Abdeljalil Saghe
email: saghe007@gmail.com

Solving a quartic equation and certain equations with degree n

Keywords: Quartic equation, cubic equation, Ferrari's method

est destinée au dépôt

Introduction

From the results of Galois (1832) [START_REF] Bourbaki | Éléments d'histoire des Mathématiques[END_REF] and Abel (1826) [START_REF] Bourbaki | Éléments d'histoire des Mathématiques[END_REF], we know that a general equation with degree n can not be solved in radicals. Now, we can solve every quartic by using the Ferrari's method [START_REF] Flament | Histoire des nombres complexes[END_REF] which published in 1545. In this paper we propose a new method of solving a general quartic which is different from the Ferrari's method, and we can determine certain equations of degree n ≥ 5 solvable by the proposed method.

Preliminary results

Lemma 1 Let P be a polynomial function with degree n ≥ 1 in C[x] such that P (x) = a n x n + a n-1 x n-1 + ... + a 1 x + a 0 If α is a root of P , and λ is a complexe number, then λα is a root of λ * P defined as

(λ * P )(x) = a n x n + a n-1 λx n-1 + a n-2 λ 2 x n-2 ... + a 1 λ n-1 x + a 0 λ n Lemma 2 Every equation of degree n ≥ 1 as a n x n + a n-1 x n-1 + ... + a 1 x + a 0 = 0 can be written as y n + b n-2 y n-2 + ... + b 1 y + b 0 = 0 Proof:
We can use this change of variable : y = x + a n-1 na n .

Notation : Now we consider this notation : √ γ is a root of z 2 = γ.

Ferrari's method

By lemma 2, all quartic equation can be reduced to a simple form with only tree parameters. That is,

(E) X 4 + T X 2 = ZX + W
where T, Z and W are real numbers. This equation is equivalent to :

(E) X 4 + T X 2 -ZX -W = 0
Let y ∈ R * , we have :

X 4 + T X 2 -ZX -W = (X 2 + y) 2 -2yX 2 + T X 2 -y 2 -ZX -W = (X 2 + y) 2 -2yX 2 + T X 2 -y 2 -ZX -W = (X 2 + y) 2 + (T -2y)X 2 -ZX -y 2 -W
We can choose that value of y = y 0 when this term (2y -T )X 2 + ZX + y 2 + W is a square, for that his discriminant is zero.

We get :

Z 2 -4(y 2 0 + W )(2y 0 -T ) = 0
Then we deduce that y 0 is a root of this cubic equation :

8y 3 -4T y 2 + 8W y -Z 2 = 0
Remark 1 To get the value of y 0 , we can apply one of methods for solving the cubic equation exmaple : Cardan's method [START_REF] Flament | Histoire des nombres complexes[END_REF] and [START_REF] Bourbaki | Éléments d'histoire des Mathématiques[END_REF])

Kulkarni's method

This method was published in 2009 [START_REF] Kulkarni | A new method for solving quartics[END_REF] ( rectified in 2011 [START_REF] Cartwright | Correction to "A new method for solving quartics[END_REF]) is similar to Ferrari's method. It's based on the possibility of writing every quartic as the difference of two squares, for that we solve a system with 4 unknowns, and we obtained a new cubic equation. We can see the proof as follows Consider this general equation

x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 = 0
Consider another quartic equation shown below.

(x 2 + b 1 x + b 0 ) 2 -p(x + c) 2 = 0
We attempt to represent the first quartic in the form of the Latter. For that we obtained the system

     2b 1 = a 3 b 2 1 + 2b 0 -p = a 2 2(b 0 b 1 -pc 0 ) = a 1 b 2 0 -pc 2 0 = a 0 Let F 1 = a 2 -( a 3
2 ) 2 , and p 1 = p + F 1 . After calculus, we obtained this cubic equation with the unknown p 1 :

p 3 1 -a 2 p 2 1 + (a 3 a 1 -4a 0 )p 1 + 4a 0 F 1 -a 2 1 = 0
We find the values of p, c, b 1 , b 0 , and finally we get this decomposition :

x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 = [x 2 + (b 1 - √ p)x + b 0 - √ pc][x 2 + (b 1 + √ p)x + b 0 + √ pc]
5 The proposed method

We use the lemma 2 for rewriting every quartic in the form :

(E) X 4 + T X 2 = ZX + W
where T, Z et W are complexe numbers. Next, we can determine two reals a and b as a fuction to T ,Z and W , such that the X is a root of :

X 2 = aX + b
For that, consider the equation

x 2 = x + ε (1)
Then :

x 4 = x 2 + 2εx + ε 2 (2)
By using the equality (1) x 2 = x + ε, we replace x 2 by x + ε in the equation ( 2), we find :

x 4 = x(2ε + 1) + ε 2 + ε
Then :

x 2 = x + ε x 4 = x(2ε + 1) + ε 2 + ε
Let α ∈ R, we have:

αx 2 = αx + αε x 4 = x(2ε + 1) + ε 2 + ε
We add the two equations, and we get :

x 4 + αx 2 = x(2ε + 1 + α) + ε 2 + ε + αε
Let λ ∈ R and X = λx, by using the lemma 1, we deduce that the X is a root of the equation :

X 4 + αλ 2 X 2 = X(2ελ 3 + 3 + αλ 3 ) + ε 2 λ 4 + ελ 4 + αελ 4
To solve the quartic (E), we can represent it in the form of the above quartic, for that we can solve this system :

(S) :

   T = αλ 2 (1 ′ ) Z = 2ελ 3 + λ 3 + αλ 3 (2 ′ ) W = ε 2 λ 4 + ελ 4 + αελ 4 (3 ′ )
a root of E is given by X = λx, where x is a root of the equation x 2 = x + ε. From (2 ′ ) Z = 2ελ 3 + λ 3 + αλ 2 λ, and T = αλ 2 from (1 ′ ), we get :

Z = 2ελ 3 + λ 3 + T λ
Then :

ελ 2 = Z -T λ -λ 3 2λ
Next, we replace ελ 2 by its value, and αλ 2 by T in equation (3 ′ ), we get :

W = ( Z -T λ -λ 3 2λ ) 2 + λ 2 ( Z -T λ -λ 3 2λ ) + T ( Z -T λ -λ 3 2λ ) 4λ 2 W = (Z -T λ -λ 3 ) 2 + 2λ 3 (Z -T λ -λ 3 ) + 2T λ(Z -T λ -λ 3 )
Then :

4λ 2 W = (Z -T λ -λ 3 )(Z -T λ -λ 3 + 2λ 3 + 2T λ) 4λ 2 W = (Z -T λ -λ 3 )(Z + λ 3 + T λ)
Finally, we deduce:

4λ 2 W = Z 2 -(λ 3 + T λ) 2 And λ 6 + 2T λ 4 + λ 2 (T 2 + 4W ) -Z 2 = 0 Conclusion 1 • λ 2 is a root of y 3 + 2T y 2 + y(T 2 + 4W ) -Z 2 = 0. • α = T λ -2 and ε = Z -T λ -λ 3 2λ 3 .
• S ε is the set of roots for the quadratic x 2 = x + ε.

• The set of the roots for (E) is given by S E = λS ε = {λx , x ∈ S ε }.

In the context of the improvement of the method

The question we can ask now about this new method, is do we improve the new method in the sense of determining a largest set of algebraic equations which we can to solve by the new method?.

It is known that the resolution of every equation by radicals is not always possible from the results given by Galois and Abel, despite that, the classification of every equation by his solvability by radicals is essential. In this paragraph, I would like to give a few remarks about the generalizability of some form sets of equations solvable by the new method. We begin by studying the general case, and after we move to the case of n = 5.

Definition 1 If we can determine a root of a equation (E) by using of the new methode, this equation is called solvable by the new method.

