Abstract

Two confluent rewriting systems in noncommutatives polynomials are constructed using the equations allowing the identification of the local coordinates (of second kind) of the graphs of the polymorphism as being (shufle or quasi-shufle) characters and bridging two algebraic structures of polyzetas. In each system, the left side of each rewriting rule corresponds to the leading monomial of the associated homogeneous in weight polynomial while the right side is canonically represented on the algebra generated by irreducible terms which encode an algebraic basis of the algebra of polyzetas.

1. Introduction

For any ≥ 1 and (1, . . . , ) ∈ N≥ 1, for any ∈ C ˜∖{︂0, 1} and ≥ 1, let 1>...>>0

1 11 . . . and

H1,..., () :=

∑︁ 1>...>>0

1 11 . . . . which are respectively called polylogarithm and harmonic sum. theorem, the following limits exist and are called polyzetas1 [ 9, 10 ]

Let ℋ be {(1, . . . , ) ∈ N

≥ 1, 1 > 1}. Then, for any (1, . . . , ) belonging to ℋ, by a Abel’s →1 (1, . . . , ) := lim Li1,..., () =

H1,..., () =

∑︁ 1>...>>0 −1 1 . . . − . (6, 2) can not be expressed on (2), ..., (8) and proved [ 6 ]

Euler earlier studied polyzetas, in particular { (1, 2)}1≥>11,2≥ 1 in classic analysis. He stated that (2, 1) = (3) and (, 1) = ( + 1) − ( + 1) ( − ) , > 1.

︁) lim →+∞ 1 (︁ 2 − 2 ∑︁ =1 (1) (2) (3) ≥ 1

The { (1, . . . , )}1>1,2,...,≥ 1 are also called multi zeta values (MZV for short) [ 13 ] or EulerZagier sums [ 2 ] and the numbers and 1 + . . . + are, respectively, depth and weight of (1, . . . , ). One can also found in their biographies some recent applications of these special values in algebraic geometry, Diophantine equations, knots invariants of Vassiliev-Kontsevich, modular forms, quantum

CEUR Workshop

ceur-ws.org ISSN1613-0073 expressed on { (2), . . . , ( + )} and are qualified as new constants (as for Euler’s (6, 2)) [ 2 ]. Such polyzetas could be Q-algebraically independent on these zeta values (see Example 1 below) and the polyzetas could be transcendent numbers (see [ 9, 10 ] for proof). Checking linear relations among { (1, . . . , )}≥1>11,2,...,≥ 1 , Zagier stated that the Q-module generated by MZV is graded (see [ 9, 10 ] 2≤ 1+...+≤ 12 for proof) and guessed (see [ 7, 8, 12 ] for other algebraic checks) 1 = 0, 2 = 3 = 1 and = − 3 + − 2, for ≥ 4.

Conjecture 1 ([ 13 ]). Let := dim and := spanQ{ ()}≥1>11,2,...,≥ 1 , for ≥ 1+...+= 1. Then

Studying Conjecture 1, in continuation with [ 3, 5 ] by a symbolic approach, this work provides more explanations and consequences regarding the algorithm LocalCoordinateIdentification , partially implemented in [ 3 ] and briefly described in [ 4 ]. ≥ 1 It applies an Abel like theorem concerning the generating series of {H1,..., }1,...,≥ 1 (resp.

≥ 1 {Li1,..., }1,...,≥ 1) [ 5 ], over the alphabet = {}≥ 1 (resp. = {0, 1}) generating the free monoid ( * , 1 * ) (resp. (* , 1* )) with respect to the concatenation (denoted by conc and omitted when there is no ambiguity), the set of Lyndon words ℒ (resp. ℒ) and the set of polynomials, Q⟨ ⟩ (resp. Q⟨⟩). This theorem exploits the indexations of polylogarithms and harmonic sums in (1) by words, i.e. [ 9, 10 ]

Li0 () = log()/!, Li01− 11...0− 11 = Li1,..., , H1 ... = H1,..., .

