=Paper=
{{Paper
|id=Vol-3754/paper06
|storemode=property
|title=Algebraic (non) relations among polyzetas
|pdfUrl=https://ceur-ws.org/Vol-3754/paper06.pdf
|volume=Vol-3754
|authors=Vincel Hoang Ngoc Minh
|dblpUrl=https://dblp.org/rec/conf/sycss/Minh24
}}
==Algebraic (non) relations among polyzetas==
https://ceur-ws.org/Vol-3754/paper06.pdf
Algebraic (non) Relations Among Polyzetas
V. Hoang Ngoc Minh1
1
University of Lille, 1 Place DΓ©liot, Lille, 59024, France
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 π polymorphism as being (shuffle
or quasi-shuffle) 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.
These polynomials are totally lexicographically ordered and generate the kernels of the π polymorphism
meaning that the free algebra of polyzetas is graded and the irreducible polyzetas are transcendent numbers,
algebraically independent.
Keywords
Polylogarithms, Hamonic Sums, Polyzetas, Rewritting Systems
1. Introduction
For any π β₯ 1 and (π 1 , . . . , π π ) β Nβ₯1 , for any π§ β C ΛοΈ
β {0, 1} and π β₯ 1, let
π
βοΈ π§ π1 βοΈ 1
Liπ 1 ,...,π π (π§) := and Hπ 1 ,...,π π (π) := . (1)
ππ 11 . . . ππ ππ ππ 11 . . . ππ ππ
π1 >...>ππ >0 π1 >...>ππ >0
which are respectively called polylogarithm and harmonic sum.
Let βπ be {(π 1 , . . . , π π ) β Nπβ₯1 , π 1 > 1}. Then, for any (π 1 , . . . , π π ) belonging to βπ , by a Abelβs
theorem, the following limits exist and are called polyzetas1 [9, 10]
πβπ
βοΈ
π(π 1 , . . . , π π ) := lim Liπ 1 ,...,π π (π§) = lim Hπ 1 ,...,π π (π) = 1
1
. . . πβπ
π .
π
(2)
π§β1 πβ+β
π1 >...>ππ >0
Euler earlier studied polyzetas, in particular {π(π 1 , π 2 )}πβ₯1
π 1 >1,π 2 β₯1 in classic analysis. He stated that
π(6, 2) can not be expressed on π(2), ..., π(8) and proved [6]
π β2
1 (οΈ βοΈ )οΈ
π(2, 1) = π(3) and π(π , 1) = π π(π + 1) β π(π + 1)π(π β π) , π > 1. (3)
2
π=1
The {π(π 1 , . . . , π π )}πβ₯1
π 1 >1,π 2 ,...,π π β₯1 are also called multi zeta values (MZV for short) [13] or Euler-
Zagier sums [2] and the numbers π and π 1 + . . . + π π are, respectively, depth and weight of π(π 1 , . . . , π π ).
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
electrodynamic, . . . .
Many new linear relations for polyzetas are detected using LLL type algorithms in high performance
computing and the truncations of {π(π 1 , . . . , π π )}πβ₯1 πβ₯1
π 1 >1,π 2 ,...,π π β₯1 , i.e. {Hπ 1 ,...,π π (π)}π 1 >1,π 2 ,...,π π β₯1 [1,
2]. In this approach, the main problem is to detect with near certainty which polyzetas can not be
SCSS 2024: 10th International Symposium on Symbolic Computation in Software Science, August 28β30, 2024, Tokyo, Japan
$ vincel.hoang-ngoc-minh@univ-lille.fr (V. Hoang Ngoc Minh)
0000-0002-3510-7639 (V. Hoang Ngoc Minh)
Β© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
1
Polyzeta is the contraction of polymorphism and of zeta (see (5)β(6) below).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
28
expressed on {π(2), . . . , π(π + π)} and are qualified as new constants (as for Eulerβs π(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
{π(π 1 , . . . , π π )}πβ₯1
π 1 >1,π 2 ,...,π π β₯1 , Zagier stated that the Q-module generated by MZV is graded (see [9, 10]
2β€π 1 +...+π π β€12
for proof) and guessed (see [7, 8, 12] for other algebraic checks)
Conjecture 1 ([13]). Let ππ := dim π΅π and π΅π := spanQ {π(π€)}πβ₯1
π 1 >1,π 2 ,...,π π β₯1 , for π β₯ 1. Then
π 1 +...+π π =π
π1 = 0, π2 = π3 = 1 and ππ = ππβ3 + ππβ2 , for π β₯ 4.
