Gérard H E Duchamp
email: gerard.duchamp@lipn.univ-paris13.fr

Ngoc Hoang
email: hoang@univ.lille2.fr

Minh

Quoc Ngo
email: quochoan_ngo@yahoo.com.vn

Hoan

The algebra of Kleene stars of the plane and polylogarithms

Keywords: Algebraically independent, Polylogarithms, Transcendent

We extend the definition and study the algebraic properties of the polylogarithm Li T , where T is rational series over the alphabet

Introduction

In all the sequel of this text, 1. We consider the differential forms ω 0 (z) = dz z and ω 1 (z) = dz 1z .

We denote Ω the cleft plane C -(] -∞, 0] ∪ [1, +∞[) and λ the rational fraction z(1z) -1 belonging to the differential unitary ring C := C[z, z -1 , (1z) -1 ] with the differential operator ∂ z := d/dz and with the unitary element

1 Ω : Ω -→ C, z -→ 1.
2. We construct, over the alphabets X = {x 0 , x 1 }, Y = {y k } k≥1 and Y 0 = Y ∪ {y 0 }, totally ordered by x 0 < x 1 and y 0 > y 1 > • • • respectively, the bialgebras 1 (C X , conc, ∆ ⊔⊔ , 1 X * , ε), (C Y , conc, ∆ , 1 Y * , ε), (C Y 0 , conc, ∆ , 1 Y * 0 , ε). These algebras, when endowed with their dual laws, are equipped with pure transcendence bases in bijection with the set of Lyndon words L yn(X), L yn(Y ) and L yn(Y 0 ) respectively.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. Let π X be the inverse of π • Y :

π X : C Y -→ C ⊕ C X x 1 , y s 1 . . . y s r -→ x s 1 -1 0 x 1 . . . x s r -1 0 x 1 .
The projectors π X 2 and π • Y are mutual adjoints :

∀p ∈ C X , ∀q ∈ C Y , π Y (p) | q = p | π X (q) .
In continuation of [START_REF] Gérard | Harmonic sums and polylogarithms at negative multiindices[END_REF][START_REF] Duchamp | Mathematical renormalization in quantum electrodynamics via noncommutative generating series[END_REF], the principal object of the present work is the polylogarithm well defined, for any r-uplet (s 1 , . . . , s r ) ∈ C r , r ∈ N + and for any z ∈ C such that | z |< 1, as follows Li s 1 ,...,s r (z) := ∑ n 1 >...>n r >0 z n 1 n s 1 1 . . . n s r r .

(

) 1
Then the Taylor expansion of the function (1z) -1 Li s 1 ,...,s r (z) is given by Li s 1 ,...,s r (z)

1 -z = ∑ N≥0 H s 1 ,...,s r (N) z N ,
where the coefficient H s 1 ,...,s r : N -→ Q is an arithmetic function, also called harmonic sum, which can be expressed as follows

H s 1 ,...,s r (N) := ∑ N≥n 1 >...>n r >0 1 n s 1 1 . . . n s r r . (2)
From the analytic continuation of polyzetas [START_REF] Goncharov | Multiple polylogarithms and mixed Tate motives[END_REF][START_REF] Zhao | Analytic continuation of multiple zeta functions[END_REF], for any r ≥ 1, if (s 1 , . . . , s r ) ∈ H r satisfies (3) then 3 , after a theorem by Abel, one obtains the polyzeta as follows

lim z→1 Li s 1 ,...,s r (z) = lim N→∞ H s 1 ,...,s r (N) = ζ (s 1 , . . . , s r ).
This theorem is no more valid in the divergent cases (for (s 1 , . . . , s r ) ∈ N r ) and require the renormalization of the corresponding divergent 2. With a little abuse of language, π X is now considered as tar- geted to C X .

3. For r ≥ 1, ζ (s 1 , . . . , s r ) is as a meromorphic function on

H r = {(s 1 , . . . , s r ) ∈ C r |∀m = 1, . . . , r, ℜ(s 1 ) + . . . + ℜ(s m ) > 1}. (3)
polyzetas. It is already done for the corresponding case of polyzetas at positive multi-indices [START_REF] Costermans | Some Results à l'Abel Obtained by Use of Techniques à la Hopf[END_REF][START_REF] Costermans | Noncommutative algebra, multiple harmonic sums and applications in discrete probability[END_REF][START_REF] Ngoc | Algebraic combinatoric aspects of asymptotic analysis of nonlinear dynamical system with singular inputs[END_REF] and it is done [START_REF] Furusho | Desingularization of multiple zeta-functions of generalized Hurwitz-Lerch type[END_REF][START_REF] Guo | Renormalization of multiple zeta values[END_REF][START_REF] Manchon | Nested sums of symbols and renormalised multiple zeta functions[END_REF] and completed in [START_REF] Gérard | Harmonic sums and polylogarithms at negative multiindices[END_REF][START_REF] Duchamp | Mathematical renormalization in quantum electrodynamics via noncommutative generating series[END_REF] for the case of polyzetas at positive multiindices.

To study the polylogarithms at negative multi-indices, one relies on [START_REF] Gérard | Harmonic sums and polylogarithms at negative multiindices[END_REF][START_REF] Duchamp | Mathematical renormalization in quantum electrodynamics via noncommutative generating series[END_REF] 1. the (one-to-one) correspondence between the multi-indices (s 1 , . . . , s r ) ∈ N r and the words y s 1 . . . y s r defined over Y 0 , 2. indexing these polylogarithms by words y s 1 . . . y s r :

Li - y s 1 ...y sr (z) = Li - s 1 ,...,s r (z) = ∑ n 1 >...>n r >0 n s 1 1 . . . n s r r z n 1 .
In the same way, for polylogarithms at positive indices, one relies on [START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF][START_REF] Ngoc | Lyndon words, polylogarithmic functions and the Riemann ζ function[END_REF] 1. the (one-to-one) correspondence between the combinatorial compositions (s 1 , . . . , s r ) and the words x s 1 -1