It follows that the isomorphism of algebras H∙ : (Q⟨ ⟩, ) →− (Q{H}∈ * , × ) (resp. Li∙ : (Q⟨⟩, ⊔⊔ ) →− (Q{Li}∈* , × )), mapping (resp. ) to H (resp. Li), induce the following surjective polymorphism [ 9, 10 ] :

(Q1* ⊕ 0Q⟨⟩1, ⊔⊔ , 1* ) (Q1 * ⊕ ( ∖ {1})Q⟨ ⟩, , 1 * ) − ↠ 011− 1 . . . 01− 1 1 . . . →−↦ (, × , 1), (1, . . . , ), where is the Q-algebra generated by polyzetas (not linearly free [ 13 ]) and the product is defined, for any , , ∈ * (resp. * ) and , ∈ (resp. , ∈ ), by (resp. ⊔⊔ ) 1 * = 1 * = and = (

) + ( ) + + ( ), (resp. ⊔⊔ 1* = 1* ⊔⊔ = and ⊔⊔ = ( ⊔⊔ ) + ( ⊔⊔ )).

The graphs of the polymorphism in (5)–(6) are expressed as follows [ 9, 10 ] (resp. ⊔⊔ )-group like series as (4) (5) (6) (7) (8) (9) = 1 ↘ ∏︁ ∈ℒ ∖{1} (Σ)Π and ⊔⊔ = ↘ ∏︁ ∈ℒ∖ () , where {Π }∈ * (resp. {}∈* ) is the PBW-Lyndon basis (of the Lie polynomilas {Π }∈ℒ (resp. {}∈ℒ ) basis) in duality with {Σ }∈* (resp. {}∈* ) (containing the basis {Σ }∈ℒ (resp. {}∈ℒ )), on the (resp. ⊔⊔ )-bialgebra [ 9, 10 ]. Finally, the identification of their local coordinates (of second kind in the group of group like series) in the equations bridging the lagebraic structures of polyzetas, i.e. [ 5 ] = 1− ∑︀≥ 2 ()(− 1)/ ⊔⊔ and ⊔⊔ = − 1+∑︀≥ 2 ()(− 1)/ , (10) provides the algebraic relations among { (Σ )}∈ℒ ∖{1} (resp. { ()}∈ℒ∖ ), independent on , leading to the algebraic bases for Im and the homogenous polynomials generating ker [ 9, 10 ] (see [ 3 ] for examples), with2 the morphism of monoids : * 1 →− * (resp. : * →− * 1) maps to 0− 11 (resp. 0− 11 to ). 2There are one-to-one correspondences over the above monoids and that generated by N≥ 1, i.e 01− 11 . . . 0− 11 ∈ * 1 ⇌ 1 . . . ∈ * ↔ (1, . . . , ) ∈ N≥* 1 2. Rewriting among {Σ}∈ℒ ∖{1} and among {}∈ℒ∖ For convenience, denotes or and if = then gDIV = and CONV = 0* 1 else gDIV = {1} and CONV = ( ∖ {1}) * . It follows that ℒ ∖ gDIV ⊂ CONV.

Expressing, i.e replacing “=” by “→”, the relations among polyzetas in [ 3 ] become the rewriting rules among polyzetas and yield the following increasing sets of irreducible polyzetas (see Example 1 below) (11) (12) (13) (14) (15) (16) (17) = {}∈ℒ ∖gDIV ℛ := spanQ ⊆ ker .

,∞ ,∞, ≥ 2 and their images by a section of (see Example 2 below) such that the following restriction is an isomorphism of algebras [ 9, 10 ]

,≤ 2 ⊂ · · · ⊂ ,≤

⊂ · · · ⊂ ,≤ 2 ℒ ⊂ · · · ⊂ ℒ ,≤

⊂ · · · ⊂ ℒ : Q[ℒ( )] →− ∞

Q[,∞] = .

Note that one also has

,∞ = ⋃︁ ,≤ and ℒ ℒ

,∞ = ⋃︁ ℒ,≤ .

ℒ ≥ 2 { () = 0}∈ℒ ∖gDIV. It is denoted by :

Now, let us describe the algorithm LocalCoordinateIdentification below which brings aditional results to [ 3 ]. It provides the rewriting systems (Q1* ⊕ 0Q⟨⟩1, ℛ) and (Q1 * ⊕ ( ∖ {1})Q⟨ ⟩, ℛ) which are without critical pairs, noetherian, confluent and precisely contains the above sets (see (11)–(12)) and, on the other hand, the set of homogenous in weight polynomials, belonging to Q[ℒ ∖ gDIV], which are image by a section of the surjective polymorphism from and generates the shufle or quasi-shufle ideal

ℛ inside ker as follows

For any ≥ 2 and ∈ ℒ := { ∈ ℒ |() = }, any nonzero homogenous in weight polynomial (belonging to ) = Σ − Υ (resp. = − ) is led by Σ (resp. ) being transcendent over Q[ℒ,≤ ] and Υ = − Σ (resp. = − ) is canonically represented in Q[ℒ,≤ ]. Then let Σ → Υ and → be the rewriting rules, respectively, of

ℛ := {Σ → Υ }∈ℒ ∖{1} and ℛ := { → }∈ℒ∖ .