• In general : Let E a set of equations with degree n solvables by the new method. Consider C = {(a 0 , a 1 , ..., a n ) / P (x) = n i=0 a i x i ∈ E}. For all number λ ∈ R * , we put :

λ * C = {(λ n a 0 , λ n-1 a 1 , ..., λa n-1 , a n ) / (a 0 , a 1 , ..., a n ) ∈ C} We have C ⊂ λ∈R λ * C (because 1 * C = C).
When C = λ∈R λ * C, the generalisation of the form concerning the équations of E from the lemma 2 it is not possible.

Example 1 In the case of the following equation

(E n ) (E n ) X n + αX n-1 = ( µ 2 γ 2 - αµ γ )X n-2 + γX + µ
We can check the following equality :

C = {(-µ, -γ, 0, ..., 0, -( µ 2 γ 2 - αµ γ ), α, 1) / (α, µ, γ) ∈ R 2 × R * }
On the other hand, we can verify λ * C ⊂ C, for every real number λ, then C = λ∈R λ * C. We deduce that it is impossible to genaralise the form of the equation E n by using the lemma2, and this remark is still checked for the equation (E4).

• For the case of n = 5, we can find many examples of the sets of the quintics which we can solve by the new method ( (E 4 ) for example ). Unfortunately, we can't improve this form of the equations by using the lemma2. Now, in the sence to generalise the form of (E 4 ), we can put the following question : Can we find the real ε which we can add it to one of coeffficients of (E 4 ) (different from the leading coefficient), and we obtained a new form of equations solvables by the new method ? For example, if ε = 1, we add the ε to the coefficient of x into (E 4 ), we get this new equation :

(E ′ 4 ) X 5 + αX 3 = βX 2 + ( µ 2 β 2 -α µ β + 1)X + µ
From the lemma 1 and 2, we can verify that the problem of resolution for every quintic reduces to problem of solving the equation of the form above, then we deduce that if the set of equations of the form (E 4 ) ′ are solvable by the new method, then every quintic is solvable by radicals, which is absurd.

Conclusion 4

We can not add a real non-zero in one of the coefficients of (E4) different from the leading coefficient for to get a another form of equations solvable by the new method.

6 Comparison between Ferrari's method and the new method

A didactic perspective

In the field of teaching and learning, the choice of a beautiful wording of the resolution of a problem Mathematics is essential to give students the generals rules of calculating which they can learn by heart. In this framework, the new method gives us a simple representation of the solving of a quartic equation over the Ferrari's method, in fact :

• For Ferrari's method : The solving of this quartic

Leads to solving the cubic 8y 3 -4T y 2 + 8W y -Z 2 = 0

• For the new method : We can to give the following simple representation : For solving the equation (E) X 4 + T X 2 = ZX + W Consider ∆ the discriminant of the quadratic with the unknown X :

to determine the real λ as defined above, for that we solve the cubic y 3 + 2T y 2 + y(T 2 + 4W ) -Z 2 = 0 which we can rewrite as the following simple form : y(y + T ) 2 = ∆

Solving certain equations with degree n

We use the Ferrari's method for solving the quartic equation, However we can use the new method for solving certain equations with degree n:

We consider the equation :

We can multiply the equation in the both sides by X 2 , we get X 5 = aX 3 + bX 2 , and we replace X 3 by aX + b, we get :

Then :

X 5 = bX 2 + a 2 X + ab We multiply the equation E 1 by c, we get :

We can do the sum of the two equations, we find :

The equation (E 2 ) is solvable by radicals. Moreover, every root of (E 1 ) is a root of (E 2 ).

Next, we can determine the set of the equations which we can rewrite in the form as (E 2 ).

Consider the equation : X 5 + αX 3 = βX 2 + γX + µ When this equation is written as (E 2 ), then we can find tree numbers a, b and c such that:

After the calcul we obtain :

We deduce that every root of the equation (E 1 ) :

, where b = β, c = α and γ = a 2 + ca.

Then : a = µ β -α.

And :

Every equation is a form as :

is solvable by radicals. Moreover, for determine the set of roots, it is sufficient to determine the roots of :

, then γ = 0 and a = 0. The equation (E 4 ) becomes :

The above equation is solvable by radicals. Moreover, tree roots of E 5 is giving by solving the fllowing equation :

• When α = 0, the equation (E 4 ) becomes :

Which are solvable by radicals. For determine the roots, we can to solve the cubic:

• Let µ = sin(ϕ), and β = cos(ϕ), where ϕ are a real number. The equation (E 5 ) becomes : (E 7 ) X 5 + tan(ϕ)X 3 = cos(ϕ)X 2 + sin(ϕ) We can determine the set of roots by solving the following equation :

The case of n ≥ 5

In this paragraph we see some examples of the set of the equations with degree n, which we can solve by the new method.

We consider the equation with unknown X :

We multiply this equation by X, we get : X n = aX n-1 + bX We replace X n-1 by aX n-2 + b, we find :

X n = a 2 X n-2 + bX + ab Next, we multiply the first equation by c and we add the two equations, we get :

Consider the equation

X n + αX n-1 = βX n-2 + γX + µ Where α = c, β = a 2 + ca,γ = b and µ = ab + bc. We deduce :

Wen can determine a root of every equation which is written as the following form :

For that, every root of X n-1 = ( µ γ -α)X n-2 + γ is a root of the above equation E n .


Nohra Hage

FINITE CONVERGENT PRESENTATIONS OF PLACTIC MONOIDS FOR SEMISIMPLE LIE ALGEBRAS 1

Keywords: Plactic algebra, Littelmann path model, standard tableau, convergent presentations

We study rewriting properties of the column presentation of plactic monoid for any semisimple Lie algebra such as termination and confluence. Littelmann described this presentation using L-S paths generators. Thanks to the shapes of tableaux, we show that this presentation is finite and convergent. We obtain as a corollary that plactic monoids for any semisimple Lie algebra satisfy homological finiteness properties.

INTRODUCTION

Using his path model, Littelmann defined in [START_REF] Littelmann | A plactic algebra for semisimple Lie algebras[END_REF] a plactic algebra for any semisimple Lie algebra. As a consequence, he gave some presentations by generators and relations of the plactic algebra of types A, B, C, D and G 2 . Using a case-by-case analysis, the plactic congruence can be also defined using Kashiwara's theory of crystal bases [START_REF] Date | Representations of U q (gl(n, C)) at q = 0 and the Robinson-Shensted [Schensted] correspondence[END_REF][START_REF] Kashiwara | Crystallizing the q-analogue of universal enveloping algebras[END_REF][START_REF] Kashiwara | Crystal graphs for representations of the q-analogue of classical Lie algebras[END_REF][START_REF] Kashiwara | On crystal bases[END_REF]. Lascoux, Leclerc and Thibon presented in [START_REF] Lascoux | Crystal graphs and q-analogues of weight multiplicities for the root system A n[END_REF] the plactic monoid of type A using the theory of crystal bases and gave a presentation of the plactic monoid of type C without proof. Shortly after that, Lecouvey in [START_REF] Lecouvey | Schensted-type correspondence, plactic monoid, and jeu de taquin for type C n[END_REF] and Baker in [START_REF] Baker | An insertion scheme for C n crystals[END_REF] described independently the plactic monoid of type C using also Kashiwara's theory of crystal bases. Plactic monoids of types B, D and G 2 were introduced by Lecouvey, see [START_REF] Lecouvey | Combinatorics of crystal graphs for the root systems of types A n , B n , C n , D n and G 2[END_REF].

The plactic monoid of rank n introduced by Lascoux and Schützenberger in [START_REF] Lascoux | ), volume 109 of Quad[END_REF], corresponds to the representations of the general linear Lie algebra of n by n matrices. This Lie algebra is of type A, so the corresponding plactic monoid is known as the plactic monoid of type A, and denoted by P n (A).