0 x 1 . . . x s r -1 0 x 1 in X * x 1 + 1 X * 2.
the indexing of these polylogarithms by words x s 1 -1

0 x 1 . . . x s r -1 0 x 1 : Li x s 1 -1 0 x 1 ...x sr -1 0 x 1 (z) = Li s 1 ,...,s r (z) = ∑ n 1 >...>n r >0 z n 1 n s 1 1 . . . n s r r .
Moreover, one obtained the polylogarithms at positive indices as image by the following isomorphism of the shuffle algebra 4 [START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF] Li

• : (C X , ⊔⊔ , 1 X * ) -→ (C{Li w } w∈X * , ×, 1 Ω ), x n 0 -→ log n (z) n! , x n 1 -→ log n (1/(1 -z)) n! , x s 1 -1 0 x 1 . . . x s r -1 0 x 1 -→ ∑ n 1 >...>n r >0 z n 1 n s 1 1 . . . n s r r .
Extending over the set of rational power series 5 on non commutative variables, C rat X , as follows

S = ∑ n≥0 S | x n 0 x n 0 + ∑ k≥1 ∑ w∈(x * 0 x 1 ) k x * 0 S | w w, Li S (z) = ∑ n≥0 S | x n 0 log n (z) n! + ∑ k≥1 ∑ w∈(x * 0 x 1 ) k x * 0 S | w Li w ,
the morphism Li • is no longer injective over C rat X but {Li w } w∈X * are still linearly independant over C [START_REF] Ngoc | Algebraic combinatoric aspects of asymptotic analysis of nonlinear dynamical system with singular inputs[END_REF][START_REF] Ngoc | Finite polyzêtas, Poly-Bernoulli numbers, identities of polyzêtas and noncommutative rational power series[END_REF].

EXAMPLE 1. i. 1 Ω = Li 1 X * = Li x * 1 -x * 0 ⊔⊔ x * 1 . ii. λ = Li (x 0 +x 1 ) * = Li x * 0 ⊔⊔ x * 1 = Li x * 1 -1 . iii. C = C[Li x * 0 , Li (-x 0 ) * , Li x * 1 ]. iv. C {Li w } w∈X * = {Li S |S ∈ C[x * 0 ] ⊔⊔ C[(-x 0 ) * ] ⊔⊔ C[x * 1 ] ⊔⊔ C X }.
Let us consider also the differential and integration operators, acting on C {Li w } w∈X * [START_REF] Ngoc | On a conjecture by Pierre Cartier about a group of associators[END_REF] :

∂ z = d dz , θ 0 = z d dz , θ 1 = (1 -z) d dz , ∀ f ∈ C , ι 0 ( f ) = z z 0 f (s)ω 0 (s) and ι 1 ( f ) = z 0 f (s)ω 1 (s).
4. As follows defined on a superset of the of Lyndon words, as pure transcendence basis, and extended by algebraic specialization [START_REF] Ngoc | Evaluation Transform[END_REF][START_REF] Ngoc | Summations of Polylogarithms via Evaluation Transform[END_REF].

5. C rat X is the closure by rational operations {+, conc, * } of C X , where, for S ∈ C X such that S | 1 X * = 0, one has [START_REF] Berstel | Rational series and their languages[END_REF]

S * = ∑ k≥0 S k .
Here, the operator ι 0 is well-defined (as in definition 1 in section 2.2) then one can check easily [START_REF] Ngoc | Differential Galois groups and noncommutative generating series of polylogarithms[END_REF][START_REF] Ngoc | Finite polyzêtas, Poly-Bernoulli numbers, identities of polyzêtas and noncommutative rational power series[END_REF][START_REF] Gérard | Harmonic sums and polylogarithms at negative multiindices[END_REF][START_REF] Duchamp | Mathematical renormalization in quantum electrodynamics via noncommutative generating series[END_REF] 1. The subspace C {Li w } w∈X * is closed under the action of {θ 0 , θ 1 } and {ι 0 , ι 1 }.

2. The operators {θ 0 , θ 1 , ι 0 , ι 1 } satisfy in particular,

θ 1 + θ 0 = θ 1 , θ 0 = ∂ z and ∀k = 0, 1, θ k ι k = Id, [θ 0 ι 1 , θ 1 ι 0 ] = 0 and (θ 0 ι 1 )(θ 1 ι 0 ) = (θ 1 ι 0 )(θ 0 ι 1 ) = Id.
3. θ 0 ι 1 and θ 1 ι 0 are scalar operators within C {Li w } w∈X * , res- pectively with eigenvalues λ and 1/λ , i.e.

(θ 0 ι 1 ) f = λ f and (θ 1 ι 0 ) f = (1/λ ) f . 4. Let w = y s 1 . . . y s r ∈ Y * (then π X (w) = x s 1 -1 0 x 1 . . . x s r -1 0 x 1 ) and u = y t 1 . . . y t r ∈ Y * 0 . The functions Li w , Li - u satisfy Li w = (ι s 1 -1 0 ι 1 . . . ι s r -1 0 ι 1 )1 Ω , Li - u = (θ t 1 +1 0 ι 1 . . . θ t r +1 0 ι 1 )1 Ω , ι 0 Li π X (w) = Li x 0 π X (w) , ι 1 Li w = Li x 1 π X (w) , θ 0 Li x 0 π X (w) = Li π X (w) , θ 1 Li x 1 π X (w) = Li π X (w) , θ 0 Li x 1 π X (w) = λ Li π X (w) , θ 0 Li x 1 π X (w) = Li π X (w) /λ .
Here, we explain the whole project of extension of Li • , study different aspects of it, in particular what is desired of ι i , i = 0, 1. The interesting problem in here is that what we do expect of ι i , i = 0, 1. In fact, the answers are -it is a section of θ i , i = 0, 1 (i.e. takes primitives for the cor- responding differential operators). -it extends ι i , i = 0, 1 (defined on C{Li w } w∈X * , and very sur- prisingly, although not coming directly from Chen calculus, they provide a group-like generating series) We will use this construction to extend Li • to C {Li w } w∈X * and, after that, we extend it to a much larger rational algebra.