On the other hand, the following assertions are equivalent (see Example 2 below) 1. = 0

,≤ (resp. ∈ ℒ,≤ ), 2. Σ ∈ ℒ 3. Σ → Σ (resp. → ).

In the other words, the ordering over ℒ induces the ordering over ℒ,∞, ℛ , ℛ and, in the systems (Q1* ⊕ 0Q⟨⟩1, ℛ) and (Q1 * ⊕ ( ∖ {1})Q⟨ ⟩, ℛ), 1. each irreducible term, in ℒ,∞, is an element of the algebraic basis {Σ }∈ℒ ∖{1} of (Q1 * ⊕ ( ∖ {1})Q⟨ ⟩, ) (resp. {Σ }∈ℒ∖ of (Q1* ⊕ 0Q⟨⟩1, ⊔⊔ )), 2. each rewriting rule, in ℛ, admits the left side being transcendent over Q[ℒ,∞] and the right side being canonically represented in Q[ℒ,∞]. The diference of these two sides belongs to the ordered ideal ℛ of Q[ℒ ∖ gDIV]. LocalCoordinateIdentification

,∞ := {}; ℒ,∞ := {}; ℛ := {}; := {}; for ranges in 2, . . . , ∞ do for ranges in the totally ordered ℒ do identify ⟨ |Π ⟩ in = (1) ⊔⊔ and ⟨⊔⊔ |⟩ in ⊔⊔ = (1)− 1 ; by elimination, obtain equations on { (Σ ′)} ′∈ℒ and on { (′)} ′∈ℒ ; ′⪯ ′⪯ express3 the equations led by (Σ ) and by () as rewriting rules; if (Σ ) → (Σ ) then ,∞ :=

,∞ ∪ { (Σ )} and ℒ,∞ := ℒ,∞ ∪ {Σ } else ℛ := ℛ ∪ {Σ → Υ } and := ∪ {Σ − Υ };

if () → () then ,∞ ∪ { ()} and ℒ,∞ := ℒ,∞ ∪ {} ,∞ := else ℛ := ℛ ∪ { → } and := ∪ { − } end_for end_for

With the notations introduced in (11)–(17), on also has4 Proposition 1 ([ 9, 10 ]).

1. ℛ = ker and Q[,∞] = = Im .

2. Q[{}∈ℒ∖ ] = ℛ ⊕ Q[ℒ,∞] and Q[{Σ }∈ℒ ∖{1}] = ℛ ⊕ Q[ℒ,∞]. Proof – 1. Let ∈ ker , ⟨|1 * ⟩ = 0. Then = 1 + 2 (with 2 ∈ Q[ℒ,∞] and 1 ∈ ℛ ). Hence, decomposing in {}∈ℒ∖ (resp. {Σ }∈ℒ ∖{1}) and reducing by ℛ, it follows that ≡ ℛ 1 ∈ ℛ and then the expected result.

Let ∈ CONV. Decomposing in {}∈ℒ∖ (resp. {Σ }∈ℒ ∖{1}) and reducing by ℛ, ∈ Q[ℒ,∞]. Applying (13) and (5)–(6), () ∈ Q[,∞] = and = Im . Extending by linearrity, it follows the expected result. 2. For any ∈ CONV, decomposing in {}∈ℒ∖ (resp. {Σ }∈ℒ ∖{1}) and reducing by ℛ, () ∈ Q[,∞]. By linearity, if ∈ Q[{}∈ℒ∖ ] (resp. Q[{Σ }∈ℒ ∖{1}]) and ∈/ ker ⊇ ℛ then ( ) ∈ Q[,∞].

On the other hand, if ∈ ℛ ∩ Q[ℒ,∞] then, by (16), () = 0 and then, by (13), = 0 yielding the expected result. □ Theorem 1 ([ 9, 10 ]). The Q-algebra is freely generated by ,∞ and = Q1 ⊕ ⨁︀≥ 2 . ,∞ and ker , being generated by the Proof – By (13) and Proposition 1, is freely generated by homogenous in weight polynomials {}∈ℒ ∖gDIV, is graded. With the notations in Conjecture 1, being isomorphic to Q1 * ⊕ ( ∖ {1})Q⟨ ⟩/ ker and to Q1* ⊕ 0Q⟨⟩1/ ker , is also graded. □ Corollary 1 ([ 9, 10 ]). Let ∈ ℒ,∞. Then ( ) is a transcendent number.

