as presented in [START_REF] Schillinger | An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces[END_REF], but they all loose the tensor product vectorization aspect, as they modify the parametric model representation.

To solve this problem, a Mortar approach can be used, to weakly enforce C 0 continuity (i.e to work with non-conformal parametric meshes at patche interfaces). The tensor product structure will then be preserved per-patch, and vectorisation will be possible, at the cost of a coupling matrix at each patch interface. In this paper, we will briefly introduce Isogeometric analysis, and the mesh refinement propagation issue, we will then explain the Isogeometric mortar method and its application to patchwise mesh refinement, and finally present numerical results to assess the efficiency of the proposed solution.

II. ISOGEOMETRIC ANALYSIS

A. NURBS

Non-Uniform Rational B-Spline (NURBS) curves are generated as a sum of n rational basis functions (R i,p ) , i=1,...,n , of degree p, and n control points P i :

C(ξ) = n i=1 R i,p (ξ)P i (1)
The rational basis functions are constructed as a rational sum of weighted elements of a Spline basis w j (N j,p ) j=1,..,n :

R i,p (ξ) = N i,p (ξ)w i n j=1 N j,p (ξ)w j (2)
And the Spline basis is constructed with a knot vector

Ξ = [ξ 1 , ..., ξ m ] with 0 = ξ 1 ≤ ξ i ≤ ξ i+1 ≤ ξ m = 1, ∀i ∈ 1, m
the n elements of the spline basis functions of degree p, (N j,p ) j=1,..,n are given by the recurrence formula:

N j,0 (ξ) = 1, if ξ j ≤ ξ < ξ j+1 0, otherwise. N j,p (ξ) = ξ -ξ j ξ j+p -ξ j N p-1 j (ξ) + ξ j+p+1 -ξ ξ j+p+1 -ξ j+1 N p-1 j+1 (ξ). ( 3
)
Tensor products of NURBS Curve can then be used to model 2D and 3D geometries, and any NURBS curve can be refined to refine the 2D/3D geometry as illustrated in figure 1. Isogeometric Analysis then consists in the use of such geometries using the spline basis illustrated in figure 1 as Galerkin projection functions, and, by construction, guarantee up to a C p-1 global continuity during the resolution.

B. Refinement propagation

As illustrated in the previous section, the refinement of a NURBS surface (patch) obtained through tensor products propagates across the whole patch. When considering multiple patches, strong C 0 continuity at interfaces can be achieved only if the shape functions are identical along the patch interfaces, which imposes the same level of refinement on every adjacent patches. Figure 2 illustrates this on a specific problem: a FEM-BEM modeling of a 3dimensional actuator. Although the region of interest is the air gap between Patches 1,2 and 1,6, the refinement propagates, and the increase in DoFs outside of the region of interest is substantial (patches 3,4,5).

a(u, v) = L(v), ∀v ∈ H 1 (Ω) (4)
Where a is continuous on (H 1 x H 1 ) , and billinear, and L is a linear form. The Mortar method consist in transforming 4 into a saddle point problem, and the problem reads: Find u ∈ H 1 and λ ∈ M such that a(u, v)

+ Ω1∩Ω2 λ(v |Ω 1 -v |Ω 2 ) = L(v) ∀v ∈ H 1 (Ω) (5) Ω1∩Ω2 τ (u |Ω 1 -u |Ω 2 ) = 0 ∀τ ∈ M(Ω) (6)
Where M is a set of Lagrange multipliers, constructed by picking "arbitrarily" one of the two patch as the slave patch (often the most refined one), and modifying the rational basis at the interface (and especially at the edges of the interface) Ω 1 ∩ Ω 2 as described in [START_REF] Brivadis | Isogeometric mortar methods[END_REF]. If Ω 2 is chosen as the slave patch, integrating Ω1∩Ω2 λv |Ω 1 and Ω1∩Ω2 τ u |Ω 1 is complicated as both quantities are expressed with rational basis that don't share their support. More information will be provided in the extended paper as this integration is a key point to allow nonconforming patches integration and non conforming interfaces (more complicated) as illustrated in 3 Fig. 3. Solution of a Laplace equation using a Mortar approach on non conforming patches and non conforming interfaces.

IV. VALIDATION AND RESULTS

The efficiency of the approach is demonstrated on the nonconformal patches geometry presented in figure 3, where an L2 Error is computed between the proposed approach and a reference solution:

An application of this method to the FEM-BEM problem mentioned in 2 will be presented in the extended paper as Boundary Element Method benefit hugely from a reduction in degrees of freedom, aswell as more details regarding the numerical integration of the mortar integrals.

