expressed on {𝜁(2), . . . , 𝜁(𝑠 + 𝑘)} and are qualified as new constants (as for Euler's 𝜁(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 {𝜁(𝑠 1 , . . . , 𝑠 𝑟 )} 𝑟≥1 𝑠 1 >1,𝑠 2 ,...,𝑠𝑟 ≥1 2≤𝑠 1 +...+𝑠𝑟 ≤12 , Zagier stated that the Q-module generated by MZV is graded (see [9,10] for proof) and guessed (see [7,8,12] for other algebraic checks) Conjecture 1 ( [13]). Let 𝑑 𝑘 := dim 𝒵 𝑘 and 𝒵 𝑘 := span Q {𝜁(𝑤)} 𝑟≥1 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].

It applies an Abel like theorem concerning the generating series of {H 𝑠 1 ,...,𝑠𝑟 } 𝑟≥1 𝑠 1 ,...,𝑠𝑟≥1 (resp. {Li 𝑠 1 ,...,𝑠𝑟 } 𝑟≥1 𝑠 1 ,...,𝑠𝑟≥1 ) [5], over the alphabet 𝑌 = {𝑦 𝑘 } 𝑘≥1 (resp. 𝑋 = {𝑥 0 , 𝑥 1 }) generating the free monoid (𝑌 * , 1 𝑌 * ) (resp. (𝑋 * , 1 𝑋 * )) with respect to the concatenation (denoted by conc and omitted when there is no ambiguity), the set of Lyndon words ℒ𝑦𝑛𝑌 (resp. ℒ𝑦𝑛𝑋) and the set of polynomials, Q⟨𝑌 ⟩ (resp. Q⟨𝑋⟩). This theorem exploits the indexations of polylogarithms and harmonic sums in (1) by words, i.e. [9,10]

It follows that the isomorphism of algebras

where 𝒵 is the Q-algebra generated by polyzetas (not linearly free [13]) and the product (resp. ⊔⊔ ) is defined, for any 𝑢, 𝑣, 𝑤 ∈ 𝑌 * (resp. 𝑋 * ) and 𝑦 𝑖 , 𝑦 𝑗 ∈ 𝑌 (resp. 𝑥, 𝑦 ∈ 𝑋), by

(resp. 𝑤 ⊔⊔ 1 𝑋 * = 1 𝑋 * ⊔⊔ 𝑤 = 𝑤 and 𝑥𝑢 ⊔⊔ 𝑦𝑣 = 𝑥(𝑢 ⊔⊔ 𝑦𝑣) + 𝑦(𝑥𝑢 ⊔⊔ 𝑣)).

The graphs of the 𝜁 polymorphism in ( 5)-( 6) are expressed as (resp. ⊔⊔ )-group like series as follows [9,10]

where {𝑆 𝑙 } 𝑙∈ℒ𝑦𝑛𝑌 )), 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]

provides the algebraic relations among {𝜁(Σ 𝑙 )} 𝑙∈ℒ𝑦𝑛𝑌 ∖{𝑦 1 } (resp. {𝜁(𝑆 𝑙 )} 𝑙∈ℒ𝑦𝑛𝑋∖𝑋 ), independent on 𝛾, leading to the algebraic bases for Im 𝜁 and the homogenous polynomials generating ker 𝜁 [9,10] (see [3] for examples), with 2 the morphism of monoids 𝜋 𝑌 :

2 There are one-to-one correspondences over the above monoids and that generated by N ≥1 , i.e 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) and their images by a section of 𝜁 (see Example 2 below)

such that the following restriction is an isomorphism of algebras [9,10]

Note that one also has

Now, let us describe the algorithm LocalCoordinateIdentification below which brings aditional results to [3]. It provides the rewriting systems

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[ℒ𝑦𝑛𝒳 ∖ gDIV], which are image by a section of the surjective 𝜁 polymorphism from {𝜁(𝑄 𝑙 ) = 0} 𝑙∈ℒ𝑦𝑛𝒳 ∖gDIV . It is denoted by 𝒬 𝒳 :

and generates the shuffle or quasi-shuffle ideal ℛ 𝒳 inside ker 𝜁 as follows

For any 𝑝 ≥ 2 and 𝑙 ∈ ℒ𝑦𝑛 𝑝 𝒳 := {𝑙 ∈ ℒ𝑦𝑛𝒳 |(𝑙) = 𝑝}, any nonzero homogenous in weight polynomial (belonging to

Then let Σ 𝑙 → Υ 𝑙 and 𝑆 𝑙 → 𝑈 𝑙 be the rewriting rules, respectively, of

On the other hand, the following assertions are equivalent (see Example 2 below)

In the other words, the ordering over ℒ𝑦𝑛𝒳 induces the ordering over ℒ 𝒳 ,∞ 𝑖𝑟𝑟 , ℛ 𝒳 , ℛ 𝒳 𝑖𝑟𝑟 and, in the systems

With the notations introduced in ( 11)-( 17), on also has4

1.

𝑖𝑟𝑟 ]. Applying ( 13) and ( 5)-( 6), 𝜁(𝑤) ∈ Q[𝒵 𝒳 ,∞ 𝑖𝑟𝑟 ] = 𝒵 and 𝒵 = Im 𝜁. Extending by linearrity, it follows the expected result. 2. For any 𝑤 ∈ CONV, decomposing in {𝑆 𝑙 } 𝑙∈ℒ𝑦𝑛𝑋∖𝑋 (resp. {Σ 𝑙 } 𝑙∈ℒ𝑦𝑛𝑌 ∖{𝑦 1 } ) and reducing by

𝑖𝑟𝑟 ] then, by (16), 𝜁(𝑄) = 0 and then, by (13), 𝑄 = 0 yielding the expected result. Proof -Let 𝑃 ∈ Q⟨𝒳 ⟩ and 𝑃 / ∈ ker 𝜁, being homogenous in weight, or 𝑃 ∈ CONV. Since 𝒵 𝑘 𝒵 𝑘 ′ ⊂ 𝒵 𝑘+𝑘 ′ (𝑘, 𝑘 ′ ≥ 1) then each monomial (𝜁(𝑃 )) 𝑘 (𝑘 ≥ 1) is of different weight and then, by Theorem 1, 𝜁(𝑃 ) could not satisfy, over Q, an algebraic equation 𝑇 𝑘 + 𝑎 𝑘−1 𝑇 𝑘−1 + . . . = 0 meaning that 𝜁(𝑃 ) is a transcendent number. Since any 𝑃 ∈ ℒ 𝒳 ,∞ 𝑖𝑟𝑟 is homogenous in weight then it follows the expected result. □ Example 1 (irreducible polyzetas, [3]).

Example 2 (Rewriting on {Σ 𝑙 } 𝑙∈ℒ𝑦𝑛𝑌 ∖{𝑦 1 } and {𝑆 𝑙 } 𝑙∈ℒ𝑦𝑛𝑋∖𝑋 , irreducible terms).

Rewriting on

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