Background

Standard topology on H (Ω)

The algebra H (Ω) is that of analytic functions defined over Ω. It is endowed with the topology of compact convergence whose seminorms are indexed by compact subsets of Ω, and defined by

p K ( f ) = || f || K = sup s∈K | f (s)|. Of course, p K 1 ∪K 2 = sup(p K 1 , p K 2 ),
and therefore the same topology is defined by extracting a fondamental subset of seminorms, here it can be choosen denumerable. As H (Ω) is complete with this topology it is a Frechet space and even, as

p K ( f g) ≤ p K ( f )p K (g),
it is a Frechet algebra (even more, as p K (1 Ω ) = 1 a Frechet algebra with unit).

With the standard topology above, an operator φ ∈ End(H (Ω)) is continuous iff (with K i compacts of Ω)

(∀K 2 )(∃K 1 )(∃M 12 > 0)(∀ f ∈ H (Ω))(||φ ( f )|| K 2 ≤ M 12 || f || K 1 ).

Study of continuity of the sections θ i and ι i

Then, C {Li w } w∈X * (and H (Ω)) being closed under the operators θ i ; i = 0, 1. We will first build sections of them, then see that they are continuous and, propose (discontinuous) sections more adapted to renormalisation and the computation of associators.

For

z 0 ∈ Ω, let us define ι z 0 i ∈ End(H (Ω)) by ι z 0 0 ( f ) = z z 0 f (s)ω 0 (s), ι z 0 1 ( f ) = z z 0 f (s)ω 1 (s).
It is easy to check that θ i ι z 0 i = Id H (Ω) and that they are continuous on H (Ω) for the topology of compact convergence because for all K ⊂ compact Ω, we have

|p K (ι z 0 i ( f )| ≤ p K ( f ) sup z∈K | z z 0 ω i (s)| ,
and, in view or paragraph (2.1), this is sufficient to prove continuity. Hence the operators ι z 0 i are also well defined on C {Li w } w∈X * and it is easy to check that

ι z 0 i (C {Li w } w∈X * ) ⊂ C {Li w } w∈X * . Due to the decomposition of H (Ω) into a direct sum of closed subspaces H (Ω) = H z 0 →0 (Ω) ⊕ C1 Ω ,
it is not hard to see that the graphs of θ i are closed, thus, the θ i are also continuous.

Much more interesting (and adapted to the explicit computation of associators) we have the operator ι i (without superscripts), men- tioned in the introduction and (rigorously) defined by means of a C-basis of

C {Li w } w∈X * = C ⊗ C C{Li w } w∈X * . As C{Li w } w∈X * ∼ = C[L yn(X)],
one can partition the alphabet of this polynomial algebra in (L yn(X) ∩ X * x 1 ) ⊔ {x 0 } and then get the decomposition

C {Li w } w∈X * = C ⊗ C C{Li w } w∈X * x 1 ⊗ C C{Li w } w∈x * 0 .
In fact, we have an algorithm to convert Li ux 1 x n 0 as Li ux 1 x n

0 (z) = ∑ m≤n P m log m (z) = ∑ m≤n w∈X * x 1 P m | w Li w (z) log m (z).
due to the identity [START_REF] Ngoc | Summations of Polylogarithms via Evaluation Transform[END_REF] ux

1 x n 0 = ux 1 ⊔⊔ x n 0 - n ∑ k=1 (u ⊔⊔ x k 0 )x 1 x n-k 0 .
This means that

B = z k Li ux 1 (z) Li x n 0 (z) (k,n,u)∈Z×N×X * ⊔ 1 (1 -z) l Li ux 1 (z) Li x n 0 (z) (l,n,u)∈N 2 + ×X * ⊔ z k Li x n 0 (z) (k,n)∈Z×N + ⊔ 1 (1 -z) l Li x n 0 (z) (k,l,n)∈N 2 + is a C-basis of C {Li w } w∈X * .
With this basis, we can define the operator ι 0 as follows DEFINITION 1. Define the index map ind :

B → Z by ind( z k (1 -z) l Li x n 0 (z)) = k, ind( z k (1 -z) l Li ux 1 (z) log n (z)) = k + |ux 1 |. Now ι 0 is computed by : 1. ι 0 (b) = z 0 b(s)ω 0 (s) if ind(b) ≥ 1. 2. ι 0 (b) = z 1 b(s)ω 0 (s) if ind(b) ≤ 0.
To show discontinuity of ι 0 with a direct example, the technique consists in exhibiting 2 sequences f n , g n ∈ C{Li w } w∈X * converging to the same limit but such that lim ι 0 ( f n ) = lim ι 0 (g n ).

Here we choose the function z which can be approached by two limits. For continuity, we should have "equality of the limits of the image-sequences" which is not the case.

z = ∑ n≥0 log n (z) n! , z = ∑ n≥1 (-1) n+1 n! log n ( 1 1 -z ).
Let then

f n = ∑ 0≤m≤n log m (z) m! and g n = ∑ 1≤m≤n (-1) m+1 m! log m ( 1 1 -z ), we have f n , g n ∈ C{Li w } w∈X * and ι 0 ( f n ) = f n+1 -1. Hence, one has lim(ι 0 ( f n )) = z -1. On the other hand lim ι 0 (g n ) = lim z 0 g n (s)ω 0 (s) = z 0 lim g n (s)ω 0 (s) = z 0 sω 0 (s) = z.
The exchange of the integral with the limit above comes from the fact that the operator

φ → z 0 φ (s)ω 0 (s), is continuous on the space H 0 (Ω ∪ B(0, 1)) of analytic functions f ∈ H (Ω ∪ B(0, 1)) such that f (0) = 0 (B(0, 1)
is the open ball of center 0 and radius 1).

Algebraic extension of Li

• to (C rat X , ⊔⊔ , 1 X * )[x * 0 , (-x 0 ) * , x * 1 ]
We will use several times the following lemma which is characteristicfree.

