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# On the Algebraic Bases of Polyzetas

Vincel Hoang Ngoc Minh

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Vincel Hoang Ngoc Minh. On the Algebraic Bases of Polyzetas. 2025. hal-05310736v2

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# On the Algebraic Bases of Polyzetas

V. Hoang Ngoc Minh

University of Lille, 1 Place Déliot, 59024 Lille, France,
vincel.hoang-ngoc-minh@univ-lille.fr

**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 $\zeta$ 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 $\mathbb{Q}$-algebra generated by irreducible terms which encode an algebraic basis of the $\mathbb{Q}$-algebra of polyzetas.

These polynomials are totally lexicographically ordered and generate the kernels of the $\zeta$ polymorphism meaning that the $\mathbb{Q}$-free algebra of polyzetas is graded and the irreducible polyzetas are transcendental numbers, $\mathbb{Q}$-algebraically independent, and then $\pi^2$ is $\mathbb{Q}$-algebraically independent on odd zeta values (so does $\pi$).

**Keywords:** Polylogarithms, Hamonic Sums, Polyzetas, Rewriting Systems.

## 1 Introduction

For any $r \ge 1$ and $(s_1, \dots, s_r) \in \mathbb{N}_{\ge 1}$, for any $z \in \mathbb{C} \setminus \{0, 1\}$ and $n \ge 1$, let

$$ \mathrm{Li}_{s_1, \dots, s_r}(z) := \sum_{n_1 > \dots > n_r > 0} \frac{z^{n_1}}{n_1^{s_1} \dots n_r^{s_r}} \quad \text{and} \quad \mathrm{H}_{s_1, \dots, s_r}(n) := \sum_{n \ge n_1 > \dots > n_r > 0} \frac{1}{n_1^{s_1} \dots n_r^{s_r}}. \quad (1) $$

which are respectively called (multiple) polylogarithm and<fnref n="1" /> (multiple) harmonic sum. They lie by the following identity

$$ (1-z)^{-1} \mathrm{Li}_{s_1, \dots, s_r}(z) = \sum_{n \ge 1} \mathrm{H}_{s_1, \dots, s_r}(n) z^n. \quad (2) $$

Let $\mathcal{H}_r$ be $\{(s_1, \dots, s_r) \in \mathbb{N}_{\ge 1}^r, s_1 > 1\}$. Then, for any $(s_1, \dots, s_r) \in \mathcal{H}_r$ the following limits exist and coincide, by a Abel's theorem, with the so-called polyzetas<fnref n="2" /> [28,29,30]

$$ \zeta(s_1, \dots, s_r) := \lim_{z \to 1} \mathrm{Li}_{s_1, \dots, s_r}(z) = \lim_{n \to +\infty} \mathrm{H}_{s_1, \dots, s_r}(n) = \sum_{n_1 > \dots > n_r > 0} \frac{1}{n_1^{s_1} \dots n_r^{s_r}}. \quad (3) $$

Hence, (1)–(3) do appear three (total or partial) functions, i.e. the functions $\mathrm{Li}_\bullet$, $\mathrm{H}_\bullet$ and $\zeta$, on the free monoid $((\mathbb{N}_{\ge 1})^*, \emptyset)$, generated by the alphabet $\mathbb{N}_{\ge 1}$.

<footnote n="1">¹The harmonic sums are viewed as arithmetic function $\mathrm{H}_{s_1, \dots, s_r} : \mathbb{N} \to \mathbb{Q}$.</footnote>

<footnote n="2">²Polyzeta is the contraction of polymorphism and of zeta (see Corollary 1 below).</footnote>
Euler earlier studied polyzetas [20], in classic analysis and in particular, the single zeta values {$\zeta(s)$}$_{s \ge 1}$ and the double zeta values {$\zeta(s_1, s_2)$}$_{s_1 > 1, s_2 \ge 1}$. For the single zeta values, using the bernoulli numbers, {$B_k$}$_{k \ge 0}$, he established the following

$$ \frac{\zeta(2k)}{(2\pi)^{2k}} = \frac{B_{2k}}{2(2k)!} \quad \text{for } k \ge 1. \qquad (4) $$

These rational ratios led to the transcendency of {$\zeta(2k)$}$_{k \ge 1}$ (see [41]). Apéry proved that $\zeta(3)$ is irrational number [1]. For the double zeta values, Euler stated that $\zeta(6, 2)$ can not be expressed on $\zeta(2), ..., \zeta(8)$ and proved<fnref n="3" />

$$ \zeta(2, 1) = \zeta(3) \text{ and } \zeta(s, 1) = \frac{1}{2} \left( s\zeta(s+1) - \sum_{j=1}^{s-2} \zeta(j+1)\zeta(s-j) \right), s > 1. \quad (5) $$

He introduced the following identity (being an origin of the quasi-shuffle product, denoted by ⊕ and recursively defined in (25) below [20])

$$ \zeta(s_1)\zeta(s_2) = \zeta(s_1, s_2) + \zeta(s_2, s_1) + \zeta(s_1 + s_2). \qquad (6) $$

For any $s$ and $k \ge 1$, denoting $\zeta(k)$ and $\zeta(s, \underbrace{1, \dots, 1}_{k \text{ times}})$ by $s_k$ and $c_{s,k}$, respectively, and hoping but doing not succeed to find a formula similar to (4) for $\zeta(2k+1)$ (see Corollary 3 below), Nielsen extended (6) and, for any $n > 0$ and $p, p' > 1$, proved the following identities [35,37]

$$ s_{p'} s_p = s_{p'+p} + c_{p',p} + c_{p,p'}, \qquad (7) $$

$$ \begin{aligned} c_{n,p} = & (-1)^n \sum_{v=0}^{p-2} \binom{n+v-1}{n-1} c_{n+v,p-v} + \sum_{v=0}^{n-2} (-1)^v \binom{p+v-1}{p-1} s_{n-v}s_{p+v} \\ & - (-1)^n \binom{p+n-2}{p-1} (s_{p+n} + c_{1,p+n-1}). \end{aligned} \qquad (8) $$

He stated that $c_{n,p}$ is homogeneous polynomials of degree $n+k$ with rational coefficients on $\zeta(2), \dots, \zeta(s+k)$ and deduced, for any $p > 1$ and $p' \ge 1$, that

$$ s_{p+1} = \sum_{v=1}^{p-1} c_{v,p-v+1} \quad \text{and} \quad c_{1,p'} = \frac{p'}{2} s_{p'+1} - \frac{1}{2} \sum_{v=2}^{p'-1} s_v s_{p'+1-v}. \qquad (9) $$

He earlier used iterated integrals [16] (satisfying the shuffle product, denoted by ⊕ and recursively defined in (17) below) to represent the following particular polylogarithms

$$ \mathrm{Li}_1(z) = \log \frac{1}{1-z} \quad \text{and} \quad \mathrm{Li}_s(z) = \int_0^z \mathrm{Li}_{s-1}(t) \frac{dt}{t}, \qquad (10) $$

$$ \mathrm{Li}_{0, \underbrace{1, \dots, 1}_{k \text{ times}}} (z) = \frac{1}{k!} \log^k \frac{1}{1-z} \quad \text{and} \quad \mathrm{Li}_{s, \underbrace{1, \dots, 1}_{k \text{ times}}} (z) = \int_0^z \mathrm{Li}_{s-1, \underbrace{1, \dots, 1}_{k \text{ times}}} (t) \frac{dt}{t}. \qquad (11) $$

<footnote n="3">³See [14] for another identities concerning double zeta values.</footnote>
The {$\zeta(s_1, \dots, s_r)$}$_{s_1 \ge 1, s_2, \dots, s_r \ge 1}^{r \ge 1}$ are also called *multi zeta values* (MZV for short) [42] or *multiple harmonic series* [32] or *Euler-Zagier sums* [6] and the numbers $r$ and $s_1 + \dots + s_r$ are, respectively, *depth* and *weight* of $\zeta(s_1, \dots, s_r)$. 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, ...

More recently, many new linear relations for polyzetas are also detected in high performance computing [4,5,6] using the algorithms of<fnref n="4" /> LLL type and the truncations at order $n$ of {$\zeta(s_1, \dots, s_r)$}$_{s_1 \ge 1, s_2, \dots, s_r \ge 1}^{r \ge 1}$, i.e. {$H_{s_1, \dots, s_r}(n)$}$_{s_1 \ge 1, s_2, \dots, s_r \ge 1}^{r \ge 1}$ satisfying the $\oplus$ product as already said above (see also Theorem 2 below). In this numerical approach, the main problem is to detect with near certainty which polyzetas can not be expressed on {$\zeta(2), \dots, \zeta(s+k)$} and are qualified as *new constants* [5,6] (as for the Euler's result on $\zeta(6,2)$). Then such polyzetas could be $\mathbb{Q}$-algebraically independent on {$\zeta(2), \dots, \zeta(s+k)$} (see Example 7 below) and the polyzetas could be transcendental numbers (see the proof of Corollary 3 below). Checking $\mathbb{Q}$-linear relations on {$\zeta(s_1, \dots, s_r)$}$_{\substack{s_1 \ge 1, s_2, \dots, s_r \ge 1 \\ 2 \le s_1 + \dots + s_r \le 12}}^{r \ge 1}$ by LLL, Zagier states that the $\mathbb{Q}$-module generated by MZV is apparently graded (see Theorem 4 below for proof) and then guess

*Conjecture 1* ([42]). For any $k \ge 1$, let

$$ \mathcal{L}_k := \operatorname{span}_{\mathbb{Q}} \{\zeta(s_1, \dots, s_r)\}_{\substack{s_1 \ge 1, s_2, \dots, s_r \ge 1 \\ s_1 + \dots + s_r = k}}^{r \ge 1} \quad \text{and} \quad d_k := \dim \mathcal{L}_k. $$

Then<fnref n="5" /> $d_1 = 0, d_2 = d_3 = 1$ and $d_k = d_{k-3} + d_{k-2}$, for $k \ge 4$.

These conjecture then each module $\mathcal{L}_k$ is $\mathbb{Q}$-linearly generated by $d_k$ transcendental zeta values and there is no $\mathbb{Q}$-linear relation among elements of different $\mathcal{L}_k$. In this vain, Ball and Rivoal proved the irrationality of an infinite number of values of the $\zeta$ function at odd integers [2,39] and Zudilin proved that at least one of the four numbers $\zeta(5), \zeta(7), \zeta(9)$ and $\zeta(11)$ is irrational [43].

Studying Conjecture 1, in continuation with the works carried out in [8,18,26] and in a symbolic approach, this work provides more explanations regarding the algorithm **LocalCoordinateIdentification**, partially implemented in [8] and briefly described in [11,31], as an application of an Abel type theorem (see Theorem 3 below) concerning the noncommutative generating series (see (37)–(38) below) of {$\mathrm{Li}_{s_1, \dots, s_r}$}$_{s_1, \dots, s_r \ge 1}^{r \ge 1}$ (resp. {$H_{s_1, \dots, s_r}$}$_{s_1, \dots, s_r \ge 1}^{r \ge 1}$), over the alphabet $X = \{x_0, x_1\}$ (resp. $Y = \{y_k\}_{k \ge 1}$) generating the free monoid $(X^*, 1_{X^*})$ (resp. $(Y^*, 1_{Y^*})$), with respect to the concatenation which is denoted by *conc* and omitted when there is no ambiguity [3,36].

This Abel type theorem exploits the one-to-one correspondences over the above monoids and that generated by integers:

$$ x_0^{s_1-1} x_1 \dots x_0^{s_r-1} x_1 \in X^* x_1 \xrightarrow{\pi_Y} y_{s_1} \dots y_{s_r} \in Y^* \leftrightarrow (s_1, \dots, s_r) \in \mathbb{N}^*_{\ge 1}, \quad (12) $$

$$ \left\{ \begin{array}{rcl} \pi_X : (Y^*, 1_{Y^*}) & \to & (X^*, 1_{X^*}), \\ y_k & \mapsto & x_0^{k-1} x_1, \end{array} \right. \quad \text{and} \quad \left\{ \begin{array}{rcl} \pi_Y : (X^* x_1, 1_{X^*}) & \to & (Y^*, 1_{Y^*}), \\ x_0^{k-1} x_1 & \mapsto & y_k. \end{array} \right. \quad (13) $$

<footnote n="4">⁴LLL is a contraction of A. Lenstra, H. Lenstra and L. Lovász.</footnote>

<footnote n="5">⁵See also [21,26,34] for other algebraic checks.</footnote>
It uses in practice the indexations of polylogarithms (resp. harmonic sums) by words over $X$ (resp. $Y$) (see (14)-(15) and Theorem 2 below), meaning that $\mathrm{Li}_\bullet$ (resp. $\mathrm{H}_\bullet$) is also a function on $X^*$ (resp. $Y^*$) and then $\zeta$ is a partial function on $X^*$ and on $Y^*$, i.e. one puts $\mathrm{Li}_{x_0}(z) = \log(z)$ and, for any$^6$x$^1_0$-$1$x$^1...x^s_0$-$1$x$^1 \in X^*$ and $y_{s_1}...y_{s_r} \in Y^*$,

$$ \mathrm{Li}_{x_0}^{s_1-1} x_1 \dots x_0^{s_r-1} x_1 = \mathrm{Li}_{s_1, \dots, s_r} \quad \text{and then} \quad \zeta(s_1, \dots, s_r) = \zeta(x_0^{s_1-1} x_1 \dots x_0^{s_r-1} x_1), \quad (14) $$

$$ \mathrm{H}_{y_{s_1} \dots y_{s_r}} = \mathrm{H}_{s_1, \dots, s_r} \quad \text{and then} \quad \zeta(s_1, \dots, s_r) = \zeta(y_{s_1} \dots y_{s_r}). \quad (15) $$

These functions on monoids yield in particular the equations on graphs of the $\zeta$ polymorphism, as grouplike series for $\sqcup$ and $\sqcup$ (see Corollary 1 below) bridging two algebraic structures of $\{\zeta(s_1, \dots, s_r)\}_{s_1>1, s_2, \dots, s_r \ge 1}^{r \ge 1}$ (see (37)-(40) and Corollary 2 below).

These bridge equations allow to identify, algorithmically, the local coordinates (of second kind on the groups of grouplike series) of the graphs of $\zeta$ and provide the $\mathbb{Q}$-algebraic relations among polyzetas and then provide the kernels of the $\zeta$ polymorphism (see Proposition 1 below) [28,29,30] which are $\sqcup$-ideal $\mathcal{R}_X$ and $\sqcup$-ideal $\mathcal{R}_Y$ (see (70) below) of the graded $\mathbb{Q}$-algebras of noncommutative polynomials, respectively, over $X$ and $Y$, i.e. $(\mathbb{Q}\langle X\rangle, \sqcup, 1_{X^*})$ and $(\mathbb{Q}\langle Y\rangle, \sqcup, 1_{Y^*})$ which admit the sets of Lyndon words, i.e. $\mathcal{L}\yn X$ and $\mathcal{L}\yn Y$, as totally ordered pure transcendence bases [36].

These ideals are constituted by homogeneous in weight polynomials, belonging to $\mathbb{Q}_X$ or to $\mathbb{Q}_Y$ (see (69) below), and viewed as confluent rewriting systems, in which each rewriting rule (belonging to $\mathcal{R}_{irr}^X$ or to $\mathcal{R}_{irr}^Y$, see (71) below) admits

1. the left side is the leading term of the associated homogeneous polynomial,

2. the right side is canonically represented on the graded $\mathbb{Q}$-algebra generated by irreducible terms (belonging to $\mathcal{L}_{irr}^{X,\infty} \subsetneq \{S_l\}_{l \in \mathcal{L}\yn X}$ or to $\mathcal{L}_{irr}^{Y,\infty} \subsetneq \{\Sigma_l\}_{l \in \mathcal{L}\yn Y}$, see Remark 9 below) which are totally ordered and encode the algebraic generators of the $\mathbb{Q}$-free algebra of polyzetas (see Proposition 1 and Theorem 4 below) [28,29,30].