The monoid P n (A) has found several applications in algebraic combinatorics and representation theory [LS81, [START_REF] Lascoux | Crystal graphs and q-analogues of weight multiplicities for the root system A n[END_REF][START_REF] Fulton | Young tableaux[END_REF][START_REF] Lothaire | Algebraic combinatorics on words[END_REF]. Similarly, plactic monoids for any types have many applications like the combinatorial description of the Kostka-Foulkes polynomials which arise as entries of the character table of the finite reductive groups [START_REF] Lecouvey | Combinatorics of crystal graphs and Kostka-Foulkes polynomials for the root systems B n , C n and D n[END_REF].

More recently, plactic monoids were investigated by rewriting methods. In [START_REF] Kubat | Gröbner-Shirshov bases for plactic algebras[END_REF], Kubat and Okninski showed that for n > 3, there is no finite completion of the presentation of the monoid P n (A) with the Knuth generators. Bokut et al. in [START_REF] Bokut | New approaches to plactic monoid via Gröbner-Shirshov bases[END_REF] and Cain et al. in [START_REF] Cain | Finite Gröbner-Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids[END_REF] constructed independently finite convergent presentation of the monoid P n (A) by adding columns generators. This presentation is called the column presentation. Having a finite convergent presentation of the monoid P n (A) had many consequences. In particular, the column presentation was used by Lopatkin in [START_REF] Viktor Lopatkin | Cohomology Rings of Plactic Monoid Algebra via Gröbner-Shirshov Basis[END_REF] to construct Anick's resolution for the monoid P n (A) and by the author and Malbos in [START_REF] Hage | Knuth's coherent presentations of plactic monoid of type A. en preparation[END_REF] to construct coherent presentations of it. Note that coherent presentation extends the notion of a presentation of the monoid by homotopy generators taking into account the relations among the relations. Using Kashiwara's theory of crystal bases, the author generalized the column presentation for type A and constructed a finite convergent presentation of plactic monoid of type C by adding admissible column generators [START_REF] Hage | Finite convergent presentation of plactic monoid of type C[END_REF]. A bit later, a similar column presentation was obtained by Cain et al. for plactic monoids of types B, C, D and G 2 [START_REF] Cain | Crystal bases, finite complete rewriting systems, and biautomatic structures for Plactic monoids of types $A_n$, $B_n[END_REF].

In the present work, we consider plactic monoids for any semisimple Lie algebra. The column presentation of these monoids was introduced by Littelmann in [START_REF] Littelmann | A plactic algebra for semisimple Lie algebras[END_REF] using L-S paths. An L-S path corresponds to a column for type A, and to an admissible column for types C, B, D and G 2 in the Lecouvey sense, see [START_REF] Lecouvey | Combinatorics of crystal graphs for the root systems of types A n , B n , C n , D n and G 2[END_REF]. We study the column presentation using rewriting methods. For this, we consider a rewriting system where the generating set contains the finite set of L-S paths. The right-hand sides of rewriting rules are standard tableaux. The rewriting system rewrites two L-S paths that their concatenation do not form a standard tableau to their corresponding standard tableau form. Using the shapes of tableaux, we show that this presentation is finite and convergent. As a consequence, we deduce that plactic monoids for any types satisfy some homological finiteness properties.

Note that the convergent column presentation of plactic monoids for any semisimple Lie algebra given in this paper using L-S paths coincides with the presentations constructed for type A in [START_REF] Cain | Finite Gröbner-Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids[END_REF][START_REF] Bokut | New approaches to plactic monoid via Gröbner-Shirshov bases[END_REF], for type C in [START_REF] Hage | Finite convergent presentation of plactic monoid of type C[END_REF] and for types B, C, D and G 2 in [START_REF] Cain | Crystal bases, finite complete rewriting systems, and biautomatic structures for Plactic monoids of types $A_n$, $B_n[END_REF].

The paper is organised as follows. In Section 2, we recall the definitions of paths, root operators and L-S paths. After we present the definition and some properties of standard tableaux as defined by Littelmann and we recall the definition of the plactic algebra for any semisimple Lie algebra. In Section 3, we recall some rewriting properties of the presentations of monoids in term of polygraphs. After, we show that the column presentation of the plactic monoid for any semisimple Lie algebra is finite and convergent.

PLACTIC ALGEBRA

L-S Paths

In this section, we recall the definitions and properties of paths, root operators and L-S paths. We refer the reader to [START_REF] Littelmann | A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras[END_REF][START_REF] Littelmann | Paths and root operators in representation theory[END_REF][START_REF] Littelmann | A plactic algebra for semisimple Lie algebras[END_REF] for a full introduction.

2.1.1. Root system. Let g be a semisimple Lie algebra. Let V be the finite vector space with standard inner product •, • spanned by Φ ⊂ V \ {0} the root system of g. Let Φ + be the set of its positive roots. A positive root α in Φ + is simple, if it can not be written as α 1 + α 2 , where α 1 , α 2 ∈ Φ + . For a root α, define by α ∨ := 2α α,α its coroot. Denote by

X = v ∈ V v, α ∨ i ∈ Z, for all i
the weight lattice of the Lie algebra g and set X R := X ⊗ Z R. A fundamental weight ω i in X satisfy ω i , α ∨ j = δ ij , for all i and j. The set of dominant weights is

X + = λ ∈ X λ, α ∨ i ≥ 0, for all i 2 2.1. L-S Paths
and the dominant chamber is

X + R = λ ∈ X R λ, α ∨ i ≥ 0, for all i .
2.1.2. Example. Let g = sl 3 . Consider V = (x 1 , x 2 , x 3 ) ∈ R 3 x 1 + x 2 + x 3 = 0 and let {ε 1 , ε 2 , ε 3 } be the canonical basis of R 3 . The simple roots of g are α 1 = ε 1ε 2 and α 2 = ε 2ε 3 . Its fundamental weights are ω 1 = ε 1 and ω 2 = ε 1 + ε 2 (we still denote by ε i the projection of ε i into V). An example of dominant weight is ω 1 + ω 2 . The dominant chamber is the hatched area on the following figure:

ε 1 = ω 1 ε 2 ε 3 α 1 ω 1 + ω 2 ω 2 α 2 0 2.1.3. Example. Let g = sp 4 . Consider V = R 2 with its canonical basis {ε 1 , ε 2 }. The simple roots are α 1 = ε 1 -ε 2 and α 2 = 2ε 2 . The fundamental weights are ω 1 = ε 1 and ω 2 = ε 1 + ε 2 . An example of dominant weight is ω 1 + ω 2 .
The dominant chamber is the hatched area on the following figure:

0 ε 2 α 2 ε 1 = ω 1 α 1 ω 2 ω 1 + ω 2
For more informations, we refer the reader to [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines[END_REF][START_REF] Humphreys | Introduction to Lie algebras and representation theory[END_REF].

Paths.

A path is a piecewise linear, continuous map π : [0, 1] -→ X R . We will consider paths up to a reparametrization, that is, the path π is equal to any path π • ϕ, where ϕ : [0, 1] -→ [0, 1] is a piecewise linear, nondecreasing, surjective, continuous map. The weight of a path π is wt(π) = π(1).