Proof – Let ∈ Q⟨ ⟩ and ∈/ ker , being homogenous in weight, or ∈ CONV. Since ′ ⊂ +′ (, ′ ≥ 1) then each monomial ( ( )) ( ≥ 1) is of diferent weight and then, by Theorem 1, ( ) could not satisfy, over Q, an algebraic equation + − 1 − 1 + . . . = 0 meaning that ( ) is a transcendent number. Since any ∈ ℒ,∞ is homogenous in weight then it follows the expected result. □ 3This step and the following ones are not yet been achieved by the implementation in [ 3 ]. 4See also [ 11 ] for further information. Example 1 (irreducible polyzetas, [ 3 ]).

,≤ 12 = { (01), (201), (401), (601), (012041), (801),

(012061), (1001), (013071), (012081), (014061)}.

,≤ 12 = { (Σ 2), (Σ 3), (Σ 5), (Σ 7), (Σ 315), (Σ 9), (Σ 317),

(Σ 11), (Σ 219), (Σ 319), (Σ 2218)}.

Example 2 (Rewriting on {Σ }∈ℒ ∖{1} and {}∈ℒ∖, irreducible terms).

Rewriting on {Σ }∈ℒ ∖{1} Rewriting on {}ℒ∖ 3 Σ 21 → 23 Σ 3 012 → 201 4 5 6 Σ 4 → 25 Σ 2 2

3 Σ 31 → 10 Σ 2 2 Σ 212 → 32 Σ 2 2 Σ 32 → 3Σ 3Σ 2 − 5Σ 5 Σ 41 → − Σ 3Σ 2 + 52 Σ 5 Σ 221 → 23 Σ 3Σ 2 − 1225 Σ 5

5 Σ 312 → 12 Σ 5 Σ 213 → 4 Σ 3Σ 2 + 54 Σ 5

1 Σ 6 → 385 Σ 2 3 Σ 51 → 72 Σ 2 3 − 21 Σ 3 2 Σ 312 → − 3107 Σ 2 3 + 94 Σ 3 2 Σ 321 → 3Σ 3 2 − 190 Σ 2 3 Σ 412 → 10 Σ 2 3 − 43 Σ 3 2

3 Σ 2212 → 1613 Σ 2 3 − 41 Σ 3 2

1 Σ 313 → 21 Σ 2 3

3. Conclusion

Thanks to a Abel like theorem and the equation bridging the algebraic structures of the Q-algebra generated by the polyzetas [ 5 ], the algorithm LocalCoordinateIdentification provides the algebraic relations5 among the local coordinates, of second kind on the groups of group-like series, of the noncommutative series ⊔⊔ (i.e. { ()}∈ℒ∖) and (i.e. { (Σ )}∈ℒ ∖{1}). These relations constitute two confluent rewriting systems in which the irreducible terms, belonging to ,∞, represent the algebraic generators for and, on the other hand, the ⊔⊔-ideal ℛ and the -ideal ℛ represent the kernels of the polymorphism (Proposition 1). These ideals are generated by the polynomials, totally ordered and homogenous in weight, {}∈ℒ ∖gDIV and are interpreted as the confluent rewriting systems in which the irreducible terms belong to ℒ,∞ and, in each rewriting rule of ℛ, the left side 5These are diferent from those among { ()}∈ℒ∖gDIV obtained by “double shufle relations” [ 8 ], for which Conjecture 1 holds, up to weight 10. is the leading monomial of , ∈ ℒ ∖ gDIV and is transcendent over Q[ℒ,∞] while the right side is canonically represented on Q[ℒ,∞]. It follows that (Q[ℒ,∞]), i.e. , as being isomorphic to Q1* ⊕ 0Q⟨⟩1/ℛ and to Q1 * ⊕ ( ∖ {1})Q⟨ ⟩/ℛ , is Q-free and graded (Theorem 1) and then irreducible polyzetas, being Q-algebraic independent, are transcendent numbers (Corollary 1). By ,≤ 12 is Q-algebraically free these results, up to weight 12, Conjecture 1 holds (see also6 [ 7, 12 ]), i.e. (Example 2). 6All these implementations base on the “double shufle relations" and provide linear relations.

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