LEMMA 1. Let (A , d) be a commutative differential ring without zero divisor, and R = ker(d) be its subring of constants. Let z ∈ A such that d(z) = 1 and S = {e α } α∈I be a set of eigenfunctions of d all different (I ⊂ R) i.e.

i. e α = 0. ii. d(e α ) = αe α ; α ∈ I. Then the family (e α ) α∈I is linearly free over R[z] 6 .

PROOF. If there is no non-trivial R[z]-linear relation, we are done. Otherwise let us consider relations

N ∑ j=1 P j (z)e α j = 0, (4)
with P j ∈ R[t] pol \{0} 7 for all j (packed linear relations). We choose one minimal w.r.t. the triplet

[N, deg(P N ), ∑ j<N deg(P j )], (5)
6. Here R[z] is understood as ring adjunction i.e. the smallest subring generated by R ∪ {z}.

7. Here R[t] pol means the formal univariate polynomial ring (the subscript is here to avoid confusion). lexicographically ordered from left to right 8 .

Remarking that d(P(z)) = P ′ (z) (because d(z) = 1), we apply the operator d -α N Id to both sides of (4) and get

N ∑ j=1 P ′ j (z) + (α j -α N )P j (z) e α j = 0. (6)
Minimality of (4) implies that ( 6) is trivial i.e.

P ′ N (z) = 0 ; (∀ j = 1..N -1)(P ′ j (z) + (α j -α N )P j (z) = 0). (7) Now relation (4) implies ∏ 1≤ j≤n-1 (α N -α j ) N ∑ j=1 P j (z)e α j = 0, (8)
which, because A has no zero divisor, is packed and has the same associated triplet ( 5) as [START_REF] Costermans | Noncommutative algebra, multiple harmonic sums and applications in discrete probability[END_REF]. From ( 7), we see that for all k < N ∏ 1≤ j≤n-1

(α N -α j )P k (z) = ∏ 1≤ j≤n-1 j =k (α N -α j )P ′ k (z), so, if N ≥ 2,
∆ ⊔⊔ ( B α α! ) = ∑ α=α 1 +α 2 B α 1 α 1 ! ⊗ B α 2 α 2 ! . Hence, if S, T ∈ (k Z , ⊔⊔ , 1 Z * ), considering S ⊔⊔ T | B α α! = S ⊗ T | ∆ ⊔⊔ ( B α α! ) = ∑ α=α 1 +α 2 S | B α 1 α 1 ! T | B α 2 α 2 ! , we see that (k Z , ⊔⊔ , 1 Z * ) ≃ (k[[I]],
., 1) (commutative algebra of formal series) which has no zero divisor).

LEMMA 3. Let A be a Q-algebra (associative, unital, commutative) and z an indeterminate, then e

z ∈ A [[z]] is transcendent over A [z].
PROOF. It is straightforward consequence of Remark [START_REF] Berstel | Rational series and their languages[END_REF]. Note that this can be rephrased as [z, e z ] are algebraically independant over A .

PROPOSITION 1. Let Z = {z n } n∈N be an alphabet, then [e z 0 , e z 1 ] is algebraically independent on C[Z] within C[[Z]].

PROOF. Using lemma 3, one can prove by recurrence that

[e z 0 , e z 1 , • • • , e z k , z 0 , z 1 , • • • , z k ],
are algebraically independent. This implies that Z ⊔ {e z } z∈Z is an algebraically independent set and, by restriction Z ⊔{e z 0 , e z 1 } whence the proposition. 8. i.e. consider the ones with N minimal and among these, we choose one with deg(P N ) minimal and among these we choose one with ∑ j<N deg(P j ) minimal.

COROLLARY 1.

i. The family {x *

0 , x * 1 } is algebraically in- dependent over (C X , ⊔⊔ , 1 X * ) within (C X rat , ⊔⊔ , 1 X * ). ii. (C X , ⊔⊔ , 1 X * )[x * 0 , x * 1 , (-x 0 ) * ] is a free module over C X , the family {(x * 0 ) ⊔⊔ k ⊔⊔ (x * 1 ) ⊔⊔ l } (k,l)∈Z×N is a C X -basis of it. iii. As a consequence, {w ⊔⊔ (x * 0 ) ⊔⊔ k ⊔⊔ (x * 1 ) ⊔⊔ l } w∈X * (k,l)∈Z×N is a C-basis of it.
PROOF. Chase denominators and use a theorem by Radford [START_REF] Reutenauer | Free Lie Algebras[END_REF] with Z = L yn(X). COROLLARY 2. There exists a unique morphism µ, from

(C X , ⊔⊔ , 1 X * )[x * 0 , (-x 0 ) * , x * 1 ] to H (Ω) defined by i. µ(w) = Li w for any w ∈ X * , ii. µ(x * 0 ) = z, iii. µ((-x 0 ) * ) = 1/z, iv. µ(x * 1 ) = 1/(1 -z).
DEFINITION 2. We call Li

(1)

• the morphism µ.

Remark that the image of (C X , ⊔⊔ , 1

X * )[x * 0 , (-x 0 ) * , x * 1 ] by Li (1) • (sect. 3) is exactly C {Li w } w∈X * .
And the operator ι 0 defined by means of Li • is the same as the one defined by tensor decomposition. We have a diagram as follows

(C X , ⊔⊔ , 1 X * ) C{Li w } w∈X * (C X , ⊔⊔ , 1 X * )[x * 0 , (-x 0 ) * , x * 1 ] C {Li w } w∈X * C X ⊔⊔ C rat x 0 ⊔⊔ C rat x 1 H (Ω) C X ⊗ C C rat x 0 ⊗ C C rat x 1 Li • Li (1) • Li (2)
• DIAGRAM 1. Arrows and spaces obtained in this project (so far). Among horizontal arrows only Li • is into (and is an isomorphism) Li

(1)

• and Li

(2)

• are not into (for example, the image of the non-zero element x * 0 ⊔⊔ x * 1 -x * 1 +1 is zero, see Example 1). The image of Li

(2)

• is presumably

Im(SP C (X)){Li w } w∈X * ≃ C{z α (1 -z) β } α,β ∈C ⊗ C C{Li w } w∈X * .