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 {Hπ 1 ,...,π π }π πβ₯1 1 ,...,π π β₯1
(resp.
πβ₯1
{Liπ 1 ,...,π π }π 1 ,...,π π β₯1 ) [5], over the alphabet π = {π¦π }πβ₯1 (resp. π = {π₯0 , π₯1 }) generating the free
monoid (π * , 1π * ) (resp. (π * , 1π * )) with respect to the concatenation (denoted by conc and omitted
when there is no ambiguity), the set of Lyndon words βπ¦ππ (resp. βπ¦ππ) and the set of polynomials,
Qβ¨π β© (resp. Qβ¨πβ©). This theorem exploits the indexations of polylogarithms and harmonic sums in (1)
by words, i.e. [9, 10]
Liπ₯π0 (π§) = logπ (π§)/π!, Liπ₯π 1 β1 π₯ ...π₯π π β1 π₯ = Liπ 1 ,...,π π , Hπ¦π 1 ...π¦π π = Hπ 1 ,...,π π . (4)
0 1 0 1
It follows that the isomorphism of algebras Hβ : (Qβ¨π β©, ) ββ (Q{Hπ€ }π€βπ * , Γ) (resp. Liβ :
(Qβ¨πβ©, ββ ) ββ (Q{Liπ€ }π€βπ * , Γ)), mapping π’ (resp. π£) to Hπ’ (resp. Liπ£ ), induce the following
surjective polymorphism [9, 10]
(Q1π * β π₯0 Qβ¨πβ©π₯1 , ββ , 1π * )
π: ββ (π΅, Γ, 1), (5)
(Q1π * β (π β {π¦1 })Qβ¨π β©, , 1π * )
π₯0 π₯π 11 β1 . . . π₯0 π₯1π π β1
β¦ββ π(π 1 , . . . , π π ), (6)
π¦π 1 . . . π¦π π
where π΅ is the Q-algebra generated by polyzetas (not linearly free [13]) and the product (resp. ββ )
is defined, for any π’, π£, π€ β π * (resp. π * ) and π¦π , π¦π β π (resp. π₯, π¦ β π), by
π€ 1π * = 1π * π€ = π€ and π¦π π’ π¦π π£ = π¦π (π’ π¦π π£) + π¦π (π¦π π’ π£) + π¦π+π (π’ π£), (7)
(resp. π€ ββ 1π * = 1π * ββ π€ = π€ and π₯π’ ββ π¦π£ = π₯(π’ ββ π¦π£) + π¦(π₯π’ ββ π£)). (8)
The graphs of the π polymorphism in (5)β(6) are expressed as (resp. ββ )-group like series as
follows [9, 10]
β
βοΈ β
βοΈ
πΎπ¦1 π(Ξ£π )Ξ π
ππΎ = π π and πββ = ππ(ππ )ππ , (9)
πββπ¦ππ β{π¦1 } πββπ¦ππβπ
where {Ξ π€ }π€βπ * (resp. {ππ€ }π€βπ * ) is the PBW-Lyndon basis (of the Lie polynomilas {Ξ π }πββπ¦ππ (resp.
{ππ }πββπ¦ππ ) basis) in duality with {Ξ£π€ }π€βπ * (resp. {ππ€ }π€βπ * ) (containing the basis {Ξ£π }πββπ¦ππ (resp.
{ππ }πββπ¦ππ )), on the (resp. ββ )-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]
π /π π
βοΈ βοΈ
ππΎ = ππΎπ¦1 β πβ₯2 π(π)(βπ¦1 ) ππ πββ and πββ = πβπΎπ₯1 + πβ₯2 π(π)(βπ₯1 ) /π ππ ππΎ , (10)
provides the algebraic relations among {π(Ξ£π )}πββπ¦ππ β{π¦1 } (resp. {π(ππ )}πββπ¦ππβπ ), independent on
πΎ, leading to the algebraic bases for Im π and the homogenous polynomials generating ker π [9, 10] (see
[3] for examples), with2 the morphism of monoids ππ : π * π₯1 ββ π * (resp. ππ : π * ββ π * π₯1 ) maps
π¦π to π₯πβ1 πβ1
0 π₯1 (resp. π₯0 π₯1 to π¦π ).