It follows that, the irreducible polyzetas (belonging to $\mathcal{L}_{irr}^{X,\infty} = \zeta(\mathcal{L}_{irr}^{X,\infty})$ or to $\mathcal{L}_{irr}^{Y,\infty} = \zeta(\mathcal{L}_{irr}^{Y,\infty})$, see (62)-(64) below), are $\mathbb{Q}$-algebraically free and are transcendental numbers (see Corollary 3 below). Hence, by Example 7 below,

1. Up to weight 12, Conjecture 1 held (see also [21,26,34]).

2. $\pi$ is $\mathbb{Q}$-algebraically independent on odd zeta values$^7 \{\zeta(2q+1)\}_{1 \le q \le 11}$ [29].

The organization of this paper is follows

(a) In Section 2, using the Cartier-Quillen-Milnor-Moore theorem (CQMM for short) and Poincaré-Birkhoff-Witt theorem (PBW for short), the isomorphies of the shuf-fle and quasi-shuffle graded $\mathbb{Q}$-bialgebras, i.e. $(\mathbb{Q}\langle X\rangle, \mathrm{conc}, 1_{X^*}, \Delta_{\sqcup})$ and $(\mathbb{Q}\langle Y\rangle, \mathrm{conc}, 1_{Y^*}, \Delta_{\sqcup})$, to the enveloping algebras of primitive elements will be recalled. In these bialgebras, dual bases are constructed (Theorem 1, (23) and (31) below) for factorizing the diagonal series $\mathcal{D}_X$ and $\mathcal{D}_Y$ ((22) and (30) below).

<footnote>$^6$If $s_1 = 1$ then $\zeta(s_1, \dots, s_r)$ is divergent.</footnote>

<footnote>$^7$The readers are also invited to have a look at [10] for a proof, with the present symbolic approach, of certain rational ratio $\zeta(s_1, \dots, s_r)/\pi^{s_1+\dots+s_r}.$</footnote>
(b) In Section 3, the isomorphies of the algebras of the complex functions, defined in (1)–(2) to the shuffle and quasi-shuffle $\mathbb{Q}$-algebras, i.e. $(\mathbb{Q}\langle X\rangle, \mathbb{1}, 1_{X^*})$ and $(\mathbb{Q}\langle Y\rangle, \mathbb{1}, 1_{Y^*})$, respectively, will be explained (Theorem 2 below). On the one hand, these isomorphies algebraically put, the noncommutative generating series of polylog-arithms and of harmonic sums in the Mélançon-Reutenauer-Schützenberger form (MRS for short) and, on the other hand, analytically establish an Abel like theorem (Theorem 3 below).

(c) In Section 4, applying the algorithm **LocalCoordinateIdentification** (which brings more achievements than in [8]), it will provide two confluent rewriting systems<fnref n="8" /> $(\mathbb{Q}1_{Y^*} \oplus (Y \setminus \{y_1\})\mathbb{Q}\langle Y\rangle, \mathcal{R}_{irr}^{Y^*})$ and $(\mathbb{Q}1_{X^*} \oplus x_0\mathbb{Q}\langle X\rangle x_1, \mathcal{R}_{irr}^{X^*})$ (without critical pairs and admitting, respectively, $\mathcal{L}_{irr}^{Y^*}$ and $\mathcal{L}_{irr}^{X^*}$ as the sets of irreducible terms). These rewriting systems provide also the image and the kernels of the $\zeta$ polymorphism (Proposition 1 below) and then the algebraic structures and the arithmetical natures of polyzetas (Theorem 4 and Corollaries 3 and 4 below).

## 2 Algebraic combinatorial framework

Throughout this work, coefficients belong to a commutative ring<fnref n="9" /> A containing $\mathbb{Q}$ and, unless explicitly stated, all tensor products will be considered over the ambient ring (or field). For all matters concerning finite or infinite alphabets, $X (=\{x_0, x_1\}$ and similar) or $Y (=\{y_k\}_{k\ge 1}$ and similar), we use a generic model noted $\mathcal{X}$ in order to state their common combinatorial features (see also Theorem 1 below). Once $\mathcal{X}$ has been totally ordered ($x_0 < x_1$ or $y_1 > y_2 > ...$), let $\mathcal{L}yn\mathcal{X}$ be the set of Lyndon words over $\mathcal{X}$ (generating the free monoid $(\mathcal{X}, 1_{\mathcal{X}^*}))^{10}$. Any pair of Lyndon words $(l_1, l_2)$ is called the standard factorization of $l \in \mathcal{L}yn\mathcal{X}$, and is denoted by $st(l)$, if $l = l_1l_2$ and $l_2$ is the longest nontrivial proper right factor of $l$ or, equivalently, its smallest such, for lexicographic ordering (see [36] for proofs). Using Notations in (12)–(13), one has [36]

$$ l \in \mathcal{L}ynX \setminus \{x_0\} \iff \pi_Y(l) \in \mathcal{L}ynY. \quad (16) $$

The A-module $A\langle \mathcal{X} \rangle$ is endowed with the unital associative concatenation product and the unital associative commutative shuffle product. The latter is recursively defined, for any $x, y \in \mathcal{X}$ and $u, v, w \in \mathcal{X}^*$, by [23]

$$ w \sqcup 1_{\mathcal{X}^*} = 1_{\mathcal{X}^*} \sqcup w = w \text{ and } xu \sqcup yv = x(u \sqcup yv) + y(xu \sqcup v). \quad (17) $$

The coproducts $\Delta_{\text{conc}}$ and $\Delta_{\text{fl}}$ are morphisms for concatenation and are defined by

$$ \forall x \in \mathcal{X}, \Delta_{\text{conc}}(x) = \Delta_{\text{fl}}(x) = 1_{\mathcal{X}^*} \otimes x + x \otimes 1_{\mathcal{X}^*}. \quad (18) $$

By a Radford's theorem [38], $\mathcal{L}yn\mathcal{X}$ forms a pure transcendence basis of the shuffle algebra $(A\langle \mathcal{X} \rangle, \mathbb{1}, 1_{\mathcal{X}^*})$. Letting $\{P_l\}_{l \in \mathcal{L}yn\mathcal{X}}$ be a homogeneous basis of the Lie algebra $\mathcal{L}ie_A\langle \mathcal{X} \rangle$, the enveloping algebra $\mathcal{U}(\mathcal{L}ie_A\langle \mathcal{X} \rangle)$ is classically endowed with the

<footnote n="8">⁸See also Remarks 2–4 below.</footnote>

<footnote n="9">⁹although some of the properties already hold for general commutative semiring [3].</footnote>

<footnote n="10">¹⁰Let $\mu$ denote the Möbius function. Then the number of Lyndon words of length $k$, over $X$, is given by a Witt's formula, i.e. $\psi_2(k) = k^{-1} \sum_{d|k} \mu(d) 2^{k/d}$ [36] and then, with Notations in Conjecture 1, one also has $d_k \le \psi_2(k) - \psi_2(k-1)$.</footnote>
graded linear basis {$P_w$}$_{w \in \mathcal{X}^*$} expanded, by PBW theorem [7,36,40], after {$P_l$}$_{l \in \mathcal{L}_{\mathcal{Y}}\mathcal{X}}$ and it is isomorphic, by CQMM theorem [13], to the following connected, graded and co-commutative bialgebra

$$
\mathcal{H}_{\perp}(\mathcal{X}) := (A\langle\mathcal{X}\rangle, \mathrm{conc}, 1_{\mathcal{X}^*}, \Delta_{\perp}) \quad (19)
$$

Remark 1. For $\mathcal{X} = X$ or $Y$ the corresponding monoids are equipped with length func-
tions, for $X$ we consider the length of words (i.e. ($w$) = $\ell(w) = |w|$) and for $Y$ the length
is given by the weight (i.e. ($w$) = $\ell(y_{i_1} \dots y_{i_n}) = i_1 + \dots + i_n$). This naturally induces a
grading of $A\langle\mathcal{X}\rangle$ and $\mathcal{L}ie_A\langle\mathcal{X}\rangle$ in free modules of finite dimensions. Hence, with the
notations used in (16), for any $\lambda \in \mathcal{L}ynX \setminus \{x_0\}$, one also has $|\lambda| = (\pi_Y(\lambda))$.

Moreover, using the following pairing

$$
A\langle\langle \mathcal{X} \rangle\rangle \otimes A\langle\mathcal{X}\rangle \to A, \quad T \otimes P \mapsto \langle T|P \rangle := \sum_{w \in \mathcal{X}^*} \langle T|w \rangle \langle P|w \rangle, \quad (20)
$$

let the graded dual basis of {$P_w$}$_{w \in \mathcal{X}^*}$ be {$S_w$}$_{w \in \mathcal{X}^*}$, containing the transcendence basis {$S_l$}$_{l \in \mathcal{L}_{\mathcal{Y}}\mathcal{X}}$ of the shuffle algebra, and let the graded dual of $\mathcal{H}_{\perp}(\mathcal{X})$ be

$$
\mathcal{H}_{\perp}^{\vee}(\mathcal{X}) := (A\langle\mathcal{X}\rangle, \mathbb{1}, 1_{\mathcal{X}^*}\Delta_{\mathrm{conc}}). \quad (21)
$$

Then one also has the following factorization of the diagonal series [40]

$$
\mathcal{D}_{\mathcal{X}} := \sum_{w \in \mathcal{X}^*} w \otimes w = \sum_{w \in \mathcal{X}^*} S_w \otimes P_w = \prod_{l \in \mathcal{L}_{\mathcal{Y}}\mathcal{X}} e^{S_l \otimes P_l}. \quad (22)
$$

The dual polynomial bases {$P_w$}$_{w \in \mathcal{X}^*}$ and {$S_w$}$_{w \in \mathcal{X}^*}$, homogeneous in weight (i.e.
length of each $w$), can be constructed recursively by (see [8], for examples by computer)

$$
\left\{
\begin{array}{lll}
P_x = x, & S_x = x, & \text{for } x \in \mathcal{X}, \\
P_l = [P_{l_1}, P_{l_2}], & S_l = y S_{l'}, & \text{for } l=yl' \in \mathcal{L}_{\mathcal{Y}}\mathcal{X} \setminus \mathcal{X}, \\
& & \text{st}(l)=(l_1, l_2), \\
P_w = P_{l_1}^{i_1} \cdots P_{l_k}^{i_k}, & S_w = \frac{S_{l_1}^{\perp} i_1 \cdots l_k \cdots i_k}{i_1! \cdots i_k!}, & \text{for } w = l_1^{i_1} \cdots l_k^{i_k}, \text{ with } l_1, \ldots, \\
& & l_k \in \mathcal{L}_{\mathcal{Y}}\mathcal{X}, l_1 \succ \ldots \succ l_k,
\end{array}
\right.
\qquad (23)
$$

Let us recall also that the polynomials {$S_l$}$_{l \in \mathcal{L}_{\mathcal{Y}}\mathcal{X}}$ (resp. {$P_l$}$_{l \in \mathcal{L}_{\mathcal{Y}}\mathcal{X}}$), homogeneous
in weight (i.e. length of each $l$), are lower (resp. upper) triangular, i.e. for any $l \in \mathcal{L}_{\mathcal{Y}}\mathcal{X}$
and $l' \in \mathcal{L}_{\mathcal{Y}}\mathcal{X}$, one has (see [40])

$$
S_l = l + \sum_{v > l, |v| = |l|} a_v v, \quad a_v \in A \text{ and } P_l = l + \sum_{v < l, |v| = |l|} c_v v, \quad c_v \in A. \quad (24)
$$

Remark 2. 1. The ordering over $\mathcal{L}ynX$ elicits the one over $\{S_l\}_{l \in \mathcal{L}ynX}$ and $\{P_l\}_{l \in \mathcal{L}ynX}$.

2. By (24), the following assertions are equivalent

(a) $\mathcal{L}ynX$ is a pure transcendence basis of $(A\langle X\rangle, \mathbb{1}, 1_{X^*})$,

(b) {$S_l$}$_{l \in \mathcal{L}_{\mathcal{Y}}\mathcal{X}}$ is a pure transcendence basis of $(A\langle X\rangle, \mathbb{1}, 1_{X^*})$.

3. One also has

$$
(A1_{X^*} \oplus x_0 A\langle X\rangle x_1, \mathbb{1}, 1_{X^*}) \cong A[\mathcal{L}_{\mathcal{Y}}\mathcal{X} \setminus X] = A[\{S_l\}_{l \in \mathcal{L}_{\mathcal{Y}}\mathcal{X} \setminus X}].
$$
*Example 1 ([24]).* For $X = \{x_0, x_1\}$, one has