Extension to

C X ⊔⊔ C rat x 0 ⊔⊔ C rat x 1 4.1 Study of the shuffle algebra SP C (X)
Indeed, the set (a 0 x 0 + a 1 x 1 ) * a 0 ,a 1 ∈C is a shuffle monoid as

(a 0 x 0 + a 1 x 1 ) * ⊔⊔ (b 0 x 0 + b 1 x 1 ) * = ((a 0 + b 0 )x 0 + (a 1 + b 1 )x 1 ) * .
As there are many relations between these elements (as a monoid it is isomorphic to C 2 , hence far from being free), we study here the vector space

SP C (X) = span C {(a 0 x 0 + a 1 x 1 ) * } a 0 ,a 1 ∈C .
It is a shuffle sub-algebra of (C) rat x 0 ⊔⊔ (C) rat x 1 which will be called star of the plane. Note that it is also a shuffle sub-algebra of the algebra (C exch X , ⊔⊔ , 1 X * ) of exchangeable series. We can give the DEFINITION 3. A series is said exchangeable iff whenever two words have the same multidegree (here bidegree) then they have the same coefficient within it. Formally for all u, v ∈ X *

(∀x ∈ X)(|u| x = |v| x ) =⇒ S | u = S | v .
On the other hand, for any S ∈ C X , we can write

S = ∑ n≥0 P n ,
where P n ∈ C[X] such that deg P n = n for every n ≥ 0. Then S is called exchangeable iff P n are symmetric by permutations of places for every n ∈ N. If S is written as above then we can write

P n = n ∑ i=0 α n,i x i 0 ⊔⊔ x n-i 1 .
DEFINITION 4. Let S ∈ C X (resp. C X ) and let P ∈ C X (resp. C X ). The left and right residual of S by P are respectively the formal power series P ⊳ S and S ⊲ P in C X defined by

P ⊳ S | w = S | wP (resp. S ⊲ P | w = S | Pw ).
For any S ∈ C X (resp. C X ) and P, Q ∈ C X (resp. C X ), we straightforwardly get

P ⊳ (Q ⊳ S) = PQ ⊳ S, (S ⊲ P) ⊲ Q = S ⊲ PQ, (P ⊳ S) ⊲ Q = P ⊳ (S ⊲ Q).
In case x, y ∈ X and w ∈ X * , we get 9

x ⊳ (wy) = δ y x w and xw ⊲ y = δ y x w.

THEOREM 1. Le δ ∈ Der(C X , ⊔⊔ , 1 X * ). Moreover, we suppose that δ is locally nilpotent 10 . Then the family (tδ ) n /n! is summable and its sum, denoted exp(tδ ), is is a one-parameter group of automorphisms of (C X , ⊔⊔ , 1 X * ). THEOREM 2. Let L be a Lie series 11 . Let δ r L and δ l L be defined respectively by δ r L (P) := P ⊳ L and δ l L (P) := L ⊲ P. Then δ r L and δ l L are locally nilpotent derivations of (C X , ⊔⊔ , 1 X * ). Therefore, exp(tδ r L ) and exp(tδ l L ) are one-parameter groups of Aut(C X , ⊔⊔ , 1 X * ) and exp(tδ r L )P = P ⊳ exp(tL) and exp(tδ l L )P = exp(tL) ⊲ P.

It is not hard to see that the algebra C X ⊔⊔ C rat x 0 ⊔⊔ C rat x 1 is closed by the shuffle derivations 12 δ l x 0 , δ l x 1 . In particular, on it, these derivations commute 13 with δ r x 0 and δ r x 1 , respectively, i.e., for any x ∈ X and S ∈ C X ⊔⊔ C rat x 0 ⊔⊔ C rat x 1 , one has δ l

x (S) = δ r x (S). Moreover, one has

(αδ l x 0 + β δ l x 1 )[(a 0 x 0 + a 1 x 1 ) * ] = (αa 0 + β a 1 )[(a 0 x 0 + a 1 x 1 ) * ], 9. For any words u, v ∈ X * , if u = v then δ v u = 1 else 0. 10. φ ∈ End(V ) is said to be locally nilpotent iff, for any v ∈ V , there exists N ∈ N s.t. φ N (v) = 0. 11. i.e. ∆ ⊔⊔ (L) = L ⊗1 + 1 ⊗L [23].
12. These operators are, in fact, the shifts of Harmonic Analysis and therefore defined as adjoints of multiplication, i.e.

∀S ∈ C x , δ l x (S) | w = S | xw .
13. Thus, in this case, the operator δ l x has the same meaning as the operator S → S x in [START_REF] Duchamp | Sweedler's duals and Schützenberger's calculus[END_REF], x -1 in the Theory of Languages and • in [START_REF] Berstel | Rational series and their languages[END_REF][START_REF] Reutenauer | Free Lie Algebras[END_REF]. from this we get that the family {(a 0 x 0 + a 1 x 1 ) * } a 0 ,a 1 ∈C is linearly free over C SP C (X) = (a 0 ,a 1 )∈C C{(a 0 x 0 + a 1 x 1 ) * }.

We can get an arrow of Li

(2)

• type (SP C (X), ⊔⊔ , 1 X * ) -→ H (Ω) by sending (a 0 x 0 + a 1 x 1 ) * = (a 0 x 0 ) * ⊔⊔ (a 1 x 1 ) * -→ z a 0 (1 -z) -a 1 .
In particular, for any n ∈ N + , one has Li - 0, . . . , 0 n times

(z) = Li (2)
(nx 0 +nx 1 ) * (z).

This arrow is a morphism for the shuffle product.

Study of the algebra

C X ⊔⊔ C rat x 0 ⊔⊔ C rat x 1
We will start by the study of the one-letter shuffle algebra, i.e. (C rat x , ⊔⊔ , 1 x * ) and use two times Lemma 1 above.