2
There are one-to-one correspondences over the above monoids and that generated by Nβ₯1 , i.e π₯π 01 β1 π₯1 . . . π₯π 0π β1 π₯1 β
π * π₯1 βππππ π¦π 1 . . . π¦π π β π * β (π 1 , . . . , π π ) β N*β₯1
29
2. Rewriting among {Ξ£π }πββπ¦ππ β{π¦1 } and among {ππ }πββπ¦ππβπ
For convenience, π³ denotes π or π and if π³ = π then gDIV = π and CONV = π₯0 π * π₯1 else
gDIV = {π¦1 } and CONV = (π β {π¦1 })π * . It follows that βπ¦ππ³ β gDIV β CONV.
Expressing, i.e replacing β=β by βββ, 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)
π³ ,β€2 π³ ,β€π π³ ,β
π΅πππ β Β· Β· Β· β π΅πππ β Β· Β· Β· β π΅πππ (11)
and their images by a section of π (see Example 2 below)
βπ³ ,β€2
πππ β Β· Β· Β· β βπ³ ,β€π
πππ β Β· Β· Β· β βπ³ ,β
πππ , (12)
such that the following restriction is an isomorphism of algebras [9, 10]
π³ ,β
π : Q[ββ
πππ (π³ )] ββ Q[π΅πππ ] = π΅. (13)
Note that one also has
βπ³ ,β
βπ³ ,β€π π³ ,β
βπ³ ,β€π
βοΈ βοΈ
πππ = πππ and βπππ = πππ . (14)
πβ₯2 πβ₯2
Now, let us describe the algorithm LocalCoordinateIdentification below which brings aditional
results to [3]. It provides the rewriting systems (Q1π * β π₯0 Qβ¨πβ©π₯1 , βπ πππ ) and (Q1π * β (π β
{π¦1 })Qβ¨π β©, βππππ ) which are without critical pairs, noetherian, confluent and precisely contains the
above sets (see (11)β(12)) and, on the other hand, the set of homogenous in weight polynomials, be-
longing to Q[βπ¦ππ³ β gDIV], which are image by a section of the surjective π polymorphism from
{π(ππ ) = 0}πββπ¦ππ³ βgDIV . It is denoted by π¬π³ :
π¬π³ = {ππ }πββπ¦ππ³ βgDIV (15)
and generates the shuffle or quasi-shuffle ideal βπ³ inside ker π as follows
βπ³ := spanQ π¬π³ β ker π. (16)
For any π β₯ 2 and π β βπ¦ππ π³ := {π β βπ¦ππ³ |(π) = π}, any nonzero homogenous in weight
polynomial (belonging to π¬π³ ) ππ = Ξ£π β Ξ₯π (resp. ππ = ππ β ππ ) is led by Ξ£π (resp. ππ ) being
transcendent over Q[βπ³ ,β€π
πππ ] and Ξ₯π = ππ β Ξ£π (resp. ππ = ππ β ππ ) is canonically represented in
Q[βπ³ ,β€π
πππ ]. Then let Ξ£π β Ξ₯π and ππ β ππ be the rewriting rules, respectively, of
βππππ := {Ξ£π β Ξ₯π }πββπ¦ππ β{π¦1 } and βπ
πππ := {ππ β ππ }πββπ¦ππβπ . (17)
On the other hand, the following assertions are equivalent (see Example 2 below)
1. ππ = 0
2. Ξ£π β βπ,β€π π,β€π
πππ (resp. ππ β βπππ ),
3. Ξ£π β Ξ£π (resp. ππ β ππ ).