<table><thead><tr><th>l</th><th>P<sub>l</sub></th><th>S<sub>l</sub></th></tr></thead><tbody><tr><td>x<sub>0</sub></td><td>x<sub>0</sub></td><td>x<sub>0</sub></td></tr><tr><td>x<sub>1</sub></td><td>x<sub>1</sub></td><td>x<sub>1</sub></td></tr><tr><td>x<sub>0</sub>x<sub>1</sub></td><td>[x<sub>0</sub>, x<sub>1</sub>]</td><td>x<sub>0</sub>x<sub>1</sub></td></tr><tr><td>x<sub>0</sub><sup>2</sup>x<sub>1</sub></td><td>[x<sub>0</sub>, [x<sub>0</sub>, x<sub>1</sub>]]</td><td>x<sub>0</sub><sup>2</sup>x<sub>1</sub></td></tr><tr><td>x<sub>0</sub>x<sub>1</sub><sup>2</sup></td><td>[[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>]</td><td>x<sub>0</sub>x<sub>1</sub><sup>2</sup></td></tr><tr><td>x<sub>0</sub><sup>3</sup>x<sub>1</sub></td><td>[x<sub>0</sub>, [x<sub>0</sub>, [x<sub>0</sub>, x<sub>1</sub>]]]</td><td>x<sub>0</sub><sup>3</sup>x<sub>1</sub></td></tr><tr><td>x<sub>0</sub><sup>2</sup>x<sub>1</sub><sup>2</sup></td><td>[x<sub>0</sub>, [[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>]]</td><td>x<sub>0</sub><sup>2</sup>x<sub>1</sub><sup>2</sup></td></tr><tr><td>x<sub>0</sub>x<sub>1</sub><sup>3</sup></td><td>[[[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>], x<sub>1</sub>]</td><td>x<sub>0</sub>x<sub>1</sub><sup>3</sup></td></tr><tr><td>x<sub>0</sub><sup>4</sup>x<sub>1</sub></td><td>[x<sub>0</sub>, [x<sub>0</sub>, [x<sub>0</sub>, x<sub>1</sub>]]]</td><td>x<sub>0</sub><sup>4</sup>x<sub>1</sub></td></tr><tr><td>x<sub>0</sub><sup>3</sup>x<sub>1</sub><sup>2</sup></td><td>[x<sub>0</sub>, [x<sub>0</sub>, [[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>]]]</td><td>x<sub>0</sub><sup>3</sup>x<sub>1</sub><sup>2</sup></td></tr><tr><td>x<sub>0</sub><sup>2</sup>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></td><td>[[x<sub>0</sub>, [x<sub>0</sub>, x<sub>1</sub>]], [x<sub>0</sub>, x<sub>1</sub>]]</td><td>2x<sub>0</sub><sup>3</sup>x<sub>1</sub><sup>2</sup> + x<sub>0</sub><sup>2</sup>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></td></tr><tr><td>x<sub>0</sub><sup>2</sup>x<sub>1</sub><sup>3</sup></td><td>[x<sub>0</sub>, [[[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>], x<sub>1</sub>]]</td><td>x<sub>0</sub><sup>2</sup>x<sub>1</sub><sup>3</sup></td></tr><tr><td>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub><sup>2</sup></td><td>[[x<sub>0</sub>, x<sub>1</sub>], [[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>]]</td><td>3x<sub>0</sub><sup>2</sup>x<sub>1</sub><sup>3</sup> + x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub><sup>2</sup></td></tr><tr><td>x<sub>0</sub>x<sub>1</sub><sup>4</sup></td><td>[[[[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>], x<sub>1</sub>], x<sub>1</sub>]</td><td>x<sub>0</sub>x<sub>1</sub><sup>4</sup></td></tr><tr><td>x<sub>0</sub><sup>5</sup>x<sub>1</sub></td><td>[x<sub>0</sub>, [x<sub>0</sub>, [x<sub>0</sub>, [x<sub>0</sub>, x<sub>1</sub>]]]</td><td>x<sub>0</sub><sup>5</sup>x<sub>1</sub></td></tr><tr><td>x<sub>0</sub><sup>4</sup>x<sub>1</sub><sup>2</sup></td><td>[x<sub>0</sub>, [x<sub>0</sub>, [[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>]]]</td><td>x<sub>0</sub><sup>4</sup>x<sub>1</sub><sup>2</sup></td></tr><tr><td>x<sub>0</sub><sup>3</sup>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></td><td>[x<sub>0</sub>, [[x<sub>0</sub>, [x<sub>0</sub>, x<sub>1</sub>]], [x<sub>0</sub>, x<sub>1</sub>]]]</td><td>2x<sub>0</sub><sup>4</sup>x<sub>1</sub><sup>2</sup> + x<sub>0</sub><sup>3</sup>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></td></tr><tr><td>x<sub>0</sub><sup>3</sup>x<sub>1</sub><sup>3</sup></td><td>[x<sub>0</sub>, [x<sub>0</sub>, [[[[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>], x<sub>1</sub>]]]</td><td>x<sub>0</sub><sup>3</sup>x<sub>1</sub><sup>3</sup></td></tr><tr><td>x<sub>0</sub><sup>2</sup>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub><sup>2</sup></td><td>[x<sub>0</sub>, [[x<sub>0</sub>, x<sub>1</sub>], [[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>]]]</td><td>3x<sub>0</sub><sup>3</sup>x<sub>1</sub><sup>3</sup> + x<sub>0</sub><sup>2</sup>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub><sup>2</sup></td></tr><tr><td>x<sub>0</sub><sup>2</sup>x<sub>1</sub><sup>2</sup>x<sub>0</sub>x<sub>1</sub></td><td>[[x<sub>0</sub>, [[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>]], [x<sub>0</sub>, x<sub>1</sub>]]</td><td>6x<sub>0</sub><sup>3</sup>x<sub>1</sub><sup>3</sup> + 3x<sub>0</sub><sup>2</sup>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub><sup>2</sup> + x<sub>0</sub><sup>2</sup>x<sub>1</sub><sup>2</sup>x<sub>0</sub>x<sub>1</sub></td></tr><tr><td>x<sub>0</sub><sup>2</sup>x<sub>1</sub><sup>4</sup></td><td>[x<sub>0</sub>, [[[[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>], x<sub>1</sub>], x<sub>1</sub>]]</td><td>x<sub>0</sub><sup>2</sup>x<sub>1</sub><sup>4</sup></td></tr><tr><td>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub><sup>3</sup></td><td>[[[x<sub>0</sub>, x<sub>1</sub>], [[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>]]]</td><td>4x<sub>0</sub><sup>2</sup>x<sub>1</sub><sup>4</sup> + x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub><sup>3</sup></td></tr><tr><td>x<sub>0</sub>x<sub>1</sub><sup>5</sup></td><td>[[[[[x<sub>0</sub>, x<sub>1</sub>], x<sub>1</sub>], x<sub>1</sub>], x<sub>1</sub>]]</td><td>x<sub>0</sub>x<sub>1</sub><sup>5</sup></td></tr></tbody></table>

*Remark 3.* Using the notations in (12)–(16) and extending Remark 1, since

$$ \forall \lambda = x_{i_1} \cdots x_{i_r} \in \mathcal{L}ynX \setminus \{x_0\}, \quad |\lambda| = r $$

and, by linear extension and by homogeneity in weight, that is length, of the polynomials $\{S_l\}_{l \in \mathcal{L}ynX}$ and $\{P_l\}_{l \in \mathcal{L}ynX}$, one has

$$ |S_{\lambda}| = r \text{ and } |P_{\lambda}| = r $$

then the depth of $\zeta(S_{\lambda})$ is the depth of $\zeta(\lambda)$, that is $r$, as previously introduced in Section 1.

In addition, $A\langle Y \rangle$ is equipped with the unital associative commutative quasi-shuffle product recursively defined, for any $u, v, w \in Y^*$ and $y_i, y_j \in Y$, by [25,26]

$$ w \boxplus 1_{Y^*} = 1_{Y^*} \boxplus w = w \text{ and } y_i u \boxplus y_j v = y_i (u \boxplus y_j v) + y_j (y_i u \boxplus v) + y_{i+j} (u \boxplus v)(25) $$

and its dual law, being a conc-morphism, is given by

$$ \forall y_k \in Y, \Delta_{\boxplus}(y_k) = y_k \otimes 1_{Y^*} + 1_{Y^*} \otimes y_k + \sum_{i+j=k} y_i \otimes y_j. \quad (26) $$
Letting $\mathrm{Prim}(\mathcal{H}_{\perp}(Y)) = \mathrm{span}_A\{\pi_1(w)\}_{w \in Y^*}$ and $\pi_1$ be the eulerian projector defined$^{11}$, for any $w \in Y^*$, by [30]

$$ \pi_1(w) = w + \sum_{k=2}^{(w)} \frac{(-1)^{k-1}}{k} \sum_{u_1, \dots, u_k \in Y^+} \langle w | u_1 | \perp \dots \perp u_k \rangle u_1 \dots u_k. \quad (27) $$

these yield another connected, graded and co-commutative bialgebra being isomorphic to the enveloping algebra of the Lie algebra of primitive elements, i.e.

$$ \mathcal{H}_{\perp}(Y) := (A\langle Y \rangle, \mathrm{conc}, 1_{Y^*}, \Delta_{\perp}) \cong \mathcal{U}(\mathrm{Prim}(\mathcal{H}_{\perp}(Y))). \quad (28) $$

Let $\{\Pi_w\}_{w \in Y^*}$ be a linear basis, homogeneous in weight (i.e. weight of each $w$), expanded by decreasing PBW after a basis $\{\Pi_l\}_{l \in \mathcal{L}_{ynY}}$ of $\mathrm{Prim}(\mathcal{H}_{\perp}(Y))$ and let $\{\Sigma_w\}_{w \in Y^*}$ (homogeneous in weight) be its dual basis containing the transcendence basis $\{\Sigma_l\}_{l \in \mathcal{L}_{ynY}}$ of the quasi-shuffle algebra. The graded dual of $\mathcal{H}_{\perp}(Y)$ is denoted as follows

$$ \mathcal{H}_{\perp}^{\vee}(Y) = (A\langle Y \rangle, \perp, 1_{Y^*}, \Delta_{\mathrm{conc}}). \quad (29) $$

The diagonal series $\mathcal{D}_Y$, on $\mathcal{H}_{\perp}(Y)$ is also factorized as follows [30]

$$ \mathcal{D}_Y := \sum_{w \in Y^*} w \otimes w = \sum_{w \in Y^*} \Sigma_w \otimes \Pi_w = \prod_{l \in \mathcal{L}_{ynY}} e^{\Sigma_l \otimes \Pi_l}. \quad (30) $$

**Theorem 1 ([30]).** Let $\varphi_{\pi_1} : (A\langle Y \rangle, \mathrm{conc}, 1_{Y^*}) \to (A\langle Y \rangle, \mathrm{conc}, 1_{Y^*})$ maps $y_k$ to $\pi_1(y_k)$. It is an automorphism of $A\langle Y \rangle$ realizing an isomorphism of bialgebras between $\mathcal{H}_{\perp}(Y)$ and $\mathcal{H}_{\perp}(Y)$. The following diagram commutes

$$ \begin{tikzcd} A\langle Y \rangle \arrow[r, "\Delta_{\perp}"] \arrow[d, "\varphi_{\pi_1}"] & A\langle Y \rangle \otimes A\langle Y \rangle \arrow[d, "\varphi_{\pi_1} \otimes \varphi_{\pi_1}"] \\ A\langle Y \rangle \arrow[r, "\Delta_{\perp}"] & A\langle Y \rangle \otimes A\langle Y \rangle \arrow[d, "\varphi_{\pi_1}^{-1}"] \end{tikzcd} $$

and $\{\Pi_w\}_{w \in Y^*}$ (resp. $\{\Sigma_w\}_{w \in Y^*}$) is image of $\{P_w\}_{w \in Y^*}$ (resp. $\{S_w\}_{w \in Y^*}$) by $\varphi_{\pi_1}$ (resp. $\varphi_{\pi_1}^{-1}$), where $\varphi_{\pi_1}$ is the adjoint of $\varphi_{\pi_1}$.

Algorithmically, the bases of homogeneous in weight polynomials $\{P_w\}_{w \in Y^*}$ and $\{S_w\}_{w \in Y^*}$ of, respectively, $\mathcal{U}(\mathrm{Prim}(\mathcal{H}_{\perp}(Y)))$ and $(A\langle Y \rangle, \perp, 1_{Y^*})$ can be directly and recursively constructed as follows (see [8], for examples by computer)

$$ \left\{ \begin{array}{ll} \Pi_{y_s} = \pi_1(y_s), & \Sigma_{y_s} = y_s \\ \Pi_l = [\Pi_{l_1}, \Pi_{l_2}], & \Sigma_l = \sum_{(*) \substack{y_{s_k} + \dots + s_{k_l} \\ i!}} \sum_{l_1 \dots l_n}, \\ \Pi_w = \Pi_{l_1}^{i_1} \dots \Pi_{l_k}^{i_k}, & \Sigma_{l_1}^{\perp} i_1 \perp \dots \perp i_k \Sigma_{l_k}^{\perp} i_k, \end{array} \right. \quad \begin{array}{l} \text{for } y_s \in Y, \\ \text{for } l \in \mathcal{L}_{ynY} \setminus Y, \\ \text{st}(l) = (l_1, l_2), \\ \text{for } w = l_1^{i_1} \dots l_k^{i_k}, \text{ with } l_1, \dots, \\ \phantom{\text{for } w = } l_k \in \mathcal{L}_{ynY}, l_1 \succ \dots \succ l_k, \end{array} \quad (31) $$

In $(*)$, the sum is taken over all $\{k_1, \dots, k_i\} \subset \{1, \dots, k\}$ and $l_1 \ge \dots \ge l_n$ such that $(y_{s_k}, \dots, y_{s_{k_i}}, l_1, \dots, l_n)$ by transitive closure of the relations on standard sequences [9,40].

<footnote>$^{11}$By (18), any letter $x \in \mathcal{X}$ is primitive, for $\Delta_{\mathrm{conc}}$ and $\Delta_{\perp}$. By (26), the polynomials $\{\pi_1(y_k)\}_{k \ge 2}$ and only the letter $y_1$ are primitive, for $\Delta_{\perp}$.</footnote>
Remark 4. 1. For any $n \ge 1$, one has

$$ (A1_{Y^*} \oplus (Y \setminus \{y_1\})A(Y), \oplus, 1_{X^*}) \cong A[\mathcal{L}ynY \setminus \{y_1\}] = A[\{\Sigma_l\}_{l \in \mathcal{L}ynY \setminus \{y_1\}}]. $$

item The ordering over $\mathcal{L}ynY$ elicits the ordering over $\{\Sigma_l\}_{l \in \mathcal{L}ynY}$ and $\{\Pi_l\}_{l \in \mathcal{L}ynY}$.

2. By (32), the following assertions are equivalent

(a) $\mathcal{L}ynY$ is a pure transcendence basis of $(A(Y), \oplus, 1_{Y^*})$

(b) $\{\Sigma_l\}_{l \in \mathcal{L}ynY}$ is a pure transcendence basis of $(A(Y), \oplus, 1_{Y^*})$.

3. One also has

$$ (A1_{Y^*} \oplus (Y \setminus \{y_1\})A(Y), \oplus, 1_{X^*}) \cong A[\mathcal{L}ynY \setminus \{y_1\}] = A[\{\Sigma_l\}_{l \in \mathcal{L}ynY \setminus \{y_1\}}]. $$