Let us now consider A = C rat x ; e α = (αx) * , α ∈ C endowed with d = δ l x (which is a derivation for the shuffle) and z = x. We are in the conditions of Lemma 1 and then ((αx) * ) α∈C is C[x]-linearly free which amounts to say that

B 0 = (x ⊔⊔ k ⊔⊔ (αx) * ) k∈N,α∈C , is C-linearly free in C rat x .
To see that it is a basis, it suffices to prove that B 0 is (linearly) generating. Consider that

C rat x = {P/Q} P,Q ∈ C[x],Q(0) =0 ,
then, as C is algebraically closed, we have a basis

B 1 ∪ B 2 = {x k } k≥0 ∪ {((αx) * ) l } α∈C * ,l≥1 ,
and it suffices to see that we can generate B 2 by elements of B 0 , which s a consequence of the two identities

x ⊔⊔ ((αx) * ) n = n+1 ∑ j=1 α(n, j)((αx) * ) ⊔⊔ j with α(n, n + 1) = 0, x k ⊔⊔ (αx) * = 1 k! (x ⊔⊔ k ⊔⊔ (αx) * ).

Now, we use again Lemma 1 with

A = C rat x 0 ⊔⊔ C rat x 1 ⊂ C x 0 , x 1 , hence without zero divisor (see Lemma 2), endowed with d = δ l x 1 then (x ⊔⊔ k 0 ⊔⊔ (αx 0 ) * ) k∈N;
α∈C , is linearly free over R = C rat x 0 . It is easily seen, using a decomposition like

S = ∑ p≥0,q≥0 S | x ⊔⊔ p 0 ⊔⊔ x ⊔⊔ q 1 x ⊔⊔ p 0 ⊔⊔ x ⊔⊔ q 1 ,
that C rat x 0 = ker(d) and one obtains then that the arrow

C rat x 0 ⊗ C C rat x 1 → C rat x 0 ⊔⊔ C rat x 1 ⊂ C rat x 0 , x 1 is an isomorphism. Hence, (x ⊔⊔ k 0 0 ⊔⊔ (α 0 x 0 ) * ⊔⊔ x ⊔⊔ k 1 1 ⊔⊔ (α 1 x 1 ) * ) k i ∈N; α i ∈C is a C-basis of A = C rat x 0 ⊔⊔ C rat x 1 . In order to extend Li • to A we send T (k 0 , k 1 , α 0 , α 1 ) = x ⊔⊔ k 0 0 ⊔⊔ (α 0 x 0 ) * ⊔⊔ x ⊔⊔ k 1 1 ⊔⊔ (α 1 x 1 ) * , to log k 0 (z)z α 0 log k 1 (1/(1 -z))(1/(1 -z)) α 1 ,
and see that the constructed arrow follows multiplication given by

T ( j 0 , j 1 , α 0 , α 1 )T (k 0 , k 1 , β 0 , β 1 ) = T ( j 0 + k 0 , j 1 + k 1 , α 0 + β 0 , α 1 + β 1 ).
Using, once more, Lemma 1, one gets PROPOSITION 2. The family {(α 0 x 0 ) * ⊔⊔ (α

1 x 1 ) * } α i ∈C is a (C X , ⊔⊔ , 1)-basis of C X ⊔⊔ C rat x 0 ⊔⊔ C rat x 1 , then we have a C-basis {w ⊔⊔ (α 0 x 0 ) * ⊔⊔ (α 1 x 1 ) * } α i ∈C w∈X * of C X ⊔⊔ C rat x 0 ⊔⊔ C rat x 1 = C X [C rat x 0 ⊔⊔ C rat x 1 ] = C X ⊔⊔ SP C (X).
PROOF. We will use a multi-parameter consequence of Lemma 1.

LEMMA 4. Let Z be an alphabet, and k a field of characteristic zero. Then, the family

{e αz } z∈Z α∈k ⊂ k[[Z]] is linearly independent over k[Z].
This proves that, in the shuffle algebra the elements

{(a 0 x 0 ) * ⊔⊔ (a 1 x 1 ) * } a 0 ,a 1 ∈C 2 are linearly independent over C X ≃ C[L yn(X)] within (C X , ⊔⊔ , 1 X * ).

Now Li

(2)

• is well-defined and this morphism is not into from

C X ⊔⊔ C rat x 0 ⊔⊔ C rat x 1 = C X [C rat x 0 ⊔⊔ C rat x 1 ] = C X ⊔⊔ SP C (X), to Im(Li (2) • ). PROPOSITION 3. Let Li (1) • : C X [x * 0 , x * 1 , (-x 0 ) * ] → H (Ω) then i. Im(Li (1) • ) = C {Li w } w∈X * . ii. ker(Li (1) • ) is the ideal generated by x * 0 ⊔⊔ x * 1 -x * 1 + 1 X * . PROOF. As C X [x * 0 , x * 1 , (-x 0 ) * ] admits {(x * 0 ) ⊔⊔ k ⊔⊔ (x * 1 ) ⊔⊔ l } l∈N k∈Z
as a basis for its structure of C X -module, it suffices to remark Li

(1) (x * 0 ) ⊔⊔ k ⊔⊔ (x * 1 ) ⊔⊔ l (z) = z k × 1 (1 -z) l is a generating system of C .
First of all, we recall the following lemma LEMMA 5. Let M 1 and M 2 be K-modules (K is a unitary ring). Suppose φ : M 1 → M 2 is a linear mapping. Let N ⊂ ker(φ ) be a submodule. If there is a system of generators in M 1 , namely {g i } i∈I , such that 1. For any i ∈ I \ J, g i ≡ ∑ j∈J⊂I c j i g j [mod N], (c j i ∈ K; ∀ j ∈ J) ; 2. {φ (g j )} j∈J is K-free in M 2 ; then N = ker(φ ).

PROOF. Suppose P ∈ ker(φ ). Then P ≡ ∑ j∈J p j g j [mod N] with {p j } j∈J ⊂ K. Then 0 = φ (P) = ∑ j∈J p j φ (g j ). From the fact that {φ (g J )} J∈J is K-free on M 2 , we obtain p j = 0 for any j ∈ J. This means that P ∈ N. Thus ker(φ ) ⊂ N. This implies that N = ker(φ ).