π³ ,β
In the other words, the ordering over βπ¦ππ³ induces the ordering over βπππ , βπ³ , βπ³
πππ and, in the
π π
systems (Q1π * β π₯0 Qβ¨πβ©π₯1 , βπππ ) and (Q1π * β (π β {π¦1 })Qβ¨π β©, βπππ ),
1. each irreducible term, in βπ³ ,β
πππ , is an element of the algebraic basis {Ξ£π }πββπ¦ππ β{π¦1 } of (Q1π * β
(π β {π¦1 })Qβ¨π β©, ) (resp. {Ξ£π }πββπ¦ππβπ of (Q1π * β π₯0 Qβ¨πβ©π₯1 , ββ )),
π³ ,β
2. each rewriting rule, in βπ³
πππ , admits the left side being transcendent over Q[βπππ ] and the right
side being canonically represented in Q[βπ³ ,β
πππ ]. The difference of these two sides belongs to the
ordered ideal βπ³ of Q[βπ¦ππ³ β gDIV].
30
LocalCoordinateIdentification
π³ ,β
π΅πππ := {}; βπ³ ,β π³
πππ := {}; βπππ := {}; π¬π³ := {};
for π ranges in 2, . . . , β do
for π ranges in the totally ordered βπ¦ππ π³ do
identify β¨ππΎ |Ξ π β© in ππΎ = π΅(π¦1 )ππ πββ and β¨πββ |ππ β© in πββ = π΅(π₯1 )β1 ππ ππΎ ;
by elimination, obtain equations on {π(Ξ£πβ² )} πβ² ββπ¦ππ π and on {π(ππβ² )} πβ² ββπ¦ππ π ;
πβ² βͺ―π πβ² βͺ―π
express3 the equations led by π(Ξ£π ) and by π(ππ ) as rewriting rules;
π,β π,β π,β π,β
if π(Ξ£π ) β π(Ξ£π ) then π΅πππ := π΅πππ βͺ {π(Ξ£π )} and βπππ := βπππ βͺ {Ξ£π }
else βπ π
πππ := βπππ βͺ {Ξ£π β Ξ₯π } and π¬π := π¬π βͺ {Ξ£π β Ξ₯π };
π,β π,β π,β π,β
if π(ππ ) β π(ππ ) then π΅πππ := π΅πππ βͺ {π(ππ )} and βπππ := βπππ βͺ {ππ }
else βπ π
πππ := βπππ βͺ {ππ β ππ } and π¬π := π¬π βͺ {ππ β ππ }
end_for
end_for
With the notations introduced in (11)β(17), on also has4
π³ ,β
Proposition 1 ([9, 10]). 1. βπ³ = ker π and Q[π΅πππ ] = π΅ = Im π.
2. Q[{ππ }πββπ¦ππβπ ] = βπ β Q[βπππ ] and Q[{Ξ£π }πββπ¦ππ β{π¦1 } ] = βπ β Q[βπ,β
π,β
πππ ].
Proof β
1. Let π β ker π, β¨π|1π³ * β© = 0. Then π = π1 + π2 (with π2 β Q[βπ³ ,β
πππ ] and π1 β βπ³ ). Hence,
decomposing in {ππ }πββπ¦ππβπ (resp. {Ξ£π }πββπ¦ππ β{π¦1 } ) and reducing by βπ³ πππ , it follows that
π β‘βπ³ π1 β βπ³ and then the expected result.
πππ
Let π€ β CONV. Decomposing in {ππ }πββπ¦ππβπ (resp. {Ξ£π }πββπ¦ππ β{π¦1 } ) and reducing by βπ³ πππ ,
π³ ,β π³ ,β
π€ β Q[βπππ ]. Applying (13) and (5)β(6), π(π€) β Q[π΅πππ ] = π΅ and π΅ = Im π. Extending by
linearrity, it follows the expected result.
2. For any π€ β CONV, decomposing in {ππ }πββπ¦ππβπ (resp. {Ξ£π }πββπ¦ππ β{π¦1 } ) and reducing by
π³ ,β
βπ³ πππ , π(π€) β Q[π΅πππ ]. By linearity, if π β Q[{ππ }πββπ¦ππβπ ] (resp. Q[{Ξ£π }πββπ¦ππ β{π¦1 } ]) and
π³ ,β
π β / ker π β βπ³ then π(π ) β Q[π΅πππ ].