Example 2 ([29]). For $Y = \{y_k\}_{k \ge 1}$, one has

<table><thead><tr><th>w</th><th>Π<sub>w</sub></th><th>Σ<sub>w</sub></th></tr></thead><tbody><tr><td>y<sub>2</sub></td><td>y<sub>2</sub> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub><sup>2</sup></td><td>y<sub>2</sub></td></tr><tr><td>y<sub>1</sub><sup>2</sup></td><td>y<sub>1</sub><sup>2</sup></td><td><sup>1</sup>&frasl;<sub>2</sub>y<sub>2</sub> + y<sub>1</sub><sup>2</sup></td></tr><tr><td>y<sub>3</sub></td><td>y<sub>3</sub> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub>y<sub>2</sub> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>2</sub>y<sub>1</sub> + <sup>1</sup>&frasl;<sub>3</sub>y<sub>1</sub><sup>3</sup></td><td>y<sub>3</sub></td></tr><tr><td>y<sub>2</sub>y<sub>1</sub></td><td>y<sub>2</sub>y<sub>1</sub> - y<sub>2</sub>y<sub>1</sub></td><td><sup>1</sup>&frasl;<sub>2</sub>y<sub>3</sub> + y<sub>2</sub>y<sub>1</sub></td></tr><tr><td>y<sub>1</sub>y<sub>2</sub></td><td>y<sub>2</sub>y<sub>1</sub> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub><sup>3</sup></td><td>v</td></tr><tr><td>y<sub>1</sub><sup>3</sup></td><td>y<sub>1</sub><sup>3</sup></td><td><sup>1</sup>&frasl;<sub>6</sub>y<sub>3</sub> + <sup>1</sup>&frasl;<sub>2</sub>y<sub>2</sub>y<sub>1</sub> + <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub>y<sub>2</sub> + y<sub>1</sub><sup>3</sup></td></tr><tr><td>y<sub>4</sub></td><td>y<sub>4</sub> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub>y<sub>3</sub> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>2</sub><sup>2</sup> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>3</sub>y<sub>1</sub> + <sup>1</sup>&frasl;<sub>3</sub>y<sub>1</sub><sup>2</sup>y<sub>2</sub> + <sup>1</sup>&frasl;<sub>3</sub>y<sub>1</sub>y<sub>2</sub>y<sub>1</sub> + <sup>1</sup>&frasl;<sub>3</sub>y<sub>2</sub>y<sub>1</sub><sup>2</sup> - <sup>1</sup>&frasl;<sub>4</sub>y<sub>1</sub><sup>4</sup></td><td>y<sub>4</sub></td></tr><tr><td>y<sub>3</sub>y<sub>1</sub></td><td>y<sub>3</sub>y<sub>1</sub> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>2</sub>y<sub>1</sub><sup>2</sup> - y<sub>1</sub>y<sub>3</sub> + <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub><sup>2</sup>y<sub>2</sub></td><td><sup>1</sup>&frasl;<sub>2</sub>y<sub>4</sub> + y<sub>3</sub>y<sub>1</sub></td></tr><tr><td>y<sub>2</sub><sup>2</sup></td><td>y<sub>2</sub><sup>2</sup> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>2</sub>y<sub>1</sub><sup>2</sup> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub><sup>2</sup>y<sub>2</sub> + <sup>1</sup>&frasl;<sub>4</sub>y<sub>1</sub><sup>4</sup></td><td><sup>1</sup>&frasl;<sub>2</sub>y<sub>4</sub> + y<sub>2</sub><sup>2</sup></td></tr><tr><td>y<sub>2</sub>y<sub>1</sub><sup>2</sup></td><td>y<sub>2</sub>y<sub>1</sub><sup>2</sup> - 2y<sub>1</sub>y<sub>2</sub>y<sub>1</sub> + y<sub>1</sub><sup>2</sup>y<sub>2</sub></td><td><sup>1</sup>&frasl;<sub>6</sub>y<sub>4</sub> + <sup>1</sup>&frasl;<sub>2</sub>y<sub>3</sub>y<sub>1</sub> + <sup>1</sup>&frasl;<sub>2</sub>y<sub>2</sub><sup>2</sup> + y<sub>2</sub>y<sub>1</sub><sup>2</sup></td></tr><tr><td>y<sub>1</sub>y<sub>3</sub></td><td>y<sub>1</sub>y<sub>3</sub> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub><sup>2</sup>y<sub>2</sub> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub>y<sub>2</sub>y<sub>1</sub> + <sup>1</sup>&frasl;<sub>3</sub>y<sub>1</sub><sup>4</sup></td><td>y<sub>4</sub> + y<sub>3</sub>y<sub>1</sub> + y<sub>1</sub>y<sub>3</sub></td></tr><tr><td>y<sub>1</sub>y<sub>2</sub>y<sub>1</sub></td><td>y<sub>1</sub>y<sub>2</sub>y<sub>1</sub> - y<sub>1</sub><sup>2</sup>y<sub>2</sub></td><td><sup>1</sup>&frasl;<sub>2</sub>y<sub>4</sub> + <sup>1</sup>&frasl;<sub>2</sub>y<sub>3</sub>y<sub>1</sub> + y<sub>2</sub><sup>2</sup></td></tr><tr><td>y<sub>1</sub><sup>2</sup>y<sub>2</sub></td><td>y<sub>1</sub><sup>2</sup>y<sub>2</sub> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub><sup>4</sup></td><td>+y<sub>2</sub>y<sub>1</sub><sup>2</sup> + <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub>y<sub>3</sub> + y<sub>1</sub>y<sub>2</sub>y<sub>1</sub></td></tr><tr><td>y<sub>1</sub><sup>4</sup></td><td>y<sub>1</sub><sup>4</sup></td><td><sup>1</sup>&frasl;<sub>2</sub>y<sub>4</sub> + y<sub>3</sub>y<sub>1</sub> + y<sub>2</sub><sup>2</sup> + y<sub>2</sub>y<sub>1</sub><sup>2</sup></td></tr><tr><td>y<sub>1</sub>y<sub>2</sub>y<sub>1</sub></td><td>y<sub>1</sub>y<sub>2</sub>y<sub>1</sub> - y<sub>1</sub><sup>2</sup>y<sub>2</sub></td><td>+y<sub>1</sub>y<sub>3</sub> + y<sub>1</sub>y<sub>2</sub>y<sub>1</sub> + y<sub>1</sub><sup>2</sup>y<sub>2</sub></td></tr><tr><td>y<sub>1</sub><sup>2</sup>y<sub>2</sub></td><td>y<sub>1</sub><sup>2</sup>y<sub>2</sub> - <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub><sup>4</sup></td><td><sup>1</sup>&frasl;<sub>2</sub>y<sub>4</sub>y<sub>3</sub> + <sup>1</sup>&frasl;<sub>6</sub>y<sub>3</sub>y<sub>1</sub> + <sup>1</sup>&frasl;<sub>4</sub>y<sub>2</sub><sup>2</sup> + <sup>1</sup>&frasl;<sub>2</sub>y<sub>2</sub>y<sub>1</sub><sup>2</sup></td></tr><tr><td>y<sub>1</sub>y<sub>2</sub>y<sub>1</sub></td><td>y<sub>1</sub>y<sub>2</sub>y<sub>1</sub> - y<sub>1</sub><sup>2</sup>y<sub>2</sub></td><td>+<sup>1</sup>&frasl;<sub>6</sub>y<sub>1</sub>y<sub>3</sub> + <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub>y<sub>2</sub>y<sub>1</sub> + <sup>1</sup>&frasl;<sub>2</sub>y<sub>1</sub><sup>2</sup>y<sub>2</sub> + y<sub>1</sub><sup>4</sup></td></tr></tbody></table>

The homogeneous in weight polynomials $\{\Sigma_l\}_{l \in \mathcal{L}ynY}$ (resp. $\{\Pi_l\}_{l \in \mathcal{L}ynY}$) are also lower (resp. upper) triangular, i.e. for any $l \in \mathcal{L}ynX$ and $l' \in \mathcal{L}ynY$, one has [28,29]

$$ \Sigma_{l'} = l' + \sum_{v > l', (v) = (l')} b_v v, \quad b_v \in A \text{ and } \Pi_{l'} = l' + \sum_{v < l', (v) = (l')} d_v v, \quad d_v \in A. \quad (32) $$

Remark 5. Extending Remark 1 and using the notations in (12)–(16), since

$$ \forall \lambda = y_{i_1} \cdots y_{i_r} \in \mathcal{L}ynY, (\lambda) = |\pi_X(\lambda)| = i_1 + \cdots + i_r $$

and, by linear extension and by homogeneity (in weight) of the polynomials $\{\Sigma_l\}_{l \in \mathcal{L}ynY}$ and $\{\Pi_l\}_{l \in \mathcal{L}ynY}$, one has

$$ (\Sigma_{\lambda}) = |\pi_X(\lambda)| = i_1 + \cdots + i_r, \text{ and } (\Pi_{\lambda}) = |\pi_X(\lambda)| = i_1 + \cdots + i_r $$

then the weight of $\zeta(\Sigma_\lambda)$ is the weight of $\zeta(\lambda)$, that is $i_1 + \cdots + i_r$, as previously introduced in Section 1.
### 3 Polylogarithms, harmonic sums and polyzetas viewed as functions on free monoids and their graphs

In all the sequel, for greater convenience,

$$ \begin{array}{l} \text{if } \mathcal{X} = X \text{ then let } \mathrm{gDIV} := X \quad \text{and } \mathrm{CONV} := x_0 X^* x_1 \\ \text{else let } \mathrm{gDIV} := \{y_1\} \quad \text{and } \mathrm{CONV} := (Y \setminus \{y_1\}) Y^*. \end{array} \quad (33) $$

It follows that

$$ \mathcal{L}yn\mathcal{X} \setminus \mathrm{gDIV} \subset \mathrm{CONV}. \quad (34) $$

With Notations in (1) and (14)–(15), one has

**Theorem 2 ([30]).** The following morphisms of algebras are injective

$$ \mathrm{H}_{\bullet} : (\mathbb{Q}\langle Y \rangle, \uplus, 1_{Y^*}) \to (\mathbb{Q}\{\mathrm{H}_w\}_{w \in Y^*}, \times, 1), $$

$$ y_{s_1} \cdots y_{s_k} \mapsto \mathrm{H}_{s_1, \dots, s_r}, $$

$$ \mathrm{Li}_{\bullet} : (\mathbb{Q}\langle X \rangle, \uplus, 1_{X^*}) \to (\mathbb{Q}\{\mathrm{Li}_w\}_{w \in X^*}, \times, 1, \widetilde{\mathbb{C}\setminus\{0,1\}}), $$

$$ x_0 \mapsto \log z, $$

$$ x_0 x_1^{s_1-1} \cdots x_0 x_1^{s_k-1} \mapsto \mathrm{Li}_{s_1, \dots, s_r}. $$

Or equivalently, $\{\mathrm{H}_w\}_{w \in Y^*}$ (resp. $\{\mathrm{H}_{\Sigma_l}\}_{l \in \mathcal{L}_{ynY}}$ and $\{\mathrm{H}_l\}_{l \in \mathcal{L}_{ynY}}$) and $\{\mathrm{Li}_w\}_{w \in X^*}$ (resp. $\{\mathrm{Li}_{S_l}\}_{l \in \mathcal{L}_{ynX}}$ and $\{\mathrm{Li}_l\}_{l \in \mathcal{L}_{ynX}}$) are linearly (resp. algebraically) independent over $\mathbb{Q}$.

**Corollary 1 ([30]).** Let

$$ \mathcal{L} := \operatorname{span}_{\mathbb{Q}}\{\zeta(w)\}_{w \in \mathrm{CONV}}. $$

Then the following polymorphism is surjective

$$ \zeta : \left\{ \begin{array}{l} (\mathbb{Q}1_{Y^*} \oplus (Y \setminus \{y_1\})\mathbb{Q}\langle Y \rangle, \uplus, 1_{Y^*}) \\ (\mathbb{Q}1_{X^*} \oplus x_0\mathbb{Q}\langle X \rangle x_1, \uplus, 1_{X^*}) \end{array} \right\} \to (\mathcal{L}, \times, 1), $$

$$ \left\{ \begin{array}{l} y_{s_1} \cdots y_{s_k} \\ x_0 x_1^{s_1-1} \cdots x_0 x_1^{s_k-1} \end{array} \right\} \mapsto \zeta(s_1, \dots, s_r). $$

Moreover, the $\mathbb{Q}$-algebra $(\mathcal{L}, \times, 1)$ is generated by $\{\zeta(l)\}_{l \in \mathcal{L}_{ynX} \setminus \mathrm{gDIV}}$, or equivalently by $\{\zeta(\Sigma_l)\}_{l \in \mathcal{L}_{ynY} \setminus \{y_1\}}$ or by $\{\zeta(S_l)\}_{l \in \mathcal{L}_{ynX} \setminus X}$.

**Remark 6.** Contrary to the harmonic sums $\{\mathrm{H}_l\}_{l \in \mathcal{L}_{ynY}}$ (resp. $\{\mathrm{H}_{\Sigma_l}\}_{l \in \mathcal{L}_{ynY}}$) and to the polylogarithms $\{\mathrm{Li}_l\}_{l \in \mathcal{L}_{ynX}}$ (resp. $\{\mathrm{Li}_{S_l}\}_{l \in \mathcal{L}_{ynX}}$), the polyzetas $\{\zeta(l)\}_{l \in \mathcal{L}_{ynX} \setminus \mathrm{gDIV}}$ are not necessary algebraically independent over $\mathbb{Q}$ (see the Euler's relations in (5)) and then it is legitimate to determine the $\mathbb{Q}$-algebraic relations among them. So does [26], in which it is conjectured that the $\mathbb{Q}$-algebra $(\mathcal{L}, \times, 1)$ is $\mathbb{Q}$-free and then the objective of this present work is to construct effectively an algebraic basis for $(\mathcal{L}, \times, 1)$ by explicitlying the non $\mathbb{Q}$-algebraic relations among $\{\zeta(\Sigma_l)\}_{l \in \mathcal{L}_{ynY} \setminus \{y_1\}}$ and $\{\zeta(S_l)\}_{l \in \mathcal{L}_{ynX} \setminus X}$ (see Proposition 1 below).
Using the notations in Conjecture 1, it follows also that

$$ \forall k, k' \ge 1, \mathcal{Z}_k \mathcal{Z}_{k'} \subset \mathcal{Z}_{k+k'}. \qquad (35) $$

Now, we are in situation to consider the graphs of $Li_\bullet$ [27] and of $H_\bullet$ [17,18], using the diagonal series $\mathcal{D}_X$ and $\mathcal{D}_Y$, as follows

$$ L := \sum_{w \in X} \mathrm{Li}_w w = (\mathrm{Li}_\bullet \otimes \mathrm{Id}) \mathcal{D}_X \quad \text{and} \quad H := \sum_{w \in Y} \mathrm{H}_w w = (\mathrm{H}_\bullet \otimes \mathrm{Id}) \mathcal{D}_Y. \qquad (36) $$

Then by (22) and (30) these graphs are grouplike series respectively, for $\sqcup$ and $\sqcup$, and are put in the MRS forms, leading to the definitions of grouplike series $Z_{\sqcup}$ [27] and $Z_{\sqcup}$ [28,29,30] respectively, for $\sqcup$ and $\sqcup$, by setting

$$ L = \prod_{l \in \mathcal{L} \mathcal{Y} n X} e^{\mathrm{Li}_l P_l} \quad \text{and let} \quad Z_{\sqcup} := \prod_{l \in \mathcal{L} \mathcal{Y} n X \setminus X} e^{\zeta(S_l) P_l}, \qquad (37) $$

$$ H = \prod_{l \in \mathcal{L} \mathcal{Y} n Y} e^{\mathrm{H}_{\Sigma_l} \Pi_l} \quad \text{and let} \quad Z_{\sqcup} := \prod_{l \in \mathcal{L} \mathcal{Y} n Y \setminus \{y_1\}} e^{\zeta(\Sigma_l) \Pi_l}. \qquad (38) $$

Let also $Z_\gamma$ be the graph of the following $\sqcup$-character [17,18]

$$ \gamma_\bullet : (\mathcal{Q}\langle Y \rangle, \sqcup, 1_{Y^*}) \to (\mathcal{L}[\gamma], \times, 1), \qquad (39) $$

mapping $y_1$ to$^{12}$ $\gamma_{y_1} = \gamma$ and $l$ to $\eta = \zeta(l)$, $l \in \mathcal{L} \mathcal{Y} n Y \setminus \{y_1\} \subset (Y \setminus \{y_1\})Y^*$:

$$ Z_{\gamma} := \sum_{w \in Y^*} \gamma_w w, \qquad (40) $$

where $\gamma_w$ denotes the finite part (f.p. for shorted) of the asymptotic expansion of the hamonic sum $H_w$ in the comparison scale $\{n^a \log^b(n)\}_{a \in \mathbb{Z}, b \in \mathbb{N}}$, for $n \to +\infty$:

$$ \gamma_w := \lim_{n \to +\infty} \mathrm{f.p.} \ H_w(n), \{n^a \log^b(n)\}_{a \in \mathbb{Z}, b \in \mathbb{N}}. \qquad (41) $$

Thus, as grouplike series, one has [28,29]

$$ Z_{\gamma} = e^{\gamma Y_1} Z_{\sqcup}. \qquad (42) $$

On the other hand, $Z_{\sqcup}$ and $Z_{\sqcup}$ are also viewed as graphs of the following characters, extending the $\zeta$ polymorphism, i.e.$^{13}$

$$ \zeta_{\sqcup} : (\mathcal{Q}\langle X \rangle, \sqcup, 1_{X^*}) \to (\mathcal{L}, \times, 1), \qquad (43) $$

$$ \zeta_{\sqcup} : (\mathcal{Q}\langle Y \rangle, \sqcup, 1_{Y^*}) \to (\mathcal{L}, \times, 1), \qquad (44) $$

<footnote>$^{12}$ $\gamma$ is the Euler-Mascheroni constant.</footnote>

<footnote>$^{13}$ With the comparison scale $\{(1-z)^a \log^b((1-z)^{-1})\}_{a \in \mathbb{Z}, b \in \mathbb{N}}$ (resp. $\{n^a H_1^b(n)\}_{a \in \mathbb{Z}, b \in \mathbb{N}}$), the finite part of the singular (resp. asymptotic) expansion of $Li_u$ (resp. $H_v$) equals $\zeta_{\sqcup}(u)$ (resp. $\zeta_{\sqcup}(v)$), while with the comparison scale $\{n^a \log^b(n)\}_{a \in \mathbb{Z}, b \in \mathbb{N}}$, the finite part of the asymptotic expansion of $H_w$ equals $\gamma_w$ (see (40) above).</footnote>
and, for any word $w$ in $Y^*$ (resp $X^*$), each polyzeta $\zeta_{\perp}(w)$ (resp. $\zeta_{\perp}(w)$) represents the finite part of the asymptotic expansion (resp. functional expansion) of the hamonic sum $H_w$ (resp. polylogarithm $Li_w$) in the comparison scale $\{n^a \log^b(n)\}_{a \in \mathbb{Z}, b \in \mathbb{N}}$ (resp. $\{(1-z)^a \log^b((1-z)^{-1})\}_{a \in \mathbb{Z}, b \in \mathbb{N}}$, for $n \to +\infty$ (resp. $z \to 1$):

$$ \zeta_{\perp}(u) = \langle Z_{\perp} | u \rangle = \underset{z \to 1}{\text{f.p.}} \mathrm{Li}_u(z), \quad \{(1-z)^a \log^b((1-z)^{-1})\}_{a \in \mathbb{Z}, b \in \mathbb{N}}, \quad (45) $$

$$ \zeta_{\perp}(v) = \langle Z_{\perp} | v \rangle = \underset{n \to +\infty}{\text{f.p.}} H_v(n), \quad \{n^a H_1^b(n)\}_{a \in \mathbb{Z}, b \in \mathbb{N}}. \quad (46) $$

Hence, in virtue of (3) and (16), the following convergent polyzetas equal:

$$ \forall l \in \mathcal{L} \mathit{ynY} \setminus \{y_1\}, \quad \zeta_{\perp}(l) = \zeta_{\perp}(\pi_X(l)) = \zeta(l) = \eta \quad (47) $$

and, on the other hand, one has for the divergent generators of weight 1,

$$ \zeta_{\perp}(x_0) = \mathrm{Li}_{x_0}(1) = \log(1) = 0 \quad (48) $$

and, as consequence of the finite parts in (45)–(46), there quite exists the simultaneously regularizations of the divergent polyzetas $\zeta(x_1)$ and $\zeta(y_1)$ to the same value, 0:

$$ \zeta_{\perp}(x_1) = \zeta_{\perp}(y_1) = 0. \quad (49) $$

*Remark 7.* 1. In [14], studying the commutative generating series of the double polyzetas $\{\zeta(s_1, s_2)\}_{s_1>1, s_2 \ge 1}$, Cartier used the convention $\zeta(1) = 0$.