Let now J be the ideal generated by x * 0 ⊔⊔ x * 1x * 1 + 1 X * . It is easily checked, from the following formulas, (for l ≥ 1) 14

w ⊔⊔ x * 0 ⊔⊔ (x * 1 ) ⊔⊔ l ≡ w ⊔⊔ (x * 1 ) ⊔⊔ l -w ⊔⊔ (x * 1 ) ⊔⊔ l-1 [J ],
14. In figure 1, (w, l, k) codes the element w ⊔⊔ (x * 0 ) ⊔⊔ l ⊔⊔ (x * 1 ) ⊔⊔ k .

(w, l, k) .

(1 X * , ×, ×) k • • (w, -l, k) l -l ⊳ ⊲ (w, l -1, k) (w, l -1, k -1) ⊲ ▽ (w, -l + 1, k) (w, -l, k -1)
w ⊔⊔ (-x 0 ) * ⊔⊔ (x * 1 ) ⊔⊔ l ≡ w ⊔⊔ (-x 0 ) * ⊔⊔ (x * 1 ) ⊔⊔ l-1 + w ⊔⊔ (x * 1 ) ⊔⊔ l [J ],
that one can rewrite [mod J ] any monomial w ⊔⊔ (x * 0 ) ⊔⊔ k ⊔⊔ (x * 1 ) ⊔⊔ l as a linar combination of such monomials with kl = 0. Then, applying lemma 5 to the map φ = Li (1)

• and the modules

M 1 = C X [x * 0 , x * 1 , (-x 0 ) * ], M 2 = H (Ω)
, N = J , the families and indices

{g i } = {w ⊔⊔ (x * 1 ) ⊔⊔ n ⊔⊔ (x * 0 ) ⊔⊔ m } (w,n,m)∈I , I = X * × N × Z, J = (X * × N × {0}) ⊔ (X * × {0} × Z),
we have the second point of proposition 3.

Of course, we also have

(x * 0 ⊔⊔ x * 1 -x * 1 + 1 X * ) ⊂ ker(Li (2)
• ), but the converse is conjectural.

Applications on polylogarithms

Let us consider also the following morphisms ℑ and Θ of algebras C X → End(C {Li w }) defined by i. ℑ(w) = Id and

Θ(w) = Id, if w = 1 X * . ii. ℑ(w) = ℑ(v)ι i and Θ(w) = Θ(v)θ i , if w = vx i , x i ∈ X, v ∈ X * .
For any n ≥ 0 and u ∈ X * , f , g ∈ C {Li w } w∈X * , one has [START_REF] Gérard | Harmonic sums and polylogarithms at negative multiindices[END_REF][START_REF] Duchamp | Mathematical renormalization in quantum electrodynamics via noncommutative generating series[END_REF] ∂

n z = ∑ w∈X n µ • (Θ ⊗ Θ)[∆ ⊔⊔ (w)], Θ(u)( f g) = µ • (Θ ⊗ Θ)[∆ ⊔⊔ (u)] • ( f ⊗ g).
By extension to complex coefficients, we obtain

H conc ∼ = (C Θ(X) , conc, Id, ∆ ⊔⊔ , ε), H ⊔⊔ ∼ = (C ℑ(X) , ⊔⊔ , Id, ∆ conc , ε).
Hence, THEOREM 3 (DERIVATIONS AND AUTOMORPHISMS).

Let

P, Q ∈ C X (resp. C[x * 0 , (-x 0 ) * , x * 1 ] ⊔⊔ C X ), T ∈ L ie C X (resp. L ie C X ). Then Θ(T ) is a derivation in (C{Li w } w∈X * , ×, 1) (resp. (C {Li w } w∈X * , ×, 1 Ω ))
and exp(tΘ(T )) is then a one-parameter group of automorphisms of

(C{Li w } w∈X * , ×, 1 Ω ) (resp. (C {Li w } w∈X * , ×, 1 Ω )).

PROOF. Because Li

P ⊔⊔ Q = Li P Li Q , Θ(T ) Li P ⊔⊔ Q = Li (P ⊔⊔ Q)⊳T and then Θ(T )(Li P Li Q ) = Li (P ⊔⊔ Q)⊳T = Li (P⊳T ) ⊔⊔ Q+P ⊔⊔ (Q⊳T ) = Li (P⊳T ) ⊔⊔ Q + Li P ⊔⊔ (Q⊳T ) = (Θ(T ) Li P ) Li Q + Li P (Θ(T ) Li Q ). THEOREM 4 (EXTENSION OF Li • ).
The following map is surjective

(C[x * 0 ] ⊔⊔ C[(-x 0 ) * ] ⊔⊔ C[x * 1 ] ⊔⊔ C X , ⊔⊔ , 1 X * ) → (C {Li w } w∈X * , ×, 1), T → ℑ(T )1 Ω .
One has, for any u ∈ Y * ,

Li - y s 1 u = θ s 1 0 (θ 0 ι 1 ) Li - u = θ s 1 0 (λ Li - u ) = s 1 ∑ k 1 =0 s 1 k 1 (θ k 1 0 λ )(θ s 1 -k 1 0 Li - u ).
Hence, successively [START_REF] Gérard | Harmonic sums and polylogarithms at negative multiindices[END_REF],

Li - y s 1 ...y sr = s 1 ∑ k 1 =0 s 1 +s 2 -k 1 ∑ k 2 =0
. . .

(s 1 +...+sr )- (k 1 +...+k r-1 ) ∑ k r =0 s 1 k 1 s 1 + s 2 -k 1 k 2 . . . s 1 + . . . + s r -k 1 -. . . -k r-1 k r (θ k 1 0 λ )(θ k 2 0 λ ) . . . (θ k r 0 λ ),
where

θ k i 0 λ (z) =      λ (z), if k i = 0, 1 1 -z k i ∑ j=1 S 2 (k i , j) j!λ j (z), if k i > 0. Hence, Li - y s 1 ...y sr = Li T = ℑ(T )1 Ω ,
where T is the following exchangeable rational series

T = s 1 ∑ k 1 =0 s 1 +s 2 -k 1 ∑ k 2 =0
. . . where R is the following exchangeable rational series

R = s 1 ∑ k 1 =0 s 1 +s 2 -k 1 ∑ k 2 =0
. . .