π³ ,β
On the other hand, if π β βπ³ β© Q[βπππ ] then, by (16), π(π) = 0 and then, by (13), π = 0
yielding the expected result.
β‘
π³ ,β β¨οΈ
Theorem 1 ([9, 10]). The Q-algebra π΅ is freely generated by π΅πππ and π΅ = Q1 β πβ₯2 π΅π .
π³ ,β
Proof β By (13) and Proposition 1, π΅ is freely generated by π΅πππ and ker π, being generated by the
homogenous in weight polynomials {ππ }πββπ¦ππ³ βgDIV , is graded. With the notations in Conjecture 1,
being isomorphic to Q1π * β (π β {π¦1 })Qβ¨π β©/ ker π and to Q1π * β π₯0 Qβ¨πβ©π₯1 / ker π, π΅ is also graded.
β‘
Corollary 1 ([9, 10]). Let π β βπ³ ,β
πππ . Then π(π ) is a transcendent number.
Proof β Let π β Qβ¨π³ β© and π β / ker π, being homogenous in weight, or π β CONV. Since
π΅π π΅πβ² β π΅π+πβ² (π, π β² β₯ 1) then each monomial (π(π ))π (π β₯ 1) is of different weight and then, by
Theorem 1, π(π ) could not satisfy, over Q, an algebraic equation π π + ππβ1 π πβ1 + . . . = 0 meaning
that π(π ) is a transcendent number. Since any π β βπ³ ,β
πππ is homogenous in weight then it follows the
expected result. β‘
3
This step and the following ones are not yet been achieved by the implementation in [3].
4
See also [11] for further information.
31
Example 1 (irreducible polyzetas, [3]).
π,β€12
π΅πππ = {π(ππ₯0 π₯1 ), π(ππ₯20 π₯1 ), π(ππ₯40 π₯1 ), π(ππ₯60 π₯1 ), π(ππ₯0 π₯21 π₯0 π₯41 ), π(ππ₯80 π₯1 ),
π(ππ₯0 π₯21 π₯0 π₯61 ), π(ππ₯10
0 π₯1
), π(ππ₯0 π₯31 π₯0 π₯71 ), π(ππ₯0 π₯21 π₯0 π₯81 ), π(ππ₯0 π₯41 π₯0 π₯61 )}.
π,β€12
π΅πππ = {π(Ξ£π¦2 ), π(Ξ£π¦3 ), π(Ξ£π¦5 ), π(Ξ£π¦7 ), π(Ξ£π¦3 π¦15 ), π(Ξ£π¦9 ), π(Ξ£π¦3 π¦17 ),
π(Ξ£π¦11 ), π(Ξ£π¦2 π¦19 ), π(Ξ£π¦3 π¦19 ), π(Ξ£π¦22 π¦18 )}.
Example 2 (Rewriting on {Ξ£π }πββπ¦ππ β{π¦1 } and {ππ }πββπ¦ππβπ , irreducible terms).