2. According to [13,33], one can consider the following simultaneous regularizations to the indeterminate $T$:

(a) for the divergent generators of weight 1:

$$ \zeta_{\perp}^T(x_1) = \zeta_{\perp}^T(y_1) = T, $$

(b) for any $u$ and $v \in X^*$ (resp. $Y^*$), one has

$$ \zeta_{\perp}^T(u \perp v) = \zeta_{\perp}^T(u) \zeta_{\perp}^T(v) \quad (\text{resp. } \zeta_{\perp}^T(u \perp v) = \zeta_{\perp}^T(u) \zeta_{\perp}^T(v)), $$

and since $T$ is transcendent then they can specialize $T=0$ or $T=\gamma$. So it have been done in [21,34] with $T=0$. But the $\mathbb{Q}$-algebra $(\mathscr{L}, \times, 1)$ is not a $\mathbb{Q}[T]$-algebra!

3. As being graphs of the morphisms $\zeta_{\perp}$ and $\zeta_{\perp}$ (see (43), the noncommutative series $Z_{\perp}$ and $Z_{\perp}$ are grouplike, for $\perp$ and $\perp$ (see (37)–(38)), respectively. Then, by (43)–(46), one obtains the regularized double shuffle relations considered in [21,34], i.e. for any $u$ and $v \in X^*$ (resp. $Y^*$), one has

$$ \zeta_{\perp}(u \perp v) = \zeta_{\perp}(u) \zeta_{\perp}(v) \quad (\text{resp. } \zeta_{\perp}(u \perp v) = \zeta_{\perp}(u) \zeta_{\perp}(v)) $$

and then, using $\pi_Y$ (resp. $\pi_X$) in (12)–(13), one deduces that

$$ \zeta_{\perp}(u \perp v) = \zeta_{\perp}(\pi_Y(u) \perp \pi_Y(v)) \quad (\text{resp. } \zeta_{\perp}(\pi_X(u) \perp \pi_X(v)) = \zeta_{\perp}(u \perp v)). $$

Using the Friedrichs criterion for $\perp$ [40] (resp. for $\perp$ [29]), the converse also holds.
4. In particular, using the algebraic bases, $\mathcal{L}ynX$ of $(\mathbb{Q}\langle X\rangle, \boxplus, 1_{X^*})$ and $\mathcal{L}ynY$ of $(\mathbb{Q}\langle Y\rangle, \boxplus, 1_{Y^*})$, these shuffle relations are nothing else the double shuffle relations earlier considered in [26], i.e. for any $l_1$ and $l_2 \in \mathcal{L}ynX$ (resp. $\mathcal{L}ynY$), one has

$$ \zeta_{\boxplus}(l_1 \boxplus l_2) = \zeta_{\boxplus}(l_1)\zeta_{\boxplus}(l_2) \text{ (resp. } \zeta_{\boxplus}(l_1 \boxplus l_2) = \zeta_{\boxplus}(l_1)\zeta_{\boxplus}(l_2)\text{)} $$

and then, by (16) and $\pi_Y$ (resp. $\pi_X$) in (12)–(13) again, one deduces that

$$ \zeta_{\boxplus}(l_1 \boxplus l_2) = \zeta_{\boxplus}(\pi_Y(l_1) \boxplus \pi_Y(l_2)) \text{ (resp. } \zeta_{\boxplus}(\pi_X(l_1) \boxplus \pi_X(l_2)) = \zeta_{\boxplus}(l_1 \boxplus l_2)\text{)} $$

Using (2), the singularities analysis on $\{\mathrm{Li}_w\}_{w \in X^*}$ lead to the following behaviors of the noncommutative generating series L, put in the MRS form as in (37) [27]

$$ L(z) \sim_0 e^{x_0 \log z} \text{ and } L(z) \sim_1 e^{-x_1 \log(1-z)} Z_{\boxplus}, \qquad (50) $$

and then by a transfer theorem [22] from $\{\mathrm{Li}_w\}_{w \in X^*}$ to $\{\mathrm{H}_w\}_{w \in Y^*}$, these lead also to the following behaviors of the noncommutative generating series H [17,18]

$$ H(n) \sim_{+\infty} e^{-\sum_{k \ge 1} \frac{\mathrm{H}_{y_k}(n)(-y_1)^k}{k}} \pi_Y(Z_{\boxplus}). \qquad (51) $$

By term by term differentiations, the noncommutative generating series L (see (36)) satisfies the following noncommutative differential equation [28,29,30]

$$ dL = (\omega_0 x_0 + \omega_1 x_1)L, \text{ where } \omega_0(z) = dz/z \text{ and } \omega_1(z) = dz/(1-z), \qquad (52) $$

in which the Haussdorf group, i.e. $e^{\mathcal{L}ie_C\langle\langle X\rangle\rangle} = \{e^C\}_{C \in \mathcal{L}ie_C\langle\langle X\rangle\rangle}$, plays the rôle of the differential Galois group of the grouplike solutions of (52) and any grouplike series in the form $Le^C$, where $C \in \mathcal{L}ie_C\langle\langle X\rangle\rangle$, satisfies (52) [28,29,30]. A well known subgroup of this Haussdorf group is the monodromy group, at 0 and 1, of L [27]:

$$ \mathcal{M}_0 L = Le^{2i\pi x_0} \text{ and } \mathcal{M}_1 L = L(Z_{\boxplus} e^{-2i\pi x_1} Z_{\boxplus}^{-1}). \qquad (53) $$

More generally, since $\mathbb{Q} \subset A \subset \mathbb{C}$ then one also has [28,30]$^{14}$

$$ dm(A) := Z_{\boxplus} e^C |_{C \in \mathcal{L}ie_A\langle\langle X\rangle\rangle}, \langle e^C | x_0 \rangle = \langle e^C | x_1 \rangle = 0 \rangle \subsetneq e^{\mathcal{L}ie_C\langle\langle X\rangle\rangle}. \qquad (54) $$

**Remark 8.** In [15], Drinfel'd stated that the noncommutative differential equation (52) admits a unique solution, $G_0$ (resp. $G_1$), satisfying the following asymptotic condition

$$ G_0(z) \sim_0 e^{x_0 \log(z)} = s^{x_0} \text{ (resp. } G_1(z) \sim_1 e^{-x_1 \log(1-z)} = (1-z)^{-x_1} \text{)} $$

and there is unique grouplike series $\Phi_{KZ} \in \mathbb{R}\langle\langle X\rangle\rangle$, called Drinfel'd series or Drinfel'd associator [12], such that $G_0 = G_1\Phi_{KZ}$ but neither have been constructed such expressions of $\Phi_{KZ}$ nor have been made explicit $G_0$ or $G_1$.

One can recognize that $Z_{\boxplus}$, operated in (50), is nothing other than the series $\Phi_{KZ}$.

The behaviors in (50)–(51) yield the following renormalizations.

<footnote>$^{14}$Note that $dm(A)$ contains $DM(A)$ introduced by Cartier in [12].</footnote>
**Theorem 3 (Abel like theorem, [17,18]).** The following limits exist (see also (3))

$$ \lim_{z \to 1} e^{y_1 \log(1-z)} \pi_Y(L(z)) = \lim_{n \to \infty} e^{\sum_{k \ge 1} H_{y_k}(n) (-y_1)^k / k} H(n) = \pi_Y(Z_{\perp}). $$

**Corollary 2 ([28,29]).** $Z_\gamma = B(y_1)\pi_Y(Z_{\perp})$, where $B(y_1) := e^{\gamma y_1 - \sum_{k \ge 2} \frac{\zeta(k)(-y_1)^k}{k}}$. By cancellation, it is equivalent to $Z_{\perp} = B'(y_1)\pi_Y(Z_{\perp})$, where$^{15}$ $B'(y_1) := e^{-\sum_{k \ge 2} \frac{\zeta(k)(-y_1)^k}{k}}$.

**Example 3 ([17,18]).** Extracting coefficients on $L(z) \sim_1 e^{-x_1 \log(1-z)} Z_{\perp}$ and then transferring, one has

$$ \text{Li}_{2,1}(z) = \zeta(3) + (1-z)\log(1-z) - (1-z)^{-1} - (1-z)\log^2(1-z)/2 + (1-z)^2(-\log^2(1-z) + \log(1-z))/4 + \dots, $$

$$ \text{Li}_{1,2}(z) = 2 - 2\zeta(3) - \zeta(2)\log(1-z) - 2(1-z)\log(1-z) + (1-z)\log^2(1-z) + (1-z)^2((\log^2(1-z) - \log(1-z))/2 + \dots), $$

$$ \text{H}_{2,1}(n) = \zeta(3) - (\log(n) + 1 + \gamma)/n + \log(n)/2n + \dots, $$

$$ \text{H}_{1,2}(n) = \zeta(2)\gamma - 2\zeta(3) + \zeta(2)\log(n) + (\zeta(2) + 2)/2n + \dots. $$

Hence, in each convergent and divergent$^{16}$ cases,

$$ \text{f.p. } H_{2,1}(n) = \text{f.p. Li}_{2,1}(z) = \zeta(2,1) = \zeta(3), \qquad (55) $$

$$ \text{f.p. } H_{1,2}(n) = \zeta(2)\gamma - 2\zeta(3) \neq \text{f.p. Li}_{1,2}(z) = 2 - 2\zeta(3). \qquad (56) $$

Identifying $\{\langle Z_\gamma \rangle_1^k w \rangle\}_{\substack{k \in \mathbb{N}_{>1} \\ w \in \mathbb{C}\mathbb{O}\mathbb{N}}}^-$ in $Z_\gamma = B(y_1)\pi_Y(Z_{\perp})$ (see Corollary 2) and using the Bell polynomials $\{b_{n,k}(t_1, \dots, t_k)\}_{n,k \in \mathbb{N}_{+}}$, the constants associated to $\{H_{y_1^k w}\}_{\substack{k \in \mathbb{N}_{>1} \\ w \in \mathbb{C}\mathbb{O}\mathbb{N}}}^-$ (depending on $\gamma$) are expressed as follows [17,18]

$$ \gamma_{y_1^k w} = \sum_{i=0}^{k} \frac{\zeta(x_0(-x_1)^{k-i} \pi_X w)}{i!} \left( \sum_{j=1}^{i} b_{i,j}(\gamma, -\zeta(2), 2\zeta(3), \dots) \right), $$

$$ \gamma_{y_1^k} = \sum_{\substack{s_1, \dots, s_k > 0 \\ s_1 + \dots + k s_k = k}} \frac{(-1)^k}{s_1! \dots s_k!} (-\gamma)^{s_1} \left(-\frac{\zeta(2)}{2}\right)^{s_2} \dots \left(-\frac{\zeta(k)}{k}\right)^{s_k}. $$

**Example 4 (Generalized Euler's gamma constant, [17,18]).**

$$ \gamma_{1,1} = \frac{1}{2}(\gamma^2 - \zeta(2)), $$

$$ \gamma_{1,1,1} = \frac{1}{6}(\gamma^3 - 3\zeta(2)\gamma + 2\zeta(3)), $$

$$ \gamma_{1,7} = \zeta(7)\gamma + \zeta(3)\zeta(5) - \frac{54}{175}\zeta(2)^4, $$

$$ \gamma_{1,1,6} = \frac{4}{35}\zeta(2)^3\gamma^2 + (\zeta(2)\zeta(5) + \frac{2}{5}\zeta(3)\zeta(2)^2 - 4\zeta(7))\gamma + \frac{19}{35}\zeta(2)^4 + \frac{1}{2}\zeta(2)\zeta(3)^2 + \zeta(6,2) - 4\zeta(3)\zeta(5). $$

<footnote>$^{15}$B'(y₁) corresponds to the Ecalle's M∩n∩o mould [19].</footnote>

<footnote>$^{16}$with $\zeta(2)\gamma = 0.9494817111149815245\dots$.</footnote>
But identifying the local coordinates, in $Z_γ = B(y_1)π_γ(Z_{⊥⊥})$ and $Z_{⊥⊥} = B(x_1)^{-1}π_X Z_γ$, one gets the algebraic relations, homogeneous in weight and independent on $γ$, among $\{\zeta(S_l)\}_{l \in \mathcal{L}_{ynX} \setminus X}$ and among $\{\zeta(\Sigma_l)\}_{l \in \mathcal{L}_{ynY} \setminus \{y_1\}}$ [8] (see also Remarks 3 and 5).

Example 5 (Polynomial relations on local coordinates, [8]).