For example, for the Lie algebra g = sl n , the paths

π ε i : t → tε i are of weight ε i , for i = 1, . . . , n. Denote by Π = π : [0, 1] -→ X R π(0) = 0 and π(1) ∈ X
the set of all paths with sources 0 such that their weight lies in X. Given two paths π 1 and π 2 in Π, the concatenation of π 1 and π 2 , denoted by π := π 1 ⋆ π 2 , is defined by:

π(t) := π 1 (2t) for 0 ≤ t ≤ 1 2 π 1 (1) + π 2 (2t -1) for 1 2 ≤ t ≤ 1 3 2. Plactic algebra
Denote by ZΠ the algebra of paths defined as the free Z-module with basis Π whose product is given by the concatenation of paths and where the unity is the trivial path

θ : [0, 1] -→ X R t → 0.
2.1.5. Example. Let g = sl 3 . Consider the paths π 1 : t → tε 1 and π 2 : t → tε 2 . The path π 1 ⋆ π 2 is the path on the following figure:

ε 1 = ω 1 ε 2 ε 3 ω 2 0 2.1.6. Root operators.
For each simple root α, one defines root operators e α , f α : Π → Π ∪ {0} as follows. Every path π in Π is cutted into three parts, i.e., π = π 1 ⋆ π 2 ⋆ π 3 . Then the new path e α (π) or f α (π) is either equal to 0 or π 1 ⋆ s α (π 2 ) ⋆ π 3 , where s α denote the simple reflection with respect to the root α. In other words, consider the function

h α : [0, 1] -→ R t → π(t), α ∨ Let Q := min(Im(h α ) ∩ Z) be the minimum attained by h α . If Q = 0, define e α (π) = 0. If Q < 0, let t 1 = min{ t ∈ [0, 1] h α (t) = Q } and t 0 = max{ t < t 1 h α (t) = Q + 1 }.
Denote by π 1 , π 2 and π 3 the paths defined by

π 1 (t) := π(tt 0 ) π 2 (t) := π(t 0 + t(t 1 -t 0 )) -π(t 0 ) π 3 (t) := π(t 1 + t(1 -t 1 )) -π(t 1 ), for t ∈ [0, 1]. By definition of the π i , we have π = π 1 ⋆ π 2 ⋆ π 3 . Then e α (π) = π 1 ⋆ s α (π 2 ) ⋆ π 3 .
Similarly, one can define the operator f α . Let

p = max{ t ∈ [0, 1] h α (t) = Q }.
Denote by P the integral part of h α (1) -Q.

If P = 0, define f α (π) = 0. If P > 0, let x > p such that h α (x) = Q + 1 and Q < h α (x) < Q + 1, for p < t < x.

L-S Paths

Denote by π 1 , π 2 and π 3 the paths defined by

π 1 (t) := π(tp) π 2 (t) := π(p + t(x -p)) -π(p) π 3 (t) := π(x + t(1 -x)) -π(x), for t ∈ [0, 1].
By definition of the π i , we have

π = π 1 ⋆ π 2 ⋆ π 3 . Then f α (π) = π 1 ⋆ s α (π 2 ) ⋆ π 3 .
These operators preserve the length of the paths. We have also that if

f α (π) = π ′ = 0 then e α (π ′ ) = π = 0 and wt(f α (π)) = wt(e α (π)) -α.
For all simple root α, let A be the subalgebra of End Z (ZΠ) generated by the root operators f α and e α . Define Π + to be the set of paths π such that the image is contained in X + R and denote by M π the A-module Aπ. Let B π be the Z-basis M π ∩ Π of M π . In other words, for a finite set I, we have

B π = f i 1 . . . f ir (π) π ∈ Π + and i 1 , . . . , i r ∈ I .
For a dominant weight λ, consider the path π λ (t) = tλ that connects the origin with λ by a straight line. Denote by M λ the A-module Aπ λ generated by the path π λ . In addition, the Z-module M λ has for a basis the set B π λ consisting of all paths in M λ .

Given two paths π and π

′ in Π + , if π(1) = π ′ (1), then the A-modules M π and M π ′ are isomorphic.

For π in Π + , let η in M π be an arbitrary path. The minimum of the function h α (t) = π(t), α ∨ is an integer for all simple roots. In addition, we have e α (η) = 0 if and only if η = π.

2.1.7. Example. Let g = sl 3 and let α 1 , α 2 be its simple roots and ω 1 , ω 2 be its fundamental weights. For λ = ω 1 + ω 2 , consider the path π λ : t → tλ. Let us compute B π λ . We have

ω 1 ε 2 ε 3 fα 2 (π λ ) α 1 fα 2 (fα 1 (π λ )) fα 2 (fα 2 (fα 1 (π λ ))) fα 1 (fα 2 (π λ )) fα 1 (fα 1 (fα 2 (π λ ))) λ π λ ω 2 fα 1 (π λ ) α 2 fα 2 (fα 1 (fα 1 (fα 2 (π λ ))))
where

π λ : t -→ tλ f α 1 (π λ ) =: π 2 : t -→ tα 2 f α 2 (π λ ) =: π 3 : t -→ tα 1 f α 2 (π 2 ) =: π 4 : t -→ -tα 2 for 0 ≤ t ≤ 1 2 (t -1)α 2 for 1 2 ≤ t ≤ 1 f α 1 (π 3 ) =: π 5 : t -→ -tα 1 for 0 ≤ t ≤ 1 2 (t -1)α 1 for 1 2 ≤ t ≤ 1 f α 2 (π 4 ) =: π 6 : t -→ -tα 2 f α 1 (π 5 ) =: π 7 : t -→ -tα 1 f α 2 (π 7 ) = f α 1 (π 6 ) =: π 8 : t -→ -tλ
Hence we obtain B π λ = {π λ , π 2 , π 3 , π 4 , π 5 , π 6 , π 7 , π 8 }.

2.1.8. Crystal graphs. For π in Π + , let G(π) be the oriented graph with set of vertices B π and an

arrow π i → π ′ means that f α i (π) = π ′ and e α i (π ′ ) = π. The graph G(π) is also called a crystal graph, see [Kas95].
For example, consider the semisimple Lie algebra g = sl 3 . For λ = ω 1 + ω 2 , the elements of B π λ obtained in Example 2.1.7 can be represented in the following crystal graph G(B π λ )

π λ 1 2 ? ? ? ? ? ? π 2 2 π 3 1 π 4 2 π 5 1 π 6
1 @ @ @ @ @ @ π 7 2 ~~~~~π 8 2.1.9. L-S paths. For a dominant weight λ, the Lakshmibai-Seshadri paths, L-S paths for short, of shape λ are the paths π of the form

π = f α 1 • . . . • f αs (π λ )
where α 1 , . . . , α s are simple roots of a semisimple Lie algebra g. That is, these paths are all the elements of B π λ .

2.1.10. Example. Consider g = sl 3 and let ω 1 and ω 2 be its fundamental weights. Let λ = ω 1 + ω 2 , the L-S paths of shape λ are all the elements of B π λ as obtained in Example 2.1.7. The L-S paths of shape ω 2 are the elments of B πω 2 , with

B πω 2 = { π ε 1 +ε 2 , π ε 1 +ε 3 , π ε 2 +ε 3 }.
For the Lie algebra sl n , the L-S paths of shape ω 1 are the paths π ε i , for i = 1, . . . , n + 1.

Tableaux

In this section, we recall the definition of standard tableaux as defined by Littelmann in [START_REF] Littelmann | A plactic algebra for semisimple Lie algebras[END_REF]. Fix an enumeration ω 1 , . . . , ω n of the fundamental weights of a semisimple Lie algebra g.

Young diagram.

A Young diagram is a finite collection of boxes, arranged in left-justified rows such that the length of each row is bigger or equal to the one below it. Note that the rows are enumerated from top to bottom and the columns from the right to the left. For example, the Young diagram with six boxes in the first row, four boxes in the second boxes and one box in the third row is the following .

L-S monomials.

Let λ = λ 1 + . . . + λ k , where λ 1 , . . . , λ k are dominant weights. If for all i = 1, . . . , k, π i is an L-S path of shape λ i , then the monomial m = π 1 ⋆ . . . ⋆ π k is called an L-S monomial of shape λ = (λ 1 , . . . , λ k ).

Young tableau.

A Young tableau of shape λ = a 1 ω 1 + . . . + a n ω n is an L-S monomial

⋆ 1≤i≤n 1≤j≤n π 1,ω j ⋆ . . . ⋆ π a i ,ω j
where π i,ω j is an L-S path of shape ω j , for 1 ≤ i ≤ n. That is, the first a 1 paths are of shape ω 1 , the next a 2 are of shape ω 2 ,. . . , the final a n paths are of shape ω n .