Rewriting on {Ξ£π }πββπ¦ππ β{π¦1 } Rewriting on {ππ }βπ¦ππβπ
3 Ξ£π¦2 π¦1 β 32 Ξ£π¦3 ππ₯0 π₯21 β ππ₯20 π₯1
2 2 2 ββ 2
Ξ£π¦4 β 5 Ξ£ π¦2 ππ₯30 π₯1 β 5 ππ₯0 π₯1
3 2 1 ββ 2
4 Ξ£π¦3 π¦1 β 10 Ξ£π¦2 ππ₯20 π₯21 β 10 ππ₯0 π₯1
2 2 2 ββ 2
Ξ£π¦2 π¦12 β 3 Ξ£ π¦2 ππ₯0 π₯31 β 5 ππ₯0 π₯1
Ξ£π¦3 π¦2 β 3Ξ£π¦3 Ξ£π¦2 β 5Ξ£π¦5 ππ₯30 π₯21 β βππ₯20 π₯1 ππ₯0 π₯1 + 2ππ₯40 π₯1
Ξ£π¦4 π¦1 β βΞ£π¦3 Ξ£π¦2 + 52 Ξ£π¦5 ππ₯20 π₯1 π₯0 π₯1 β β 23 ππ₯40 π₯1 + ππ₯20 π₯1 ππ₯0 π₯1
5 Ξ£π¦22 π¦1 β 32 Ξ£π¦3 Ξ£π¦2 β 25
12 Ξ£π¦5 ππ₯20 π₯31 β βππ₯20 π₯1 ππ₯0 π₯1 + 2ππ₯40 π₯1
5 1
Ξ£π¦3 π¦12 β 12 Ξ£π¦5 ππ₯0 π₯1 π₯0 π₯21 β 2 ππ₯40 π₯1
Ξ£π¦2 π¦13 β 14 Ξ£π¦3 Ξ£π¦2 + 54 Ξ£π¦5 ππ₯0 π₯41 β ππ₯40 π₯1
8 3 8 ββ 3
Ξ£π¦6 β 35 Ξ£π¦2 ππ₯50 π₯1 β 35 ππ₯0 π₯1
4 6 ββ 3 1 ββ 2
Ξ£π¦4 π¦2 β Ξ£π¦3 2 β 21 Ξ£ π¦2 3 ππ₯40 π₯21 β 35 ππ₯0 π₯1 β 2 ππ₯20 π₯1
2 3 1 2 4 ββ 3
Ξ£π¦5 π¦1 β 7 Ξ£ π¦2 β 2 Ξ£ π¦3 ππ₯30 π₯1 π₯0 π₯1 β 105 ππ₯0 π₯1
17 23 ββ 3 ββ 2
Ξ£π¦3 π¦1 π¦2 β β 30 Ξ£π¦2 3 + 94 Ξ£π¦3 2 ππ₯30 π₯31 β 70 ππ₯0 π₯1 β ππ₯20 π₯1
9 2 ββ 3
Ξ£π¦3 π¦2 π¦1 β 3Ξ£π¦3 2 β 10 Ξ£π¦2 3 ππ₯20 π₯1 π₯0 π₯21 β 105 ππ₯0 π₯1
3 3 89 ββ 3 ββ 2
6 Ξ£π¦4 π¦12 β 3
10 Ξ£π¦2 β 4 Ξ£π¦3
2 ππ₯20 π₯21 π₯0 π₯1 β β 210 ππ₯0 π₯1 + 32 ππ₯2 π₯
0 1
11 3 1 2 6 ββ 3 1 ββ 2
Ξ£π¦22 π¦12 β 63 Ξ£π¦2 β 4 Ξ£π¦3 ππ₯20 π₯41 β 35 ππ₯ π₯
0 1 β π 2
2 π₯0 π₯1
1 3 8 ββ 3 ββ 2
Ξ£π¦3 π¦13 β 21 Ξ£π¦2 ππ₯0 π₯1 π₯0 π₯31 β 21 ππ₯0 π₯1 β ππ₯20 π₯1
17 3 3 2 8 ββ 3
Ξ£π¦2 π¦14 β 50 Ξ£π¦2 + 16 Ξ£π¦3 ππ₯0 π₯51 β 35 ππ₯0 π₯1
βπ,β€12
πππ = {ππ₯0 π₯1 , ππ₯20 π₯1 , ππ₯40 π₯1 , ππ₯60 π₯1 , ππ₯0 π₯21 π₯0 π₯41 , ππ₯80 π₯1 ,
ππ₯0 π₯21 π₯0 π₯61 , ππ₯10
0 π₯1
, ππ₯0 π₯31 π₯0 π₯71 , ππ₯0 π₯21 π₯0 π₯81 , ππ₯0 π₯41 π₯0 π₯61 }.
π,β€12
βπππ = {Ξ£π¦2 , Ξ£π¦3 , Ξ£π¦5 , Ξ£π¦7 , Ξ£π¦3 π¦15 , Ξ£π¦9 , Ξ£π¦3 π¦17 , Ξ£π¦11 , Ξ£π¦2 π¦19 , Ξ£π¦3 π¦19 , Ξ£π¦22 π¦18 }.