<table><thead><tr><th></th><th>Relations on {ζ(Σ<sub>l</sub>)<sub>l∈L<sub>ynX</sub>\{y<sub>1</sub></sub></th><th>Relations on {ζ(S<sub>l</sub>)<sub>l∈L<sub>ynX</sub>\X</sub></th></tr></thead><tbody><tr><td>3</td><td>ζ(Σ<sub>y<sub>2</sub>y<sub>1</sub></sub>) = <sup>3</sup>&frasl;<sub>2</sub>ζ(Σ<sub>y<sub>3</sub></sub>)</td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) = ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)</td></tr><tr><td rowspan="3">4</td><td>ζ(Σ<sub>y<sub>4</sub></sub>) = <sup>2</sup>&frasl;<sub>5</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) = <sup>2</sup>&frasl;<sub>5</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>3</sub>y<sub>1</sub></sub>) = <sup>3</sup>&frasl;<sub>10</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) = <sup>1</sup>&frasl;<sub>10</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>2</sub>y<sub>1</sub></sub>) = <sup>2</sup>&frasl;<sub>3</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) = <sup>2</sup>&frasl;<sub>5</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td rowspan="5">5</td><td>ζ(Σ<sub>y<sub>3</sub>y<sub>2</sub></sub>) = 3ζ(Σ<sub>y<sub>3</sub></sub>)ζ(Σ<sub>y<sub>2</sub></sub>) - 5ζ(Σ<sub>y<sub>5</sub></sub>)</td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) = -ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) + 2ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)</td></tr><tr><td>ζ(Σ<sub>y<sub>4</sub>y<sub>1</sub></sub>) = -ζ(Σ<sub>y<sub>3</sub></sub>)ζ(Σ<sub>y<sub>2</sub></sub>) + <sup>5</sup>&frasl;<sub>2</sub>ζ(Σ<sub>y<sub>5</sub></sub>)</td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) = -<sup>3</sup>&frasl;<sub>2</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) + ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)</td></tr><tr><td>ζ(Σ<sub>y<sub>2</sub>y<sub>1</sub></sub>) = <sup>3</sup>&frasl;<sub>2</sub>ζ(Σ<sub>y<sub>3</sub></sub>)ζ(Σ<sub>y<sub>2</sub></sub>) - <sup>25</sup>&frasl;<sub>12</sub>ζ(Σ<sub>y<sub>5</sub></sub>)</td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) = -ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) + 2ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)</td></tr><tr><td>ζ(Σ<sub>y<sub>3</sub>y<sub>1</sub></sub>) = <sup>5</sup>&frasl;<sub>12</sub>ζ(Σ<sub>y<sub>5</sub></sub>)</td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) = <sup>1</sup>&frasl;<sub>2</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)</td></tr><tr><td>ζ(Σ<sub>y<sub>2</sub>y<sub>1</sub></sub>) = <sup>1</sup>&frasl;<sub>4</sub>ζ(Σ<sub>y<sub>3</sub></sub>)ζ(Σ<sub>y<sub>2</sub></sub>) + <sup>5</sup>&frasl;<sub>4</sub>ζ(Σ<sub>y<sub>5</sub></sub>)</td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) = ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)</td></tr><tr><td rowspan="10">6</td><td>ζ(Σ<sub>y<sub>6</sub></sub>) = <sup>8</sup>&frasl;<sub>35</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) = <sup>8</sup>&frasl;<sub>35</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>4</sub>y<sub>2</sub></sub>) = ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup> - <sup>4</sup>&frasl;<sub>21</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) = <sup>6</sup>&frasl;<sub>35</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup> - <sup>1</sup>&frasl;<sub>2</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>5</sub>y<sub>1</sub></sub>) = <sup>2</sup>&frasl;<sub>7</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup> - <sup>1</sup>&frasl;<sub>2</sub>ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) = <sup>4</sup>&frasl;<sub>105</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>3</sub>y<sub>1</sub>y<sub>2</sub></sub>) = -<sup>17</sup>&frasl;<sub>30</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup> + <sup>9</sup>&frasl;<sub>4</sub>ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) = <sup>23</sup>&frasl;<sub>70</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup> - ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>3</sub>y<sub>2</sub>y<sub>1</sub></sub>) = 3ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup> - <sup>9</sup>&frasl;<sub>10</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) = <sup>2</sup>&frasl;<sub>105</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>4</sub>y<sub>1</sub></sub>) = <sup>3</sup>&frasl;<sub>10</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup> - <sup>3</sup>&frasl;<sub>4</sub>ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) = -<sup>89</sup>&frasl;<sub>210</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup> + <sup>3</sup>&frasl;<sub>2</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>2</sub>y<sub>1</sub></sub>) = <sup>11</sup>&frasl;<sub>63</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup> - <sup>1</sup>&frasl;<sub>4</sub>ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) = <sup>6</sup>&frasl;<sub>35</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup> - <sup>1</sup>&frasl;<sub>2</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>3</sub>y<sub>1</sub></sub>) = <sup>1</sup>&frasl;<sub>21</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) = <sup>8</sup>&frasl;<sub>21</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup> - ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>2</sub>y<sub>1</sub></sub>) = <sup>17</sup>&frasl;<sub>50</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup> + <sup>3</sup>&frasl;<sub>16</sub>ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) = <sup>8</sup>&frasl;<sub>35</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup></td></tr></tbody></table>

Now, for$^{17}$ $\bar{Z}_{⊥⊥} := Z_{⊥⊥}e^C \in dm(A)$, with $e^C \in \mathcal{L}ie_A(\langle X \rangle)$, let $\bar{L} := Le^C$ and then

$$ \bar{H}(n) := \sum_{w \in Y^*} \bar{H}_w(n)w, \text{ where } \bar{H}_w(n) = \langle \frac{\langle \bar{L}(z) | \pi_X(w) \rangle}{1-z} | z^n \rangle. \quad (57) $$

As previously, $\bar{L}$ satisfies (52) and, as in (50)–(51), one also has [28,29,30]

$$ \bar{L}(z) \sim_0 e^{x_0 \log z} e^C \text{ and } \bar{L}(z) \sim_1 e^{-x_1 \log(1-z)} \bar{Z}_{⊥⊥}, \quad (58) $$

$$ \bar{H}(n) \sim_{+\infty} e^{-\sum_{k \ge 1} \frac{\mathrm{H}_{y_k}(n)(-y_1)^k}{k}} \pi_Y(\bar{Z}_{⊥⊥}). \quad (59) $$

These yield similar results as in Theorem 3 and in Corollary 2 [28,29,30]:

$$ \lim_{z \to 1} e^{y_1 \log(1-z)} \pi_Y(\bar{L}(z)) = \pi_Y(\bar{Z}_{⊥⊥}) = \lim_{n \to \infty} e^{\sum_{k \ge 1} \frac{\mathrm{H}_{y_k}(n)(-y_1)^k}{k}} \bar{H}(n), \quad (60) $$

$$ \bar{Z}_{\gamma} = B(y_1)\pi_Y(\bar{Z}_{\perp}) \iff \bar{Z}_{\perp} = B'(y_1)\pi_Y(\bar{Z}_{\perp}). \quad (61) $$

It follows that $\bar{L}$ is the unique solution of (52) satisfying the asymptotic conditions in (50), i.e. with $e^C = 1_{X^*}$, and then $Z_{⊥⊥}$ is unique [30].

<footnote>$^{17}$Note that $\bar{Z}_{⊥⊥}$ is grouplike series, for $⊥⊥$, and it satisfies $\langle \bar{Z}_{⊥⊥} | x_0 \rangle = \langle \bar{Z}_{⊥⊥} | x_1 \rangle = 0$).</footnote>
<table><caption>Example 7 (Rewitting among local coordinates<fnref n="19" /> and irreducible terms, [8]).</caption><thead><tr><th></th><th>Rewriting on {ζ(Σ<sub>l</sub>)<sub>l∈LynX\{y<sub>1</sub></sub>}</th><th>Rewriting on {ζ(S<sub>l</sub>)<sub>l∈LynX\X</sub></th></tr></thead><tbody><tr><td>3</td><td>ζ(Σ<sub>y<sub>2</sub>y<sub>1</sub></sub>) → <sup>3</sup>&frasl;<sub>2</sub>ζ(Σ<sub>y<sub>3</sub></sub>)</td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) → ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)</td></tr><tr><td rowspan="3">4</td><td>ζ(Σ<sub>y<sub>4</sub></sub>) → <sup>2</sup>&frasl;<sub>5</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) → <sup>2</sup>&frasl;<sub>5</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>3</sub>y<sub>1</sub></sub>) → <sup>3</sup>&frasl;<sub>10</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) → <sup>1</sup>&frasl;<sub>10</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>2</sub>y<sub>1</sub></sub>) → <sup>2</sup>&frasl;<sub>3</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) → <sup>2</sup>&frasl;<sub>5</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td rowspan="5">5</td><td>ζ(Σ<sub>y<sub>3</sub>y<sub>2</sub></sub>) → 3ζ(Σ<sub>y<sub>3</sub></sub>)ζ(Σ<sub>y<sub>2</sub></sub>) - 5ζ(Σ<sub>y<sub>5</sub></sub>)</td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) → -ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) + 2ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)</td></tr><tr><td>ζ(Σ<sub>y<sub>4</sub>y<sub>1</sub></sub>) → -ζ(Σ<sub>y<sub>3</sub></sub>)ζ(Σ<sub>y<sub>2</sub></sub>) + <sup>5</sup>&frasl;<sub>2</sub>ζ(Σ<sub>y<sub>5</sub></sub>)</td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) → -<sup>3</sup>&frasl;<sub>2</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) + ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)</td></tr><tr><td>ζ(Σ<sub>y<sub>2</sub>y<sub>1</sub></sub>) → <sup>3</sup>&frasl;<sub>2</sub>ζ(Σ<sub>y<sub>3</sub></sub>)ζ(Σ<sub>y<sub>2</sub></sub>) - <sup>25</sup>&frasl;<sub>12</sub>ζ(Σ<sub>y<sub>5</sub></sub>)</td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) → -ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) + 2ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)</td></tr><tr><td>ζ(Σ<sub>y<sub>3</sub>y<sub>1</sub></sub>) → <sup>5</sup>&frasl;<sub>12</sub>ζ(Σ<sub>y<sub>5</sub></sub>)</td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) → <sup>1</sup>&frasl;<sub>2</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)</td></tr><tr><td>ζ(Σ<sub>y<sub>2</sub>y<sub>1</sub></sub>) → <sup>1</sup>&frasl;<sub>4</sub>ζ(Σ<sub>y<sub>3</sub></sub>)ζ(Σ<sub>y<sub>2</sub></sub>) + <sup>5</sup>&frasl;<sub>4</sub>ζ(Σ<sub>y<sub>5</sub></sub>)</td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) → ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)</td></tr><tr><td rowspan="10">6</td><td>ζ(Σ<sub>y<sub>6</sub></sub>) → <sup>8</sup>&frasl;<sub>35</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) → <sup>8</sup>&frasl;<sub>35</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>4</sub>y<sub>2</sub></sub>) → ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup> - <sup>4</sup>&frasl;<sub>21</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>) → <sup>6</sup>&frasl;<sub>35</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup> - <sup>1</sup>&frasl;<sub>2</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>5</sub>y<sub>1</sub></sub>) → <sup>2</sup>&frasl;<sub>7</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup> - <sup>1</sup>&frasl;<sub>2</sub>ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) → <sup>4</sup>&frasl;<sub>105</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>3</sub>y<sub>1</sub>y<sub>2</sub></sub>) → -<sup>17</sup>&frasl;<sub>30</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup> + <sup>9</sup>&frasl;<sub>4</sub>ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) → <sup>23</sup>&frasl;<sub>70</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup> - ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>3</sub>y<sub>2</sub>y<sub>1</sub></sub>) → 3ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup> - <sup>9</sup>&frasl;<sub>10</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub></sub>) → <sup>2</sup>&frasl;<sub>105</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>4</sub>y<sub>1</sub><sup>2</sup></sub>) → <sup>3</sup>&frasl;<sub>10</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup> - <sup>3</sup>&frasl;<sub>4</sub>ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub><sup>2</sup>x<sub>0</sub>x<sub>1</sub></sub>) → -<sup>89</sup>&frasl;<sub>210</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup> + <sup>3</sup>&frasl;<sub>2</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>2</sub>y<sub>1</sub><sup>2</sup></sub>) → <sup>11</sup>&frasl;<sub>63</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup> - <sup>3</sup>&frasl;<sub>4</sub>ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub><sup>4</sup></sub>) → <sup>6</sup>&frasl;<sub>35</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup> - <sup>1</sup>&frasl;<sub>2</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>3</sub>y<sub>1</sub><sup>3</sup></sub>) → <sup>1</sup>&frasl;<sub>21</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub>x<sub>0</sub>x<sub>1</sub><sup>3</sup></sub>) → <sup>8</sup>&frasl;<sub>21</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup> - ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>2</sup></td></tr><tr><td>ζ(Σ<sub>y<sub>2</sub>y<sub>1</sub><sup>4</sup></sub>) → <sup>17</sup>&frasl;<sub>50</sub>ζ(Σ<sub>y<sub>2</sub></sub>)<sup>3</sup> + <sup>3</sup>&frasl;<sub>16</sub>ζ(Σ<sub>y<sub>3</sub></sub>)<sup>2</sup></td><td>ζ(S<sub>x<sub>0</sub>x<sub>1</sub><sup>5</sup></sub>) → <sup>8</sup>&frasl;<sub>35</sub>ζ(S<sub>x<sub>0</sub>x<sub>1</sub></sub>)<sup>3</sup></td></tr></tbody></table>

$$ \mathcal{F}_{irr}^{X, \le 12} = \{ \zeta(S_{x_0 x_1}), \zeta(S_{x_0^2 x_1}), \zeta(S_{x_0^4 x_1}), \zeta(S_{x_0^6 x_1}), \zeta(S_{x_0^2 x_1 x_0 x_1^4}), \zeta(S_{x_0^8 x_1}), \\ \zeta(S_{x_0 x_1^2 x_0 x_1^6}), \zeta(S_{x_0^{10} x_1}), \zeta(S_{x_0 x_1^2 x_0 x_1^7}), \zeta(S_{x_0 x_1^2 x_0 x_1^8}), \zeta(S_{x_0 x_1^4 x_0 x_1^6}) \}. $$

$$ \mathcal{F}_{irr}^{Y, \le 12} = \{ \zeta(\Sigma_{y_2}), \zeta(\Sigma_{y_3}), \zeta(\Sigma_{y_5}), \zeta(\Sigma_{y_7}), \zeta(\Sigma_{y_3 y_1}^5), \zeta(\Sigma_{y_9}), \zeta(\Sigma_{y_3 y_1}^7), \\ \zeta(\Sigma_{y_2 y_1}^9), \zeta(\Sigma_{y_3 y_1}^9), \zeta(\Sigma_{y_2 y_1}^8) \}. $$

Now, let us describe the algorithm **LocalCoordinateIdentification** (partially implemented in [8] and briefly described in [11,31]), applying Theorem 3 and Corollary 2, bringing additional informations and results to [8]. It uses the notations concerning

1. the shuffle and quasi-shuffle subalgebras, recalled in Section 2, i.e.

$$ (\mathbb{Q}1_{X^*} \oplus x_0 \mathbb{Q}\langle X\rangle x_1, \uparrow, 1_{X^*}) \cong (\mathbb{Q}[\{S_l\}_{l \in \mathcal{L}ynX\setminus X}], \uparrow, 1_{X^*}), \quad (65) $$

$$ (\mathbb{Q}1_{Y^*} \oplus (Y \setminus \{y_1\})\mathbb{Q}\langle Y\rangle, \uparrow, 1_{Y^*}) \cong (\mathbb{Q}[\{\Sigma_l\}_{l \in \mathcal{L}ynY\setminus\{y_1\}}], \uparrow, 1_{Y^*}), \quad (66) $$

2. the algebras of polyzas, recalled in Section 3, i.e.

$$ (\mathbb{Q}[\{\zeta(S_l)\}_{l \in \mathcal{L}ynX\setminus X}], \times, 1) \text{ and } (\mathbb{Q}[\{\zeta(\Sigma_l)\}_{l \in \mathcal{L}ynY\setminus\{y_1\}}], \times, 1), \quad (67) $$

3. the rewriting systems, introduced in (69)–(71) below, i.e.

$$ (\mathbb{Q}1_{X^*} \oplus x_0 \mathbb{Q}\langle X\rangle x_1, \mathcal{R}_{irr}^X) \text{ and } (\mathbb{Q}1_{Y^*} \oplus (Y \setminus \{y_1\})\mathbb{Q}\langle Y\rangle, \mathcal{R}_{irr}^Y), \quad (68) $$

which are being without critical pairs, noetherian, confluent.