Standard tableau.

Let m be a Young tableau of shape λ = a 1 ω 1 + . . . + a n ω n . The Young tableau m is called standard of shape λ if m ∈ A(π ω 1 ⋆ . . . ⋆ π ωn ).

Example. For type A n , consider the ordered alphabet

A n+1 = {1 < • • • < n+1}.
For a dominant weight λ = a 1 ω 1 + . . . + a n ω n , set p i = a i + . . . + a n . Consider the Young diagram with p 1 boxes in the first row, p 2 boxes in the second row, etc. A Young tableau of shape λ and type A is a filling of the boxes of this Young diagram with elements of the alphabet A n+1 such that the entries are striclty increasing in the column from top to bottom. A standard tableau of shape λ = ω 1 + . . . + ω k is a Young tableau of shape λ and type A such that the entries are weakly increasing in the rows from left to right. In other words all the standard tableaux of shape λ are the elements of B πω 1 ⋆...⋆πω k , where the L-S monomial π ω 1 ⋆ . . . ⋆ π ω k corresponds to the Young tableau with only 1's in the first row, 2's in the second row,. . . , k's in the k-th row. Note that the standard tableaux of type A are also called semistandard tableaux in [START_REF] Fulton | Young tableaux[END_REF].

For type A 3 , the following Young tableau is standard of shape ω 1 + 2ω 2 + ω 3 1 1 1 2 2 3 3 3 .

2.2.6. Example. For type C n , consider the ordered alphabet C n = {1 < . . . < n < n < . . . < 2 < 1}.

For a dominant weight λ = a 1 ω 1 + . . . + a n ω n , set p 1 = a 1 + 2a 2 + . . . + 2a n and for i ≥ 2

3. Finite convergent presentation of plactic monoids set p i = 2a i + . . . + 2a n . Consider the Young diagram with p 1 boxes in the first row, p 2 boxes in the second row, etc. A Young tableau of shape λ and type C is a filling of the boxes of this Young diagram with elements of the alphabet C n such that the entries are striclty increasing in the column from top to bottom, but i and i are never entries in the same column. In addition, for each pair of columns (C a 1 +2j-1 , C a 1 +2j ), j = 1, . . . , a 2 + . . . + a n , either these columns are equal or the column C a 1 +2j is obtained from C a 1 +2j-1 by exchanging an even number of times an entry k, 1

≤ k ≤ 1, in C a 1 +2j-1 by k, see [GL12, Section 4.2].
A standard tableau of shape λ = ω 1 + . . . + ω k is a Young tableau of shape λ and type C such that the entries are weakly increasing in the rows from left to right. Note that the standard tableaux of type C are also called symplectic tableaux in [START_REF] Lecouvey | Schensted-type correspondence, plactic monoid, and jeu de taquin for type C n[END_REF].

For type C 3 , the following Young tableau is a standard tableau of shape ω 1 + 2ω 2 1 1 1 2 3 2 2 3 3 3 3 2 1 .

Plactic algebra for any semisimple Lie algebra

Let us recall the definition of the plactic algebra for any semisimple Lie algebra and we refer the reader to [START_REF] Littelmann | A plactic algebra for semisimple Lie algebras[END_REF] for more details. Let ZΠ 0 be the A-submodule AΠ + of ZΠ generated by the paths in Π + . For two paths π 1 and π 2 in ZΠ 0 , denote by π + 1 and π + 2 the unique paths in Π + such that π 1 ∈ M π + 1 and π 2 ∈ M π +

2

. One can define a relation ∼ on ZΠ 0 by : π 1 ∼ π 2 if, and only if, wt(π + 1 ) = wt(π + 2 ) and ψ(π 1 ) = π 2 under the isomorphism

ψ : M π + 1 -→ M π + 2
. The plactic algebra for g is the quotient

ZP := ZΠ 0 / ∼ .

FINITE CONVERGENT PRESENTATION OF PLACTIC MONOIDS

Rewriting properties of 2-polygraphs

In this section, we recall some rewriting properties of the presentations of monoids. These presentations are studied in terms of polygraphs. For more informations, we refer the reader to [START_REF] Guiraud | Polygraphs of finite derivation type[END_REF].

A 2-polygraph (with only one 0-cell) is a pair Σ = (Σ 1 , Σ 2 ) where Σ 1 is the set of generating 1-cells and Σ 2 is the set of generating 2-cells, that is rules of the form u ⇒ v, with u and v are words in the free monoid Σ * 1 . A monoid M is presented by a 2-polygraph Σ if M is isomorphic to the quotient of the free monoid Σ * 1 by the congruence generated by Σ 2 . A 2-polygraph Σ is finite if Σ 1 and Σ 2 are finite. A rewriting step of Σ is a 2-cell of the form wϕw ′ : wuw ′ % 9 wvw ′ , where ϕ : u % 9 v is a 2-cell in Σ 2 and w and w ′ are words of Σ * 1 . A rewriting sequence of Σ is a finite or infinite sequence of rewriting steps. We say that u rewrites into v if Σ has a nonempty rewriting sequence from u to v. A word of Σ * 1 is a normal form if Σ has no rewriting step with source u. A normal form of u is a word v of Σ * 1 that is a normal form and such that u rewrites into v. We say that Σ terminates if it has no infinite rewriting sequences. We say that Σ is confluent if for any words u, u ′ and u" of Σ * 1 , such that u rewrites into u ′ and u", there exists a word v in Σ * 1 such that u ′ and u" rewrite into v. We say that Σ is convergent if it terminates and it is confluent. Note that a terminating 2-polygraph is convergent if every word admits a unique normal form.

Column presentation

Let g be a semisimple Lie algebra and let ω 1 , . . . , ω n be its fundamental weights. Let B i be the set of L-S paths of shape ω i and B = ∪ n i=1 B i . Denote by B * the free monoid over B. A word on B * is a concatenation of L-S paths.

For every L-S paths c 1 and c 2 in B such that c 1 ⋆ c 2 is not a standard tableau, we define a 2-cell

c 1 ⋆ c 2 γ c 1 ,c 2 % 9 T
where T is the unique standard tableau such that where λ 1 and λ 2 are dominant weights. That is, λ 1λ 2 is a nonnegative integral sum of positive roots.

Using this order, one can find for each dominant weight, a finite number of dominant weights that are smaller than it, then the partial order is a well-founded order.

Let us define an order on the tableaux of B * as follows. For two tableaux m and m ′ of shape λ and λ ′ respectively, we have m m ′ if and only if λ λ ′ .

3.2.2. Remark. In the proof of Theorem B in [START_REF] Littelmann | A plactic algebra for semisimple Lie algebras[END_REF], Littelmann showed that for every tableau m of shape λ that it is not standard, the shape of the standard tableau m ′ such that [m] = [m ′ ] is strictly smaller than λ.

In the homological way, a monoid M is of homological type FP ∞ when there exists a resolution of M by projective, finitely generated ZM-modules. By Corollary 5.4.4 in [START_REF] Guiraud | Higher-dimensional normalisation strategies for acyclicity[END_REF] the property FDT ∞ implies the property FP ∞ . Hence we have 3.3.2. Proposition. For any semisimple Lie algebra g , plactic monoids for g satisfy the homological finiteness property type FP ∞ .

For π ∈

 π ZΠ 0 , we denote by [π] ∈ ZP its equivalence class. The classes [m] of standard Young tableaux form a basis of the plactic algebra ZP, see [Lit96, Theorem 7.1]. 2.3.1. Example ([Lit96, Theorem C]). For type A, consider the alphabet {1, . . . , n}. The plactic congruence coincides with the congruence generated by the following families of relations on the word algebra Z{1, . . . , n} : (a) xzy = zxy for 1 ≤ x < y ≤ z ≤ n. (b) yxz = yzx for 1 ≤ x ≤ y < z ≤ n. (c) 12 . . . n is the trivial word.