Fig. 1 .

 1 Fig. 1. Left: Tensor product of two NURBS Curves (Grey), their associated polynomial basis, to assemble the NURBS Surface (Green). Right: Refinement of one of the NURBS Curves via Knot vector insertion, and the resulting tensor surface.

Fig. 2 .

 2 Fig. 2. Left: Invalid C 0 patch gluing for a 3D actuator geometry. Patches 2,3 and 5,6 are not compatible. Right: Minimal patch refinement required for the C 0 patch gluing, increasing the degree of freedom outside of the region of interest III. ISOGEOMETRIC MORTARING To overcome this drawback, imposing weakly the C 0 continuity across patch interface is possible using a Mortar method. Given a domain Ω, and two patches Ω 1 , Ω 2 such that Ω 1 ∩Ω 2 ̸ = ∅ and Ω 1 ∪ Ω 2 = Ω, we consider a very general problem: Find u in H 1 (Ω) such that :


Catherine Baskiotis

Josselin Aval

Marwa El Bouz

Ayman Al Falou

SELECTING HYPERSPECTRAL BANDS FOR LEAF MASS PER AREA PREDICTION BY MEANS OF NEURAL NETWORKS

Keywords: Leaf Mass per Area, hyperspectral data, band selection, neural-networks, vegetation index

Accurate prediction of Leaf Mass per Area (LMA) from leaf-level hyperspectral reflectance measurements is fundamental for plant health watching.

Limiting the number of required bands to a very low number while maintaining a good accuracy of LMA prediction is of very high interest for cost-effective implementations. For such purpose, the best current model consists in a combination of vegetation indices and have been obtained by performing computationally intensive brute force approach. Further accuracy improvements are, at present, limited by current computational capacity limit.

We propose to use a neural network-based methodology for selecting a very low number of hyperspectral bands for further improving the LMA prediction accuracy. Testing on the publicly available Angers dataset, we diminished the Root Mean Square Error of prediction by 26% and 19% as compared to the best literature vegetation index and combination of vegetation indices, respectively.

INTRODUCTION

Leaf Mass per Area (LMA) is a functional trait whose precise knowledge is fundamental for numerous plant health monitoring activities such as, for example, growth plant analysis, drought detection or nutrient stress detection [START_REF] Poorter | Causes and consequences of variation in leaf mass per area (LMA): a meta-analysis[END_REF].

Despite the possibility of using destructive bio-chemical methods for accurate LMA determination, the only route for cost-effective and intensive LMA watching consists in nondestructive methods, such as hyperspectral remote sensing.

Numerous current hyperspectral remote sensing methods are employing more than 100 entry bands for LMA prediction, such as the PROSPECT physical model [START_REF] Féret | Estimating leaf mass per area and equivalent water thickness based on leaf optical properties: Potential and limitations of physical modeling and machine learning[END_REF] or Machine Learning models [START_REF] Féret | Estimating leaf mass per area and equivalent water thickness based on leaf optical properties: Potential and limitations of physical modeling and machine learning[END_REF]. Reducing the number of required bands is of very high interest for cost reduction. For such purpose, vegetation indices are the simplest approach but suffers from a lack of precision [START_REF] Le Maire | Calibration and validation of hyperspectral indices for the estimation of broadleaved forest leaf chlorophyll content, leaf mass per area, leaf area index and leaf canopy biomass[END_REF][START_REF] Chen | Optimized Estimation of Leaf Mass per Area with a 3D Matrix of Vegetation Indices[END_REF].

Current most precise vegetation indices have been developed by a brute force approach, consisting in extensive one by one try/catch explorations of numerous band combinations [START_REF] Le Maire | Calibration and validation of hyperspectral indices for the estimation of broadleaved forest leaf chlorophyll content, leaf mass per area, leaf area index and leaf canopy biomass[END_REF][START_REF] Chen | Optimized Estimation of Leaf Mass per Area with a 3D Matrix of Vegetation Indices[END_REF]. At present, the further improvements of vegetation indices are, therefore, limited by current (c) Indicative reflectance spectrum of one leaf.

computational capacity limit.