<footnote n="19">¹⁹This table is “handly” obtained from the one of Example 5 in [8].</footnote>
This algorithm comprises an initialization phase followed by two nested loops. The first loop focuses on the weight of Lyndon words, and the second on the ordered set of Lyndon words at the same weight. The last yields equations on polyzetas, as in [8], and then transforms them to corresponding rewriting rules or, exclusively, irreducible terms. This algorithm is illustrated by Examples 5–8, provides (68) and precisely contains

1. the above increasing set {$\mathcal{L}_{irr}^{\mathcal{X}, \le p}$}$_{p \ge 2}$ and its image, {$\mathcal{L}_{irr}^{\mathcal{X}, \le p}$}$_{p \ge 2}$, by the restriction of the $\zeta$ polymorphism, described in (62) which is bijective,

2. the following set of homogeneous in weight polynomials being image by a section of $\zeta$ from {$\zeta(Q_l) = 0$}$_{l \in \mathcal{L}_{\text{yn},\mathcal{X}} \setminus \text{gDIV}}$:

$$ \mathcal{D}^{\mathcal{X}} = \{Q_l\}_{l \in \mathcal{L}_{\text{yn},\mathcal{X}} \setminus \text{gDIV}}, \quad (69) $$

3. the shuffle or quasi-shuffle ideal $\mathcal{R}^{\mathcal{X}}$ inside ker $\zeta$, generated by $\mathcal{D}^{\mathcal{X}}$ as follows

$$ \mathcal{R}_{\mathcal{X}} := (\operatorname{span}_{\mathcal{Q}} \mathcal{D}_{\mathcal{X}, \perp}, 1_{X^*}) \subseteq \operatorname{ker} \zeta \text{ and } \mathcal{R}_{\mathcal{Y}} := (\operatorname{span}_{\mathcal{Q}} \mathcal{D}_{\mathcal{Y}, \perp}, 1_{Y^*}) \subseteq \operatorname{ker} \zeta \quad (70) $$

4. for any $l \in \mathcal{L}_{\text{yn},\mathcal{X}}^{p}$ : $\{l \in \mathcal{L}_{\text{yn},\mathcal{X}}|(l) = p \ge 2\}$, any nonzero homogeneous in weight polynomial $Q_l = \Sigma_l - \Upsilon_l$ (resp. $Q_l = S_l - U_l$), of $\mathcal{D}^{\mathcal{X}}$, is led by $\Sigma_l$ (resp. $S_l$), which does not belong to $\mathcal{Q}[\mathcal{L}_{irr}^{\mathcal{X}, \le p}]$, and $\Upsilon_l$ (resp. $U_l$) is canonically represented in $\mathcal{Q}[\mathcal{L}_{irr}^{\mathcal{X}, \le p}]$. Then let $\Sigma_l \to \Upsilon_l$ and $S_l \to U_l$ be the rewriting rules, respectively, of

$$ \mathcal{R}_{irr}^{Y} := \{\Sigma_l \to \Upsilon_l\}_{l \in \mathcal{L}_{\text{yn},Y \setminus \{y_1\}}} \text{ and } \mathcal{R}_{irr}^{X} := \{S_l \to U_l\}_{l \in \mathcal{L}_{\text{yn},X \setminus X}}. \quad (71) $$

On the other hand, denoting $\mathcal{L}_{irr}^{\mathcal{X}, \le p} := \mathcal{L}_{irr}^{\mathcal{X}, 2} \cup \dots \cup \mathcal{L}_{irr}^{\mathcal{X}, p}$, the following assertions are equivalent (see Example 8 below)

(a) $Q_l = 0$

(b) $\Sigma_l \in \mathcal{L}_{irr}^{Y, \le p}$ (resp. $S_l \in \mathcal{L}_{irr}^{X, \le p}$),

(c) $\Sigma_l \to \Sigma_l$ (resp. $S_l \to S_l$).

### LocalCoordinateIdentification

$$ \mathcal{D}_{irr}^{Y, \infty} := \{\}; \mathcal{L}_{irr}^{Y, \infty} := \{\}; \mathcal{R}_{irr}^{Y} := \{\}; \mathcal{D}_{Y} := \{\}; $$

$$ \mathcal{D}_{irr}^{X, \infty} := \{\}; \mathcal{L}_{irr}^{X, \infty} := \{\}; \mathcal{R}_{irr}^{X} := \{\}; \mathcal{D}_{X} := \{\}; $$

for $p$ ranges in $2, \dots, \infty$ do

for $l$ ranges in the totally ordered $\mathcal{L}_{\text{yn},p}Y$ do

$$ \lambda := \pi_X(l); $$

identify $\langle Z_{\gamma} | \Pi_{l} \rangle$ in $Z_{\gamma} = B(y_{1})\pi_{Y}Z_{\perp}$ and $\langle Z_{\perp} | P_{\lambda} \rangle$ in $Z_{\perp} = B(x_{1})^{-1}\pi_{X}Z_{\gamma}$;

by elimination, obtain equations on $\{\zeta(\Sigma_l)\}_{l' \in \mathcal{L}_{\text{yn},p}Y}$ and on $\{\zeta(S_{\lambda'})\}_{\lambda' \in \mathcal{L}_{\text{yn},p}X}$; $\lambda' \le \lambda$

express these equations, led by $\zeta(\Sigma_l)$ and by $\zeta(S_\lambda)$, as rewriting rules;

if $\zeta(\Sigma_l) \to \zeta(\Sigma_l)$ then $\mathcal{D}_{irr}^{Y, \infty} := \mathcal{D}_{irr}^{Y, \infty} \cup \{\zeta(\Sigma_l)\}$ and $\mathcal{L}_{irr}^{Y, \infty} := \mathcal{L}_{irr}^{Y, \infty} \cup \{\Sigma_l\}$

else $\mathcal{R}_{irr}^{Y} := \mathcal{R}_{irr}^{Y} \cup \{\Sigma_l \to \Upsilon_l\}$ and $\mathcal{D}_{Y} := \mathcal{D}_{Y} \cup \{\Sigma_l - \Upsilon_l\}$;

if $\zeta(S_\lambda) \to \zeta(S_\lambda)$ then $\mathcal{D}_{irr}^{X, \infty} := \mathcal{D}_{irr}^{X, \infty} \cup \{\zeta(S_\lambda)\}$ and $\mathcal{L}_{irr}^{X, \infty} := \mathcal{L}_{irr}^{X, \infty} \cup \{S_\lambda\}$

else $\mathcal{R}_{irr}^{X} := \mathcal{R}_{irr}^{X} \cup \{S_\lambda \to U_\lambda\}$ and $\mathcal{D}_{X} := \mathcal{D}_{X} \cup \{S_\lambda - U_\lambda\}$

end_for

end_for

<footnote>a This step and the following ones are not yet been achieved by the implementation in [8].</footnote>
In the other words, the orderings over $\mathcal{L}ynY$ and $\mathcal{L}ynX$ induce the orderings over $\{S_\lambda\}_{\lambda \in \mathcal{L}ynX \setminus X}$ and $\{\Sigma_l\}_{l \in \mathcal{L}ynY \setminus \{y_1\}}$ (see Remarks 2 and 4), over $\mathcal{L}_{irr}^{\mathcal{X}, \infty}$ and $\mathcal{R}_{\mathcal{X}}$, and then over $\mathcal{R}_{irr}^{\mathcal{X}}$.

Moreover,

1. each irreducible term, in $\mathcal{L}_{irr}^{\mathcal{X}, \infty}$ (resp. $\mathcal{L}_{irr}^{\mathcal{Y}, \infty}$), is an element of the algebraic basis $\{S_l\}_{l \in \mathcal{L}ynX \setminus X}$ (resp. $\{\Sigma_l\}_{l \in \mathcal{L}ynY \setminus \{y_1\}}$) of the subalgebra in (65) (resp. (66)),

2. each rewriting rule, in $\mathcal{R}_{irr}^{\mathcal{X}}$, admits the left side doing not belong to $\mathcal{Q}[\mathcal{L}_{irr}^{\mathcal{X}, \infty}]$ and the right side being canonically represented in $\mathcal{Q}[\mathcal{L}_{irr}^{\mathcal{X}, \infty}]$. The difference of these two sides belongs to the totally ordered ideal $\mathcal{R}_{\mathcal{X}}$ of $\mathcal{Q}[\mathcal{L}yn\mathcal{X}] \setminus \text{gDIV}$.

**Example 8** (Rewriting among $\{\Sigma_l\}_{l \in \mathcal{L}ynX \setminus \{y_1\}}$ and $\{S_l\}_{l \in \mathcal{L}ynX \setminus X}$, irreducible terms$^{20}$ [11,31]).

<table><thead><tr><th></th><th>Rewriting on {S<sub>l</sub>}<sub>l∈LynX\{y<sub>1</sub></sub></th><th>Rewriting on {S<sub>l</sub>}<sub>l∈LynX\X</sub></th></tr></thead><tbody><tr><td>3</td><td>Σ<sub>y<sub>2</sub>y<sub>1</sub></sub> → <sup>3</sup>&frasl;<sub>2</sub>Σ<sub>y<sub>3</sub></sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub> → S<sub>x<sub>0</sub>x<sub>1</sub></sub></td></tr><tr><td rowspan="3">4</td><td>Σ<sub>y<sub>4</sub></sub> → <sup>2</sup>&frasl;<sub>5</sub>Σ<sub>y<sub>2</sub></sub><sup>±</sup><sub>2</sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>3</sup> → <sup>2</sup>&frasl;<sub>5</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>2</sub></td></tr><tr><td>Σ<sub>y<sub>3</sub>y<sub>1</sub></sub> → <sup>3</sup>&frasl;<sub>10</sub>Σ<sub>y<sub>2</sub></sub><sup>±</sup><sub>2</sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>2</sup> → <sup>1</sup>&frasl;<sub>10</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>2</sub></td></tr><tr><td>Σ<sub>y<sub>2</sub>y<sub>1</sub></sub><sup>2</sup> → <sup>2</sup>&frasl;<sub>3</sub>Σ<sub>y<sub>2</sub></sub><sup>±</sup><sub>2</sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>3</sup> → <sup>2</sup>&frasl;<sub>5</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>2</sub></td></tr><tr><td rowspan="5">5</td><td>Σ<sub>y<sub>3</sub>y<sub>2</sub></sub> → 3Σ<sub>y<sub>3</sub></sub><sup>±</sup><sub>2</sub>Σ<sub>y<sub>2</sub></sub> - 5Σ<sub>y<sub>5</sub></sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>3</sup> → -S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>2</sub> S<sub>x<sub>0</sub>x<sub>1</sub></sub> + 2S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>4</sup></td></tr><tr><td>Σ<sub>y<sub>4</sub>y<sub>1</sub></sub> → -Σ<sub>y<sub>3</sub></sub><sup>±</sup><sub>2</sub>Σ<sub>y<sub>2</sub></sub> + <sup>5</sup>&frasl;<sub>2</sub>Σ<sub>y<sub>5</sub></sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>2</sup>x<sub>0</sub>x<sub>1</sub> → -<sup>3</sup>&frasl;<sub>2</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>4</sup> + S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>2</sup></td></tr><tr><td>Σ<sub>y<sub>2</sub>y<sub>1</sub></sub> → <sup>3</sup>&frasl;<sub>2</sub>Σ<sub>y<sub>3</sub></sub><sup>±</sup><sub>2</sub>Σ<sub>y<sub>2</sub></sub> - <sup>25</sup>&frasl;<sub>12</sub>Σ<sub>y<sub>5</sub></sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>2</sup> → -S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>2</sub> S<sub>x<sub>0</sub>x<sub>1</sub></sub> + 2S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>4</sup></td></tr><tr><td>Σ<sub>y<sub>3</sub>y<sub>1</sub></sub><sup>2</sup> → <sup>5</sup>&frasl;<sub>12</sub>Σ<sub>y<sub>5</sub></sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>2</sup>x<sub>0</sub>x<sub>1</sub> → <sup>1</sup>&frasl;<sub>2</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>4</sup></td></tr><tr><td>Σ<sub>y<sub>2</sub>y<sub>1</sub></sub><sup>3</sup> → <sup>1</sup>&frasl;<sub>4</sub>Σ<sub>y<sub>3</sub></sub><sup>±</sup><sub>2</sub>Σ<sub>y<sub>2</sub></sub> + <sup>5</sup>&frasl;<sub>4</sub>Σ<sub>y<sub>5</sub></sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>4</sup> → S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>2</sub></td></tr><tr><td rowspan="8">6</td><td>Σ<sub>y<sub>6</sub></sub> → <sup>8</sup>&frasl;<sub>35</sub>Σ<sub>y<sub>2</sub></sub><sup>±</sup><sub>3</sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>5</sup> → <sup>8</sup>&frasl;<sub>35</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>3</sub></td></tr><tr><td>Σ<sub>y<sub>4</sub>y<sub>2</sub></sub> → Σ<sub>y<sub>3</sub></sub><sup>±</sup><sub>2</sub> - <sup>4</sup>&frasl;<sub>21</sub>Σ<sub>y<sub>2</sub></sub><sup>±</sup><sub>3</sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>4</sup> → <sup>6</sup>&frasl;<sub>35</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>3</sub> - <sup>1</sup>&frasl;<sub>2</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>2</sub></td></tr><tr><td>Σ<sub>y<sub>5</sub>y<sub>1</sub></sub> → <sup>2</sup>&frasl;<sub>7</sub>Σ<sub>y<sub>2</sub></sub><sup>±</sup><sub>3</sub> - <sup>1</sup>&frasl;<sub>2</sub>Σ<sub>y<sub>3</sub></sub><sup>±</sup><sub>2</sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>3</sup>x<sub>0</sub>x<sub>1</sub> → <sup>4</sup>&frasl;<sub>105</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>3</sub></td></tr><tr><td>Σ<sub>y<sub>3</sub>y<sub>1</sub>y<sub>2</sub></sub> → -<sup>17</sup>&frasl;<sub>30</sub>Σ<sub>y<sub>2</sub></sub><sup>±</sup><sub>3</sub> + <sup>9</sup>&frasl;<sub>4</sub>Σ<sub>y<sub>3</sub></sub><sup>±</sup><sub>2</sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>3</sup><sub>1</sub> → <sup>23</sup>&frasl;<sub>70</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>3</sub> - S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>2</sub></td></tr><tr><td>Σ<sub>y<sub>3</sub>y<sub>2</sub>y<sub>1</sub></sub> → 3Σ<sub>y<sub>3</sub></sub><sup>±</sup><sub>2</sub> - <sup>9</sup>&frasl;<sub>10</sub>Σ<sub>y<sub>2</sub></sub><sup>±</sup><sub>3</sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>2</sup>x<sub>0</sub>x<sub>1</sub> → <sup>2</sup>&frasl;<sub>105</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>3</sub></td></tr><tr><td>Σ<sub>y<sub>4</sub>y<sub>2</sub><sup>2</sup></sub> → <sup>3</sup>&frasl;<sub>10</sub>Σ<sub>y<sub>2</sub></sub><sup>±</sup><sub>3</sub> - <sup>3</sup>&frasl;<sub>4</sub>Σ<sub>y<sub>3</sub></sub><sup>±</sup><sub>2</sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>2</sup>x<sub>0</sub>x<sub>1</sub> → -<sup>89</sup>&frasl;<sub>210</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>3</sub> + <sup>3</sup>&frasl;<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>2</sub></td></tr><tr><td>Σ<sub>y<sub>2</sub>y<sub>1</sub><sup>2</sup></sub> → <sup>11</sup>&frasl;<sub>63</sub>Σ<sub>y<sub>2</sub></sub><sup>±</sup><sub>3</sub> - <sup>1</sup>&frasl;<sub>4</sub>Σ<sub>y<sub>3</sub></sub><sup>±</sup><sub>2</sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>2</sup>x<sub>0</sub>x<sub>1</sub> → <sup>6</sup>&frasl;<sub>35</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>3</sub> - <sup>1</sup>&frasl;<sub>2</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>2</sub></td></tr><tr><td>Σ<sub>y<sub>3</sub>y<sub>1</sub><sup>3</sup></sub> → <sup>1</sup>&frasl;<sub>21</sub>Σ<sub>y<sub>2</sub></sub><sup>±</sup><sub>3</sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>2</sup>x<sub>0</sub>x<sub>1</sub> → <sup>8</sup>&frasl;<sub>21</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>3</sub> - S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>2</sub></td></tr><tr><td>Σ<sub>y<sub>2</sub>y<sub>1</sub><sup>4</sup></sub> → <sup>17</sup>&frasl;<sub>50</sub>Σ<sub>y<sub>2</sub></sub><sup>±</sup><sub>3</sub> + <sup>3</sup>&frasl;<sub>16</sub>Σ<sub>y<sub>3</sub></sub><sup>±</sup><sub>2</sub></td><td>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>5</sup> → <sup>8</sup>&frasl;<sub>35</sub>S<sub>x<sub>0</sub>x<sub>1</sub></sub><sup>±</sup><sub>3</sub></td></tr></tbody></table>

$$ \mathcal{L}_{irr}^{\mathcal{X}, \leq 12} = \{S_{x_0 x_1}, S_{x_0^2 x_1}, S_{x_0^4 x_1}, S_{x_0^6 x_1}, S_{x_0 x_1^2 x_0 x_1^4}, S_{x_0^8 x_1}, S_{x_0 x_1^2 x_0 x_1^6}, S_{x_0^{10} x_1}, S_{x_0 x_1^2 x_0 x_1^7}, S_{x_0 x_1^2 x_0 x_1^8}, S_{x_0 x_1^4 x_0 x_1^6}\} $$

$$ \mathcal{L}_{irr}^{\mathcal{Y}, \leq 12} = \{\Sigma_{y_2}, \Sigma_{y_3}, \Sigma_{y_5}, \Sigma_{y_7}, \Sigma_{y_3 y_1^5}, \Sigma_{y_9}, \Sigma_{y_3 y_1^7}, \Sigma_{y_{11}}, \Sigma_{y_2 y_1^9}, \Sigma_{y_3 y_1^9}, \Sigma_{y_2 y_1^8}\} $$

With the notations introduced in (63)–(71), let us return to some relevant results already mentioned in [28,29,30], for which the previous algorithm is the harvested fruit (briefly described in [11,31]).