  [T ] = [c 1 ⋆ c 2 ]. Let denote by Path(n) the 2-polygraph whose set of 1-cells is B and the set of 2-cells is Path 2 (n) = c 1 ⋆ c 2 γ c 1 ,c 2 % 9 T c 1 , c 2 ∈ B and c 1 ⋆ c 2 is not a standard tableau .This presentation is called the column presentation. It is a presentation of the plactic monoid for any semisimple Lie algebra [Lit96, Theorem B]. Let us prove that it is terminating and confluent.3.2.1. Order on the tableaux. Consider the partial order of dominant weights defined by λ 1 λ 2 if and only if λ 1λ 2 ∈ NΦ +

3.2.3. Lemma. The 2-polygraph Path(n) is terminating.

Proof. Let us show that the 2-polygraph Path(n) is compatible with the order . We have to prove that if h % 9 h ′ , then h ′ ≺ h, where h is not a sandard tableau and h ′ is a standard tableau of shape λ ′ . There is two cases depending on whether or not h is a tableau. Suppose that h is a tableau of shape λ that it is not standard, then by Remark 3.2.2 we have λ ′ < λ. Thus we obtain that h ′ ≺ h. Suppose that h is an L-S monomial of shape λ that it is not a tableau. By decomposing each L-S path of shape ω i , for i = 1, . . . , n, in h into concatenation of L-S paths of shape ω 1 , we transform the L-S monomial h into a tableau of shape kω 1 , with k ∈ N. Again by Remark 3.2.2, we have λ ′ < kω 1 . Thus we obtain that h ′ ≺ h.

Hence, rewriting an L-S monomial that is not a standard tableau always decreases it with repsect to the order . Since every application of a 2-cell of Path(n) yields a ≺-preceding tableau, it follows that any sequence of rewriting using Path(n) must terminate.

and the standard tableaux form a cross-section of the plactic monoid for a semisimple Lie algebra g, we have k = l and c i = c ′ i , for all i = 1, . . . , k. Thus T = T ′ . Since the 2-polygraph Path(n) is terminating, and rewriting any non-standard tableau must terminate with a unique normal form, Path(n) is confluent.

Every semisimple Lie algebra g admits a finite number of fundamental weights, then there is a finite number of L-S paths of shape ω i , for i = 1, . . . , n. Thus the 2-polygraph Path(n) is finite. Hence, by Lemmas 3.2.3 and 3.2.4, we obtain the following theorem.

3.2.5. Theorem. For any semisimple Lie algebra g, the 2-polygraph Path(n) is a finite convergent presentation of the plactic monoid for g.

Finiteness properties of plactic monoids

A monoid is of finite derivation type (FDT 3 ) if it admits a finite presentation whose relations among the relations are finitely generated, see [START_REF] Squier | A finiteness condition for rewriting systems[END_REF]. The property FDT 3 is a natural extension of the properties of being finitely generated (FDT 1 ) and finitely presented (FDT 2 ). Using the notion of polygraphic resolution, one can define the higher-dimensional finite derivation type properties FDT ∞ , see [START_REF] Guiraud | Higher-dimensional normalisation strategies for acyclicity[END_REF]. They generalise in any dimension the finite derivation type FDT 3 . A monoid is said to be FDT ∞ if it admits a finite polygraphic resolution. By Corollary 4.5.4 in [START_REF] Guiraud | Higher-dimensional normalisation strategies for acyclicity[END_REF], a monoid with a finite convergent presentation is FDT ∞ . Then by Theorem 3.2.5, we have 3.3.1. Proposition. For any semisimple Lie algebra g , plactic monoids for g satisfy the homotopical finiteness condition FDT ∞ .


Isabelle Blanc

How to calculate the environmental impact of renewable energy

How to calculate the environmental impact of renewable energy

Isabelle Blanc, professeur à MINES ParisTech

The development of renewable energy since the early 2000s should continue and intensify in the coming years, changing significantly the electricity mix of the future while reducing the associated environmental impacts. It is therefore crucial to study the environmental impact of the different production sectors.

The energy transition and, more specifically, the electricity mix of the future has sparked many debates about its environmental impacts, both on locally and globally. Direct emissions of greenhouse gases caused by the production of electricity and heat from fossil fuels alone account for 25% of global emissions in 2010. To fight against climate change while ensuring a sustainable energy supply, the European Union has set itself the goal of "3 x 20" for 2020: reduce greenhouse gas emissions by 20% compared to the 1990 level, reach 20% of energy savings and a 20% share in renewable energy (RE) in the European energy mix. Recently, the EU set even more ambitious targets for 2030 with 40% reduction in greenhouse gas emissions, 27% in energy savings and a 27% share of renewable energy.

While some renewable energy systems have little or no impact during their operational phase, they nevertheless have an effect on the environment: upstream in the production chain, during the manufacturing phase, and downstream, at the end the system's service life.

To study the environmental impact of different production chains, we can use tools such as life-cycle assessment and other related indicators, such as energy payback time.

Life-cycle assessment and energy payback time

The life-cycle assessment (LCA) examines the environmental impacts of a product or service throughout its life cycle. It is based on the compilation of measured environmental impacts at each stage, from the manufacturing of the components that form the system to the end of service life of the system itself. For example, in the case of the production of electricity by photovoltaic panels installed in France, the assessment will take into account the manufacturing of its polycrystalline cells in Asia, the mounting of its panels in Germany and its installation in France. LCA is takes into account an inventory of pollutants and the consumption of resources. This holistic approach, "from cradle to grave," has undergone extensive standardization by the ISO (14040 and 14044 standards) and the European Commission. It is considered as a powerful supporting tool for decision-making and environmental optimization.

Thanks to this type of analysis, direct and indirect impacts of energy production can be qualified and quantified according to various indicators. The impact categories assessed globally include global warming potential, human toxicity, acidification, primary energy consumption, but also the depletion of resources or anticipated shortages of mineral and fossil resources. This latter point should not be overlooked. For example, wind turbines depend on neodymium and dysprosium, two rare earth metals which act as permanent magnets in the alternator. These illustrate this sensitive issue of mineral resources: given the increase in the current demand, there will be a dysprosium shortage around 2020. Another example with very promising photovoltaic technologies such as CIGS (copper, indium, gallium, selenium) faces the same challenges: the "production reserves" ratio of indium is estimated to last only 20 years.

The life-cycle analysis is based on various indicators. Among them, the "energy payback time" is a particularly appropriate indicator to quantity the environmental performance of renewable energy. These sectors are characterized by a major energy investment in the manufacturing phase but low energy consumption during the use phase. Energy payback time indicates the time required by a sector to produce a primary energy quantity equivalent to that which was necessary for its manufacturing. It is expressed in years and calculated based on the ratio of the energy consumed by the manufacturing of the system and the energy produced by the system over a year. The latter quantity is based on the primary energy necessary to produce the electric energy in the country where the renewable system is installed. This type of calculation is therefore specific to the electricity mix in the concerned country.

As an illustration, the following table provides the energy payback time for two solar photovoltaic panels with a peak power of 3kW (monocrystalline and polycrystalline technology) produced in Europe and China and installed in Europe. The geographical information is critical, because the return time depends on the energy content in kWh of the installation area. For Europe, the average electricity mix has a value of 11.4 MJ/kWh of primary energy per kWh of electricity produced. The return time would be higher for identical facilities located in countries where the energy mixes have a lower energy content.

Life-cycle inventories of these systems are necessary to calculate the manufacturing of photovoltaic energy, covering all the sub-systems and the induced transportation. In this example, the life-cycle inventory values of these systems, manufactured in Europe or China, come from the Ecoinvent v2.2 database, based on the situation in 2011.

The energy payback time is less than two years for both technologies when the panels are manufactured in Europe and over a few months when they are manufactured in China for the monocrystalline technology. This "return time" approach also allows to assess the "environmental performance," for example, by applying the same reasoning to the carbon footprint of the life cycle in the renewable sectors: this is referred to as "climate return time."