Other approaches for LMA prediction from only a few bands, founded on numerical methods, such as Partial Least Square Regression (PLSR), have been tested but have shown to be of poorest accuracy as compared to current vegetation indices [START_REF] Chen | Optimized Estimation of Leaf Mass per Area with a 3D Matrix of Vegetation Indices[END_REF].

We propose a new neural network-based methodology for selecting a very few bands in the purpose of LMA prediction and demonstrate that using the selected bands as entries of a neural network allows to reach better precision than current literature vegetations indices.

MATERIAL

We consider the Angers publicly available database acquired in Angers (France) in 2003 [5,[START_REF] Féret | PROSPECT-4 and 5: advances in the leaf optical properties model separating photosynthetic pigments[END_REF]. This database contains N=276 leaves of 49 different dicotyledon species. One specie: Sycamore Maple, with 181 samples, is representing 65% of the total leaves number.

The LMA was measured by extracting small disks of fresh leaves, immediately weighting them, then drying them in an oven and immediately weighting them at the withdraw out of the oven [START_REF] Féret | Estimating leaf mass per area and equivalent water thickness based on leaf optical properties: Potential and limitations of physical modeling and machine learning[END_REF]. LMA covers a large range of values from 1.66 mg/cm 2 to 33.10 mg/cm 2 (cf. figure 1.(a-b)).

The available leaf reflectance measurements span over the wavelength range [400nm, 2450nm], cf. figure 1.(b-c).

EVALUATION PROTOCOL AND METRICS

We divided the dataset into two non-overlapping subsets: the calibration and the test subset with, respectively 207 samples and 69 samples (75% and 25% of the dataset). The calibration subset was used to construct the models, while the test subset was used to evaluate the models.

We assessed the accuracy of the fit and the prediction on respectively the calibration and test subset through the Root Mean Square Error (RMSE) [START_REF] Le Maire | Calibration and validation of hyperspectral indices for the estimation of broadleaved forest leaf chlorophyll content, leaf mass per area, leaf area index and leaf canopy biomass[END_REF].

We performed 20 evaluations randomly drawings the calibration and test subsets. We determined, for each of the constructed model, the mean RMSE obtained over all the drawings on the calibration and test subsets. We used feed-forward neural networks with a number of hidden layers comprised in between 2 and 4. We used, respectively, the "tanh" and the "sigmoid" function as transfer function for, respectively, the hidden layers and the output layer. The trainings were performed using the backpropagation procedure [START_REF] Goodfellow | Deep Learning[END_REF].

Overfitting management

To avoid the over-fitting phenomenon, we further divided the calibration subset into training and validation subset with 138 samples and 69 samples, respectively [START_REF] Goodfellow | Deep Learning[END_REF].

The parameters of the neural network (namely the weight and bias of each neuron) were assessed by minimizing the RMSE on the training subset. The hyperparameters (number of hidden layers, number of neurons on each layer, number of epochs, batch and learning rate) were chosen by minimizing the mean value between the RMSE on the training subset and the RMSE on the validation subset (tuning RMSE). In particular, we choose the number of epochs with the early-stopping procedure [START_REF] Goodfellow | Deep Learning[END_REF].

Band selection scheme 4.3.1 Input weights definition

We firstly trained the network by using all the available hyperspectral bands. We assessed the relative weight Si,0 of each band i on the output of the neural network following the methodology proposed in [START_REF] Yacoub | Features selection and architecture optimization in connectionist systems[END_REF].

The figure 2.(a) presents an indicative Si,0 spectrum obtained, for one of the 20 subsets drawings. We observe several bands at which Si,0 exhibits a peak of relatively high value in a background where Si,0 is almost constant and very small. We called P peaks the set constituted by these bands.

Band selection algorithm

We followed the protocol described in [START_REF] Yacoub | Features selection and architecture optimization in connectionist systems[END_REF], which consists in the following backward elimination loop:

1. Train the network and stop the training by early stopping. 2. Compute Si,0 for all the input bands. We, then, as second step, performed a sampling of the remaining bands with various sampling steps. After congregating the sampled bands and the P peaks, we evaluated the tuning RMSE for the different sampling steps and selected the sampling step which exhibits the minimum tuning RMSE (cf. figure 2.(b)). It is worth noting that, due to the "curse of dimensionality" [START_REF] Goodfellow | Deep Learning[END_REF], a too small sampling step leads to a relatively high value for the tuning RMSE. We, thirdly, started, with the congregation of the P peaks and the S sampled bands, the band selection scheme described at the section 4.3.2 (step 3, cf. figure 2.(c)). When removing one by one the bands at the step 3, the tuning RMSE is keeping approximately the same value until reaching a given number of bands below which it increases dramatically (cf. figure 3).