(
F k i =      x * 1 -1 X * , if k i = 0, x * 1 ⊔⊔ k i ∑ j=1 S 2 (k i , j) j!(x * 1 -1 X * ) ⊔⊔ j , if k i > 0.
Since ℑ(x * 1 )1 Ω = 1/(1z) then this proves once again that [START_REF] Gérard | Harmonic sums and polylogarithms at negative multiindices[END_REF][START_REF] Duchamp | Mathematical renormalization in quantum electrodynamics via noncommutative generating series[END_REF] Li - y s 1 ...

Conclusion

We have studied the structure of the algebra

C X ⊔⊔ C rat x 0 ⊔⊔ C rat x 1 ,
where X = {x 0 , x 1 } is an alphabet. We have also considered the ways for denoting the polylogarithms. By the results on the algebra C X ⊔⊔ C rat x 0 ⊔⊔ C rat x 1 , we have given an extension of the polylogarithms and have obtained polylogarithmic transseries With this extension, we have constructed several shuffle bases of the algebra of polylogarithms. In the special case of the "Laurent subalgebra"

(C X , ⊔⊔ , 1 X * )[x * 0 , (-x * 0 ), x * 1 ] ⊂ C X ⊔⊔ C rat x 0 ⊔⊔ C rat x 1 ,
we have completely characterized the kernel of the polylogarithmic map Li • , providing a rewriting process which terminates to a normal form.

1 0x 1 . . . x s r - 1 0x 1 -

 1111 c 2016 ACM. ISBN 978-1-4503-2138-9. DOI: 10.1145/12351. Which are all Hopf save the last one due to y 0 which is infiltration-like[START_REF] Bui | Pure) transcendence bases in φ -deformed shuffle bialgebras[END_REF].Let us consider also the following morphismπ • Y : (C ⊕ C X x 1 , conc) -→ (C Y , .), x s 1 -→ y s 1 . . .y s r , for r ≥ 1 and, for any a ∈ C, π • Y (a) = a. The extension of π • Y over C X is denoted by π Y : C X -→ C Y satisfying, for any p ∈ C X x 0 , π Y (p) = 0. Hence, ker(π Y ) = C X x 0 and Im (π Y ) = C Y .

  we would get a relation of lower triplet[START_REF] Gérard | Harmonic sums and polylogarithms at negative multiindices[END_REF]. This being impossible, we get N = 1 and (4) boils down to P N (z)e N = 0 which, as A has no zero divisor, implies P N (z) = 0, contradiction.Then the(e α ) α∈I are R[z]-linearly independent. REMARK 1. If A is a Q-algebra or only of characteristic zero (i.e., n1 A = 0 ⇒ n = 0), then d(z) = 1 implies that z is transcendent over R.First of all, let us prove LEMMA 2. Let k be a field of characteristic zero and Z an alphabet. Then (k Z , ⊔⊔ , 1 Z * ) is a k-algebra without zero divisor.PROOF. Let B = (b i ) i∈I be an ordered basis of L ie k Z and ( B α α! ) α∈N (I) the divided corresponding PBW basis. One has

Figure 1 :

 1 Figure 1: Rewriting modJ of {w ⊔⊔ (x * 0 ) ⊔⊔ l ⊔⊔ (x * 1 ) ⊔⊔ k } k∈N,l∈Z w∈X *

(s 1 ∑ k r =0 s 1 k 1 s 1 + s 2 -k 1 k 2 . . . s 1 +S 2

 111212 +...+sr )-(k 1 +...+k r-1 ) . . .+ s rk 1 -. . .k r-1 k r T k 1 ⊔⊔ . . . ⊔⊔ T k r , + x 1 ) * , if k i = 0, (k i , j) j!((x 0 + x 1 ) * ) ⊔⊔ j , if k i > 0. Due to surjectivity of Li • , from C[x * 0 ] ⊔⊔ C[(-x 0 ) * ] ⊔⊔ C[x * 1 ] ⊔⊔ C X to C {Li w } w∈X * , one also has Li - y s 1 ...y sr = Li R = ℑ(R)1 Ω ,

  y sr = Li T = L R = Li F ∈ C[1/(1z)] C .One can deduce finally that COROLLARY 3.C {Li w } w∈X * C[1/(1z)]{Li w } w∈X * = span C    ∑ n 1 >...>n r >0 n s 1 1 . . . n s r r z n 1 | (s 1 , . . . , s r ) ∈ Z r , r ∈ N +    .

C{z α ( 1 -

 1 z) β Li w } w∈X * α,β ∈C .

  s 1 +...+sr )-(k 1 +...+k r-1 ) 1 + . . . + s rk 1 -. . .k r-1 k r R k 1 ⊔⊔ . . . ⊔⊔ R k r , 0 ⊔⊔ x * 1 ) ⊔⊔ j , if k i > 0,and again (see Example 1)Li - y s 1 ...y sr = Li F = ℑ(F)1 Ω ,where F is the following rational series onx 1 s 1 +...+sr )-(k 1 +...+k r-1 ) 1 + s 2k 1 k 2 . . . s 1 + . . . + s rk 1 -. . .k r-1 k r F k 1 ⊔⊔ . . . ⊔⊔ F k r ,

							F =	s 1 ∑ k 1 =0	s 1 +s 2 -k 1 ∑ k 2 =0	. . .	∑ k r =0	s 1 k 1
				∑ k r =0	s 1 k 1	s 1 + s 2 -k 1 k 2	. . .
	R k i =	    	x * 1 ⊔⊔	k i ∑	x * 0 ⊔⊔ x * 1 , if k i = 0,

s j=1 S 2 (k i , j) j!(x * (s