A comparative view of electrical production channels based on life-cycle analysis

The characterization of energy systems by life-cycle analysis and related return time indicators enables the comparison between channels and the identification of improvement solutions. It also feeds the technical and social debate around the expansion of renewables in the energy mix.

In this perspective, the IPCC conducted a study in 2011 that compiles scientific articles on the carbon performance indicator. This indicator (in g of CO 2-eq /kWh) compares the emission of greenhouse gases throughout the life cycle of the industry to the production of electricity. The aim of the study was to compare fossil energies (coal, gas and natural gas), nuclear energy and renewables. This compilation revealed the extent of the distribution of the results for each of the sectors, reflecting the great variability of local conditions and technological characteristics of systems in the available studies. It would therefore be unrealistic to try to identify each sector with a unique and absolute value of carbon performance. This overly simplistic interpretation would offer a partial view of the problem.

The variation range of the carbon performance indicator highlights the best performance of renewables. This is a crucial point. But carbon is only one indicator among others. What about other environmental impacts, such as the depletion of natural resources or human health? The study is limited to the carbon performance indicator and would deserve to be extended to other indicators.

To contribute to the debate on energy transition, these environmental assessments must be conducted properly i.e. by taking into account technological and local specificities. This will allow a comparison of power generation in energy mixes, as in the following example: Guadeloupe.

Life-cycle analysis on a regional scale

The "Observation, Impacts, Energy" center of Mines ParisTech conducted a first life-cycle analysis on a regional scale, the EVALGTHDOM research project, to assess the environmental impact of prospective electric mixes for Guadeloupe. If the French authorities set an objective of 23% renewables in of the country's electricity mix by 2020, for the Departments and Regions of Overseas, the goal is to reach 50% by the same date, and ultimately, a self-sufficient 100% by 2030.

The choice of Guadeloupe is no coincidence. This volcanic island is a laboratory for geothermal energy: the central of Bouillante, the first geothermal electricity plant in France, boasts a capacity of 15 MW (photo BRGM).

The study compares the reference scenario of the electricity mix in 2013 with three scenarios for 2030, by incorporating the official objectives concerning the control of consumption and promotion of renewables. It is based on 13 indicators of impacts (climate change, human toxicity, reduction of fossil resources, acidification, eutrophication…) that assess each sector of power generation, including six renewables: geothermal, biomass (bagasse, fibrous sugarcane), wind, photovoltaic, hydropower, waste recycling and biogas. The definition of the four scenarios of development of Guadeloupe's electricity mix is based on the work accomplished by the Guadeloupe Region within the framework of PRERURE (Plan énergétique régional pluriannuel de prospection et d'exploitation des énergies renouvelables et d'utilisation rationnelle de l'énergie de la Guadeloupe: Regional multi-year energy plan for the exploration and exploitation of renewable energy and rational use of energy in Guadeloupe):

. The baseline scenario used as a reference represents the state of Guadeloupe's electricity mix in 2013, with 83% of fossil fuels, imported for the most part. It serves as a comparison with future scenarios and assesses the impacts related to energy planning decisions. Renewables account for 17% of the electricity mix.

. The trend scenario follows the trends observed in recent years in terms of energy demand without any special effort in terms of developing renewable energy or strengthening energy demand management (also known as "demand side management" or DSM).

. The PRERURE scenario reflects an increased effort to control consumption and promote the development of renewable sectors (up to 75% of the electricity mix). It is designed to promote the diversification of the energy mix. According to this scenario, geothermal energy is growing to reach a power of 85 MW in 2030.

. The moderate scenario, specifically established as part of the project, reflects a moderate effort to control consumption on the one hand and the development of renewables, on the other. The geothermal energy sector reaches a capacity of 45 MW.

The different phases of the life cycle of the electrical system in Guadeloupe include the construction, production and transportation of energy. Impacts related to the final use of electricity are not included, nor the storage of electricity or processes of recycling and end-oflife. Inventories of energy conversion technologies have been compiled within the limits of specific details available for each technology.

The results of this study can be illustrated through four representative environmental impacts:

. The global warming potential, which identifies greenhouse gases per kWh of produced electricity, reaches virtually a reduction factor of 4 times in greenhouse gases between the current scenario and the PRERURE scenario, thanks to the very significant reduction of fossil fuels and the abandonment of coal . Acidification (the ability of substances to create and release H + ions expressed in a SO 2 sulfur dioxide equivalent) can have direct and indirect impacts on ecosystems (acid rain and soil leaching). This impact follows the same reduction scheme for the examined scenarios than the global warming potential.

. Ecotoxicity, expressed in CTUe (Comparative Toxic Units) is an estimate of the fraction of potentially affected species, integrated in time and space, per unit of chemical mass unit emitted. A decrease of the impact of the bagasse sector between the baseline scenario and the trend scenario, while its share in the power generation mix remains very similar (respectively 2.6% and 2.3%). This difference is due to the hypothesis of a decrease in the use of pesticides (complete stop in the use of diuron in three prospective scenarios, following its recent ban in Guadeloupe). Impacts are greater in moderate and PRERURE scenarios and are the direct consequence of the development of the cane and fibrous sugarcane industry for both scenarios.

. Marine eutrophication (in kg of nitrogen) is used to evaluate the potential for the eutrophication of the marine environment. Eutrophication is a process of accumulation of nutrients and can lead to problems (loss of diversity, degradation) in the affected environment.

The study shows a strong decrease of this impact in all three scenarios. It is mainly due to the important efforts in mitigating the emissions of nitrogen oxides (NO x ) and ammonium (NH 4 + ) in thermal power plants equipped with a denitrification system that reduces the NO x emissions in fumes by 85%.

Beyond its results, this study on life-cycle analysis of the energy mix at a regional scale participates in the debate on the implementation of the energy transition: it offers a first assessment of the impacts generated by each sector within this mix by pinpointing the sectors that have the greatest impact according to different selected indicators. LCA also reveals the contradictory trends of these choices on the different impacts and contributes to influence decision-makers in the Guadeloupe region.

Based on these LCA studies, a sustainable energy development plan can be designed at regional scale: the optimization of local resources (whether renewable or not) to meet the energy demand; the control of import flows of fossil fuels, biomass or electricity; and of course, the mitigation of "environmental leakage," such as carbon leakage.


Hui Li

Alain Fourmigue

Sébastien Le Beux

Ian O'connor
email: ian.oconnor@ec-lyon.fr

Gabriela Nicolescu

A thermal-Aware Laser Tuning Approach for Silicon Photonic Interconnects

Keywords:

Optical interconnect is an emerging technology considered as one of the key solutions for future generation onchip interconnects. However, silicon photonic devices are highly sensitive to temperature variation. Under a given chip activity, this leads to a lower laser efficiency and a drift of wavelengths of optical devices (on-chip lasers and Microring Resonators (MRs)), which results in a higher Bit Error Ratio (BER). In this paper, for the first time, we propose to tune the laser injection current in order to align the wavelength of the emitted signals with the resonant wavelengths of the MRs. Our approach allows significant improvements of the power consumption with regard to the related approaches.

INTRODUCTION

Technology scaling down to the ultra-deep submicron domain enables the integration of hundreds of cores. In order to enhance connecting many cores, chip-level communication needs a new interconnect solution to bring down the power budget. On-chip silicon photonic interconnects are an emerging technology considered as one of the key solutions for future generations of many cores, providing high bandwidth and low latency. They are composed of CMOS-compatible on-chip laser sources, Microring Resonators (MRs), waveguide and photodetector. However, silicon photonic devices are highly sensitive to temperature variation induced by thermal effect over the chip, which leads to a drift of the laser wavelength and MRs resonant wavelength. Consequently, the Signal to Noise Ratio (SNR) of the signals received by the photodetector decreases, which leads to a higher Bit Error Ratio (BER). This is further accentuated by the significant reduction of on-chip lasers efficiency as the temperature increases.