We defined a stopping test for the step 3: does the suppression of one band implies an increase of the tuning RMSE greater than 5% as regard to its minimum value? In case of a positive answer the step 3 is stopped, and the remaining bands are selected.

Ultimate sets of bands

After performing the step 3, in the remaining set of bands, redundant bands may still be present [START_REF] Adil | Hybrid Method HVS-MRMR for Variable Selection in Multilayer Artificial Neural Network Classifier[END_REF]. To remove potential redundant bands, we applied a fourth step which consists in, first, testing the impact on the tuning RMSE of the removing of each of the remaining bands and then remove the band which produces the lower increase in RMSE. This step is ended with the same stopping criterion as those of the step 3.

At the end of the step 4, we obtain the ultimate set of bands which is specific to each of the 20 subsets drawings and contains from 3 to 9 bands.

For each ultimate set, the optimum neural network architectures were exhibiting 2 hidden layers with 6 to 10 neurons in the first hidden layer and 7 to 9 neurons in the second hidden layer.

We evaluated the mean RMSE over all the drawings on the training, validation and test subset (mean training, validation and test RMSE) using the ultimate sets and the optimum neural network architectures. . 4.4 Unifying the set of selected bands We constructed an unified set of bands by taking the four most frequent bands: 1720 nm, 1300 nm, 1400 nm and 400nm with an occurrence number of respectively 18, 16, 14 and 11 among the 20 subsets drawings.

VEGETATION INDICES REFERENCE

A vegetation index (VI) which predicts the LMA is a combination of the reflectance of the leaf at few different bands, chosen in such a way that the LMA can be approximated by a simple function (fitting function) of this combination [START_REF] Le Maire | Calibration and validation of hyperspectral indices for the estimation of broadleaved forest leaf chlorophyll content, leaf mass per area, leaf area index and leaf canopy biomass[END_REF][START_REF] Chen | Optimized Estimation of Leaf Mass per Area with a 3D Matrix of Vegetation Indices[END_REF][START_REF] Erudel | Criteria Comparison for Classifying Peatland Vegetation Types Using In Situ Hyperspectral Measurements[END_REF].

We tested 24 VI dedicated to LMA extracted from references [START_REF] Le Maire | Calibration and validation of hyperspectral indices for the estimation of broadleaved forest leaf chlorophyll content, leaf mass per area, leaf area index and leaf canopy biomass[END_REF][START_REF] Chen | Optimized Estimation of Leaf Mass per Area with a 3D Matrix of Vegetation Indices[END_REF][START_REF] Erudel | Criteria Comparison for Classifying Peatland Vegetation Types Using In Situ Hyperspectral Measurements[END_REF], which consist either in a single index (single VI) either in a combination of two or three indices (2-VI or 3-VI combination).

For the fitting function, we used a polynomial function of second degree in one dimension when fitting a single index and in two (or three) dimensions when fitting a combination of two (or three) indices. We do not observe any improvement when increasing the degree of the polynomial function.

For each of the 20 subsets drawings, we determined the polynomial function coefficients of each VI by fitting the calibration subset.

We selected the VI which leads to the lower mean test RMSE over the 20 drawings. Among, respectively, the single vegetation indices, the 2-VI combinations and the 3-VI combinations, we found out, respectively, MND ([R2285 -R1335]/[R2285 + R1335 -2 . R2400]), ND-R2300 (combination of ND=[R1368-R1722]/[R1368+R1722] and R2300) and MSR-ND-R2300 (combination of MSR=[R2265-R2400]/[R1620-R2400], ND=[R1368-R1722]/[R1368+R1722] and R2300).

RESULTS

The mean calibration and test RMSE obtained, over the 20 drawings, with the best vegetation indices and our neural network-based scheme are depicted in table 1 [START_REF] Chen | Optimized Estimation of Leaf Mass per Area with a 3D Matrix of Vegetation Indices[END_REF]. We explain this property by the fact that PLSR is restricted to the investigation of linear relationships between band reflectances and LMA, while neural networks include non-linear relationships in their exploration.