3. Conclusion
Thanks to a Abel like theorem and the equation bridging the algebraic structures of the Q-algebra π΅
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 πββ (i.e. {π(ππ )}πββπ¦ππβπ ) and π (i.e. {π(Ξ£π )}πββπ¦ππ β{π¦1 } ). These relations
π³ ,β
constitute two confluent rewriting systems in which the irreducible terms, belonging to π΅πππ , represent
the algebraic generators for π΅ and, on the other hand, the ββ -ideal βπ and the -ideal βπ represent
the kernels of the π polymorphism (Proposition 1). These ideals are generated by the polynomials, totally
ordered and homogenous in weight, {ππ }πββπ¦ππ³ βgDIV and are interpreted as the confluent rewriting
systems in which the irreducible terms belong to βπ³ ,β π³
πππ and, in each rewriting rule of βπππ , the left side
5
These are different from those among {π(π)}πββπ¦ππ³ βgDIV obtained by βdouble shuffle relationsβ [8], for which Conjecture 1
holds, up to weight 10.
32
is the leading monomial of ππ , π β βπ¦ππ³ β gDIV and is transcendent over Q[βπ³ ,β
πππ ] while the right
side is canonically represented on Q[βπ³ ,β π³ ,β
πππ ]. It follows that π(Q[βπππ ]), i.e. π΅, as being isomorphic to
Q1π * β π₯0 Qβ¨πβ©π₯1 /βπ and to Q1π * β (π β {π¦1 })Qβ¨π β©/βπ , is Q-free and graded (Theorem 1) and
then irreducible polyzetas, being Q-algebraic independent, are transcendent numbers (Corollary 1). By
π³ ,β€12
these results, up to weight 12, Conjecture 1 holds (see also6 [7, 12]), i.e. π΅πππ is Q-algebraically free
(Example 2).
References
[1] J. BlΓΌmlein, D. J. Broadhurst, and J. A. M. Vermaseren.β The multiple zeta value data mine, Computer
Physics Communications, 181(3), pp. 582-625, 2010.
[2] D. Borwein, J.M. Borwein & R. Girgensohn.β Explicit evaluation of Euler sums. Proc. Edin. Math.
Soc., 38 (1995), pp. 277-294.
[3] V.C. Bui, G.H.E. Duchamp, V. Hoang Ngoc Minh.β Structure of Polyzetas and Explicit Representation
on Transcendence Bases of Shuffle and Stuffle Algebras, J. Sym. Comp. 83 (2017).
[4] V.C. Bui, V. Hoang Ngoc Minh, Q.H. Ngo and V. Nguyen Dinh.β On The Kernels Of The Zeta
Polymorphism, to appear in the proceeding of βXV International Workshop Lie Theory and Its
Applications in Physics".
[5] C. Costermans, V. Hoang Ngoc Minh.β Noncommutative algebra, multiple harmonic sums and
applications in discrete probability, J. Sym. Comp. (2009), 801-817.
[6] L. Euler.β Meditationes circa singulare serierum genus, Novi. Comm. Acad. Sci. Petropolitanae, 20
(1775), 140-186.
[7] M. Espie, J.-C. Novelli, G. Racinet.β Formal Computations About Multiple Zeta Values, IRMA Lect.
Math. Theor. Phys. 3, de Gruyter, Berlin, 2003, pp. 1-16.
[8] V. Hoang Ngoc Minh, M. Petitot.β A Lyndon words, polylogarithms and the Riemann π function,
Proceedings of FPSACβ98, 1998.
[9] V. Hoang Ngoc Minh.β Structure of polyzetas and Lyndon words, Vietnamese Math. J. (2013), 41,
Issue 4, 409-450.
[10] V. Hoang Ngoc Minh.β On the solutions of universal differential equation with three singularities, in
Confluentes Mathematici, Tome 11 (2019) no. 2, p. 25-64.
[11] V. Hoang Ngoc Minh.β On the Algebraic Bases of Polyzetas, in submission.
[12] M. Kaneko, M. Noro, and K. Tsurumaki.β On a conjecture for the dimension of the space of the
multiple zeta values, IMA Volumes in Mathematics and its Applications 148, Springer, New York,
2008, pp. 47-58.
[13] D. Zagier.β Values of zeta functions and their applications, in βFirst European Congress of Mathe-
matics", vol. 2, BirkhΓ€user, pp. 497-512, (1994).
6
All these implementations base on the βdouble shuffle relations" and provide linear relations.
33