This paper proposes a thermal-aware tuning approach to overcome the wavelength variation induced by temperature variation, while achieving the targeted BER. The novelty of the approach relies on the tuning of the laser driver current, which ideally complements traditional methods such as MRs tuning and channel remapping [2][3].

II. PROPOSED LASER TUNING APPROACH

Figure 1 illustrates our approach in the context of a MWSR-(Multiple Writer Single Reader) architecture [START_REF] Vantrease | Corona: System Implications of Emerging Nanophotonic Technology[END_REF] with a single laser source, 2 writers and one reader. In [START_REF] Chen | Sharing and Placement of On-chip Laser Sources in Silicon-Photonic NoCs[END_REF], the authors explore the placement of shared on-chip lasers on a layer located on top of the optical interconnect. In our work, we work on distributed lasers that are located in the same layer with MRs, waveguide and photodetectors. For this purpose, we assume the use of CMOS-compatible laser sources such as Vertical-Cavity Surface-Emitting Lasers (VCSELs), for which the size is similar to the one of MRs [START_REF] Li | Thermal Aware Design Method for VCSEL-based On-Chip Optical Interconnect[END_REF]. As illustrated in Figure 1-a, ONI m communicates with ONI r : the MR in the intermediate ONI i is turned OFF while the MR in ONI m is in the modulation state (state OFF and ON to modulate data '1' and '0' respectively). Figure 1-b illustrates the ideal transmission of a data '1' that occurs when there is no temperature variation along the communication path. The optical signal injected by the laser (which is characterized by a power OP ideal and a wavelength λ laser ) crosses ONI m and ONI i , propagates along the waveguide until ONI r where it will be dropped to the photodetector (as illustrated by the blue transmission lines in the Figure 1-b). The power of the optical signal decreases along the path due to the waveguide propagation losses and the MRs crossing losses (blue line in Figure 1-b). From the received power OP pd,ideal , the receiver sensitivity and the crosstalk (induced by other transmitting signals at different wavelengths, not illustrated here for sake of clarity but taken into account in our models), the BER is estimated.

1) Traditional tuning approach: In case of temperature gradient over the communication path, the resonant wavelength of the MRs will drift (see the red transmission lines in Figure 1-c) while the wavelength of the emitted signal (λ laser ) remains the same in case off-chip lasers are considered. Without compensating effect of this drift, the misalignment between the signal wavelength and the MRs resonant wavelengths leads to significantly increased BER. To overcome this effect, the MRs along the waveguide are tuned back to their initial positions (the grey transmission lines in Figure 1-c) using thermal tuning or voltage tuning. The posttuning signal transmission is illustrated by the blue line in Figure 1-c: the received optical power is slightly lower than in the ideal scenario due to marginal wavelengths misalignment. The MRs tuning power depends on their temperature drift (∆T MR,m , ∆T MR,i and ∆T MR,r ) and the thermal sensitivity coefficient ρ MR . The total power consumption of the link is given by the sum of the laser power consumption P laser and the MRs tuning power ∑P MR .

2) Proposed tuning approach: In order to reduce the total link power consumption, we consider tuning the driver current I laser in the context of on-chip lasers. As illustrated in Figure 1d, the laser power consumption will move from P laser to P' laser and the emitted signal is tuned from to λ laser to λ' laser . Therefore, under the same temperature gradient considered in Figure 1-c, the MRs wavelengths need to be tuned to λ' laser , instead of λ laser . (see the green transmission lines in Figure 1-d). Since the distance is lower, the MRs tuning power is lowered. As a drawback, the power of the emitted signal is reduced, meaning that a tradeoff needs to be defined in order to reach the target BER while decreasing the total power consumption of the channel. It is worth noticing that, although the approach is illustrated with a 1-wavelength channel, it is generic and it can be applied to WDM channels (we investigate 4-wavelength channels in the results section). Moreover, it is complementary to related approaches providing channel remapping [START_REF] Ye | System-Level Modeling and Analysis of Thermal Effects in WDM-Based Optical Networks-on-Chip[END_REF][3] that we have adapted to minimize the tuning power instead of the tuning distance.

III. RESULTS

We evaluate our approach with a MWSR network used to interconnect 12 interfaces (i.e. 12 channels with 11 writers and 1 reader per channel) and 4 wavelengths (i.e. 4 on-chip lasers per channel and 4 MRs per writer). Following the method described in [START_REF] Li | Thermal Aware Design Method for VCSEL-based On-Chip Optical Interconnect[END_REF], the silicon photonic interconnect is placed on top of the SCC chip and thermal simulations are performed using IcTherm [START_REF] Fourmigue | Efficient Transient Thermal Simulation of 3D ICs with Liquid-Cooling and Through Silicon Vias[END_REF]. From the resulting thermal map, the proposed laser tuning approach is applied in order to i) evaluate the BER using the transmission models from [START_REF] Li | Thermal Aware Design Method for VCSEL-based On-Chip Optical Interconnect[END_REF] and ii) evaluate the total link power consumption P tuning (including the laser power consumption P VCSEL and the MR tuning power P MR ). We first run thermal simulations under a homogeneous 25% chip activity, with a laser power consumption P VCSEL ranging from 1mW to 20mW (obtained by tuning the laser injection current I laser ). From the thermal map, we evaluate the total power required to align the MRs resonant wavelengths with the emitted signals and we evaluate the worst-case BER among all the channels in the network, as illustrated in Figure 2. For small laser injection current (i.e. P VCSEL ranging from 1mW to 4mW), the optical power of the emitted signal is too low to compensate the channel losses, which results in a high BER. Then, the communication quality improves as the laser injection current increases: the BER reaches its optimal value at P VCSEL =9mW. By considering a static design method, the corresponding injection current will be selected to ensure the best communication quality. Above 9mW, the laser efficiency decreases due to the increase of the local temperature, leading to a higher BER. The channel power consumption is also given in the figure. It is composed of the laser power consumption (simply reported from the x-axis) and the MRs tuning power. We evaluate the potential gain of the proposed laser tuning approach with the traditional one (for which only MRs tuning is possible). For this purpose, we first set the laser power consumption to 9mW in order to reach a targeted 10 -12 BER for the highest considered chip activity (30% in this example). We then reduce the chip activity to 25%, 20% and 10%, and we evaluate the achievable power reduction while still reaching 10 -12 BER. As illustrated in Table 1, up to 52% reduction is obtained for a 10% chip activity. In case a 10 -9 BER turns to be acceptable (i.e., in order to match the requirements of an application to be executed), the power consumption can be further decreased. These results demonstrate the significant power reductions achievable by using chip activity-aware methods.

IV. CONCLUSION

In this paper, we propose a laser tuning approach in fully integrated silicon photonic interconnects. The proposed approach allows drastic reductions of the power consumption when the chip activity decreases and when the application BER requirements are lowered. In future work, we will investigate run-time approaches to calibrate the laser injection current.

Figure 1 :

 1 Figure 1: a) a MWSR channel with one wavelength and two writers, b) transmission without thermal variation (ideal case), c) transmission considering traditional MR tuning approach with off-chip laser sources, and d) transmission with our proposed on-chip laser tuning approach.

Figure 2 :

 2 Figure 2: Tuning power (Ptuning) and BER for a MWSR interconnect with 12 interfaces, 4 wavelengths per channel, with FSR=59nm, m=16, thermal efficiency of the voltage tuning TEVT=0.13mW/nm, thermal efficiency of the thermal tuning TETT=0.24mW/nm [7][8].

Table 1 : Power consumption reduction: w/ laser tuning wrt. w/o laser tuning

 1

	Chip activity Target BER	25%	20%	10%
	10 -12	21%	42%	52%
	10 -9	32%	42%	62%