Moreover, we explain the superiority of our neural network-based scheme on literature vegetation indices by the fact that current literature vegetation indices have been built by investigating a very little number of relationship types between band reflectance and LMA [START_REF] Le Maire | Calibration and validation of hyperspectral indices for the estimation of broadleaved forest leaf chlorophyll content, leaf mass per area, leaf area index and leaf canopy biomass[END_REF][START_REF] Chen | Optimized Estimation of Leaf Mass per Area with a 3D Matrix of Vegetation Indices[END_REF], while the neural networks explore a dramatically wider space of possible functions linking band reflectances and LMA [START_REF] Goodfellow | Deep Learning[END_REF].

Subset of predictive bands

The bands used by each model are detailed in figure 4.(c). All the bands of best literature LMA-VI (whatever the number of indices they are combining) are belonging to Short Wave InfraRed (SWIR) domain. Moreover, one band close to the 1450 nm water peak absorption is systematically present in the best literature LMA-VI (1335nm, 1368nm and 1368nm for the best single VI, 2-VI combination and 3-VI combination, respectively).

Neural network selected band set and best literature VI (ND-R2300) band set share one band in common: 1720 nm. This band is close to an absorption peak of the lignin, the starch and the nitrogen [START_REF] Jacquemoud | Leaf Optical Properties[END_REF]. As compared to the band set used in the best literature VI (ND-2300), the neural network selected band set presents two specific characteristics: firstly, the presence of two bands (1300 nm and 1400 nm) and not only one band near the 1450 nm water absorption peak; secondly the presence of a band in the visible domain (400 nm) at which the absorption of pigments is high and the absorption of water and dry matter is low.

CONCLUSION

We propose to use a neural-network based methodology for selecting an over-restrained number of leaf-level hyperspectral reflectance bands for the prediction of LMA. This methodology takes accounts of the nonlinear relationship between the band reflectances and the LMA.

By performing tests in cross-prediction configuration on the Angers dataset, we found out that the neural-network based methodology allows to improve the RMSE of prediction by 26%, 19% and 21% as compared to current literature single VI, 2-VI combination and 3-VI combination vegetation indices. Very simple neural networks architectures with only two hidden layers are enough for achieving such result.

The eventually selected bands by our neural network scheme are: 1720nm, 1300nm, 1400nm and 400nm. Current works targeting to build our own measurements database are ongoing to further investigate the involvement of those bands in LMA modelling.

ACKNOWLEDGMENENTS

This work was funded by the Brittany Region (France) and the L@bISEN, an Yncrea Ouest laboratory (France).

Figure 1 :

 1 Figure 1: (a) Statistical repartition of LMA of the Angers dataset. (b) Table of characteristics of the Angers dataset.(c) Indicative reflectance spectrum of one leaf.

Figure 2 :

 2 Figure 2: (a) LMA Si,0 spectrum for an indicative random subsets drawing. (b) Tuning RMSE as a function of different sampling steps. (c) Summary of our band selection algorithm.

4 .

 4 NEURAL NETWORK BAND SELECTION 4.1 Choices for the neural network The hyperspectral bands were set to be the multiple inputs of a neural network, with one unique output: the LMA value of the sample under consideration.

3 .

 3 Remove the band with the lowest coefficient Si,0. 4. Iterate to step 1 until reaching one band only. 4.3.3 Si,0 spectrum peaks and sampled bands congregation Following the above-detailed algorithm, we should suppress one by one the bands starting by the whole reflectance spectrum (2050 bands). To save computing time, we decreased the number of starting bands by first seleting the P peaks (step 1 of our algorithm, cf. figure 2.(c)).

Figure 3 :

 3 Figure 3: Details of the step 3 for an indicative random subsets drawing.

Figure 4 :

 4 Figure 4: (a-b) Predicted to measured LMA values comparison (for the samples of the test subset) of an indicative random subsets drawing for the best literature vegetation index (a) and our neural network-based scheme (b). (c) Bands used by the different models.

  Table of characteristics of the Angers dataset.

Table 1 :

 1 . The figure 4.(a-b) presents the predicted to measured LMA values comparison for the samples of the test subset of an indicative random subsets drawing.It is worth noting that the best literature VI is the 2-VI RMSE in mg/cm 2 obtained with the best literature vegetation indices and neural networks.

	Model	Calibration	Test	Nb. of bands
	Single VI	1.15	1.25	3
	2 VI-combination	0.97	1.14	3
	3 VI-combination	0.94	1.16	6
	Neur. Net. (ultimate sets)	0.77	0.92	3-9
	Neur. Net. (unified set)	0.79	0.94	4