<footnote>$^{20}$There is some typographical errors in [11,31].</footnote>
**Proposition 1.** *One has*

1. $\mathrm{Im} \zeta = (\mathcal{L}, \times, 1) \cong \mathbb{Q}[\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \infty}]$ and $\mathcal{R}_{\mathcal{X}} = \mathrm{ker} \zeta$.

2. $\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \le p}$ is $\mathbb{Q}$-algebraically independent, for $p \ge 2$. So is $\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \infty}$.

3. $\mathbb{Q}[\{S_l\}_{l \in \mathcal{L}_{\mathrm{yn}X \setminus X}}] = \mathcal{R}_X \oplus \mathbb{Q}[\mathcal{L}_{\mathrm{irr}}^{X, \infty}]$ and $\mathbb{Q}[\{\Sigma_l\}_{l \in \mathcal{L}_{\mathrm{yn}Y \setminus \{y_1\}}}] = \mathcal{R}_Y \oplus \mathbb{Q}[\mathcal{L}_{\mathrm{irr}}^{Y, \infty}]$.

*Proof.* 1. By Corollary 1 and the constructions described in (62)–(70) (see also Remark 9), one obtains

$$ \mathrm{Im} \zeta = (\mathcal{L}, \times, 1) \cong \mathbb{Q}[\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \infty}] \text{ and } \mathcal{R}_{\mathcal{X}} \subseteq \mathrm{ker} \zeta \subseteq \mathbb{Q}[\mathcal{L}_{\mathrm{yn}X} \setminus \mathrm{gDIV}]. $$

Now, if $Q \in \mathrm{ker} \zeta$ such that $\langle Q | 1_{\mathcal{X}^*} \rangle = 0$ then there is $Q_1 \in \mathcal{R}_{\mathcal{X}}$ and $Q_2 \in \mathbb{Q}[\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \infty}]$ such that $Q = Q_1 + Q_2$. Thus, decomposing in the algebraic basis $\{S_l\}_{l \in \mathcal{L}_{\mathrm{yn}X} \setminus X}$, or $\{\Sigma_l\}_{l \in \mathcal{L}_{\mathrm{yn}Y} \setminus \{y_1\}}$, and then reducing by the rewriting rules in $\mathcal{R}_{\mathrm{irr}}^{\mathcal{X}}$ and by hypothesis, one obtains $Q \equiv_{\mathcal{R}_{\mathrm{irr}}^{\mathcal{X}}} Q_1 \in \mathcal{R}_{\mathcal{X}}$.

Finally, using the restriction of the $\zeta$ polymorphism, described in (62) which is bijective, it follows then the expected result.

2. Extracted from the algebraic basis $\{S_l\}_{l \in \mathcal{L}_{\mathrm{yn}X}}$, or $\{\Sigma_l\}_{l \in \mathcal{L}_{\mathrm{yn}Y}}$, the family $\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \le p}$ is $\mathbb{Q}$-algebraically independent. By the restriction of the $\zeta$ polymorphism, described in (62) which is bijective, the family $\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \le p}$ is $\mathbb{Q}$-algebraically independent. The proof is similar for $\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \infty}$.

3. Let $w \in x_0 X^* x_1$ (resp. $(Y \setminus \{y_1\}) Y^*$). Then decomposing in the algebraic basis $\{S_l\}_{l \in \mathcal{L}_{\mathrm{yn}X} \setminus X}$ (resp. $\{\Sigma_l\}_{l \in \mathcal{L}_{\mathrm{yn}Y} \setminus \{y_1\}}$) of $(\mathbb{Q}1_{X^*} \oplus x_0 \mathbb{Q}\langle X \rangle x_1, \pi, 1_{X^*})$ (resp. $(\mathbb{Q}1_{Y^*} \oplus (Y \setminus \{y_1\}) \mathbb{Q}\langle Y \rangle, \oplus, 1_{Y^*})$) and then reducing by the rewriting rules in $\mathcal{R}_{\mathrm{irr}}^{\mathcal{X}}$, described in (71), and using the restriction of the $\zeta$ polymorphism, described in (62) which is bijective, one has $\zeta(w) \in \mathbb{Q}[\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \infty}]$. Hence, let $P \in \mathbb{Q}[\{S_l\}_{l \in \mathcal{L}_{\mathrm{yn}X} \setminus X}]$, or $\mathbb{Q}[\{\Sigma_l\}_{l \in \mathcal{L}_{\mathrm{yn}Y} \setminus \{y_1\}}]$, such that $P \notin \mathrm{ker} \zeta \supseteq \mathcal{R}_{\mathcal{X}}$. Then, by linearity, $\zeta(P) \in \mathbb{Q}[\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \infty}]$. On the other hand, if $Q \in \mathcal{R}_{\mathcal{X}} \cap \mathbb{Q}[\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \infty}]$ then, by (70), $\zeta(Q) = 0$ and then, by (62) again, one obtains $Q = 0$ leading to the expected result.

**Theorem 4.** *The $\mathbb{Q}$-algebra $(\mathcal{L}, \times, 1)$ is freely generated by $\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \infty}$ and, as $\mathbb{Q}$-module,*

$$ \mathcal{L} = \mathbb{Q}1 \oplus \bigoplus_{k \ge 2} \mathcal{L}_k. $$

*Proof.* By Proposition 1, $\mathbb{Q}$-algebra $\mathcal{L}$ is freely generated by $\mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \infty}$ and $\mathrm{ker} \zeta$, being generated by the homogeneous in weight polynomials $\{Q_l\}_{l \in \mathcal{L}_{\mathrm{yn}X} \setminus \mathrm{gDIV}}$, is graded.

With the notations in Conjecture 1, the $\mathbb{Q}$-algebra $\mathcal{L}$ is also graded, as being $\mathrm{Im} \zeta$ which is isomorphic to $\mathbb{Q}1_{Y^*} \oplus (Y \setminus \{y_1\}) \mathbb{Q}\langle Y \rangle / \mathrm{ker} \zeta$ and to $\mathbb{Q}1_{X^*} \oplus x_0 \mathbb{Q}\langle X \rangle x_1 / \mathrm{ker} \zeta$.

**Corollary 3.** *One has*

1. Let $P \in \mathbb{Q}\langle \mathcal{X} \rangle$ and $P \notin \mathrm{ker} \zeta$, homogeneous in weight, or $P \in \mathrm{CONV}$. Then $\zeta(P)$ is a transcendental number.

2. Let $P \in \mathcal{L}_{\mathrm{irr}}^{\mathcal{X}, \infty}$. Then $\zeta(P)$ is a transcendental number.

3. $\zeta(2q+1)/\pi^{2p} \notin \mathbb{Q}$ and $\zeta(2p+1)/\zeta(2q+1) \notin \mathbb{Q}$, for $p, q \ge 1, p \ne q$.
*Proof.* 1. By (35), each monomial $(\zeta(P))^k$ ($k \ge 1$) is of different weight and then, by Theorem 4, $\zeta(P)$ could not satisfy an algebraic equation $T^k + a_{k-1}T^{k-1} + \dots = 0$ with coefficients in $\mathbb{Q}$, i.e. the expected result.

2. Any $P \in \mathcal{L}_{irr}^{\mathcal{X},\infty}$ is homogeneous in weight and then, by the previous item, $\zeta(P)$ is a transcendental number.

3. $\zeta(2q+1) \in \mathcal{L}_{2q+1}$ and $\zeta(2p+1) \in \mathcal{L}_{2p+1}$ ($p \ne q$) and, by the rationality of $\zeta(2n)/\pi^{2n}$, one has $\pi^{2n} \in \mathcal{L}_{2p}$. By Theorem 4, i.e. there is no linear relation among elements of different component, it follows then the expected results.

**Corollary 4.** Let $\mathcal{P}$ (resp. $\mathcal{Q}$) be the $\mathbb{Q}$-algebra generated by $\mathcal{L}_{irr}^{\mathcal{X},\infty} \setminus \{\zeta(2)\}$ (resp. $\{\zeta(2q+1)\}_{q \ge 1}$). Then

1. $\pi$ is transcendental over $\mathcal{P}$.

2. If $\{\zeta(2q+1)\}_{q \ge 1} \subset \mathcal{L}_{irr}^{\mathcal{X},\infty}$ then $\pi$ is transcendental over $\mathcal{Q}$.

*Proof.* One has, on the one hand, as said above $\zeta(2) = \pi^2/6$ is irreducible (see also Example 8) and then, by Proposition 1 (item 2), $\pi^2$ is transcendental over $\mathcal{P}$. On the other hand, for any $p \ge 1$, by Corollary 3,

$$ \zeta(2p+1) \notin \bigoplus_{n \ge 1} \mathbb{Q}[\pi^{2n}] $$

meaning that $\zeta(2p+1)$ is a transcendental over $\mathbb{Q}[\pi^2]$ and then, by hypothesis and by Proposition 1 (item 2), $\pi^2$ is transcendental over $\mathcal{Q}$.

Hence, if there exists $P \in \mathcal{P}[T]$ (resp. $\mathcal{Q}[T]$) such that $P \ne 0$ and $P(\pi) = 0$ then let $R \in \mathcal{P}[T]$ be defined by

$$ R(T) := P(T)P(-T). $$

It follows that $R(T) = R(-T)$ and, by hypothesis, $R(\pi) = 0$ meaning that $\pi^2$ satisfies an algebraic equation with coefficients in $\mathcal{P}$ (resp. $\mathcal{Q}$) contradicting with two previous assertions. Thus, such $P$ does not exist and then $\pi$ is transcendental over $\mathcal{P}$ (resp. $\mathcal{Q}$).

Ending this note, let $l \in \mathcal{L}_{yn,\mathcal{X}}$ such that $(l) = n > 1$. Then

$$ \Sigma_l > \Sigma_{y_n} = y_n \text{ and } S_l > S_{x_0^{n-1}x_1} = x_0^{n-1}x_1. \qquad (72) $$

But unfortunately

$$ \{\Sigma_{y_n}\}_{n>1} \not\subset \mathcal{L}_{irr}^{Y,\infty} \text{ and } \{S_{x_0^{n-1}x_1}\}_{n>1} \not\subset \mathcal{L}_{irr}^{X,\infty}. \qquad (73) $$

Indeed, $\zeta(\Sigma_{y_2}) = \zeta(S_{x_0x_1}) = \zeta(2) = \pi^2/6$ is irreducible then so are $\Sigma_{y_2}$ and $S_{x_0x_1}$. The rationality of $\zeta(2n)/\pi^{2n}$ (see (4)) means that $\Sigma_{y_{2n}} = y_{2n}$ and $S_{x_0^{2n-1}x_1} = x_0^{2n-1}x_1$ are not irreducible, for $n > 1$. On the other hand, $\{\zeta(2n+1)\}_{1 \le n \le 5}$, or equivalently $\{\Sigma_{2n+1}\}_{1 \le n \le 5}$ and $\{S_{x_0^{2n}x_1}\}_{1 \le n \le 5}$, are irreducible (see Examples 5–8). It could remain that $\{\zeta(2n+1)\}_{n>5}$, or equivalently $\{\Sigma_{2n+1}\}_{n>5}$ and $\{S_{x_0^{2n}x_1}\}_{n>5}$, could be irreducible.
# 5 Conclusion

Thanks to an Abel like theorem (see Theorem 3) and the equations bridging the algebraic structures (see Corollary 2) of the $\mathbb{Q}$-algebra $(\mathcal{L}, \times, 1)$ generated by the polyzas, i.e. the zeta values $\{\zeta(w)\}_{w \in \text{CONV}}$ (see the notations in (33))$^{21}$, the algebraic relations among the local coordinates (of second kind on the groups of grouplike series) of the grouplike series $Z_{\perp}$ (i.e. among $\{\zeta(S_l)\}_{l \in \text{LynX} \times X}$) and of $Z_{\perp}$ (i.e. among $\{\zeta(\Sigma_l)\}_{l \in \text{LynY} \setminus \{y_1\}}$) are provided by the algorithm **LocalCoordinateIdentification**. These algebraic relations are interpreted as two confluent, without critical pairs rewriting systems $(\mathbb{Q}1_{Y^*} \oplus (Y \setminus \{y_1\})\mathbb{Q}\langle Y\rangle, \mathcal{R}_{irr}^Y)$ and $(\mathbb{Q}1_{X^*} \oplus x_0\mathbb{Q}\langle X\rangle x_1, \mathcal{R}_{irr}^X)$ admitting $\mathcal{L}_{irr}^{\mathcal{X}, \infty}$ as the set of irreducible terms, in which the irreducible terms represent the algebraic generators for $(\mathcal{L}, \times, 1)$ and, on the other hand, the $\perp$-ideal $\mathcal{R}_X$ and the $\perp$-ideal $\mathcal{R}_Y$ represent the kernels of the $\zeta$ polymorphism (see Proposition 1). These ideals are generated by the polynomials, totally ordered and homogeneous in weight, $\{Q_l\}_{l \in \text{LynX} \setminus \text{gDIV}}$, and are interpreted as the confluent rewriting systems in which the irreducible terms belong to $\mathcal{L}_{irr}^{\mathcal{X}, \infty}$ and, in each rewriting rule of $\mathcal{R}_{irr}^X$, the left side is the leading monomial of $Q_l$ ($l \in \text{LynX} \setminus \text{gDIV}$) and does not belong to $\mathbb{Q}[\mathcal{L}_{irr}^{\mathcal{X}, \infty}]$ while the right side is canonically represented on $\mathbb{Q}[\mathcal{L}_{irr}^{\mathcal{X}, \infty}]$. It follows that $\mathcal{L} = \zeta(\mathbb{Q}[\mathcal{L}_{irr}^{\mathcal{X}, \infty}])$, being isomorphic to $\mathbb{Q}1_{X^*} \oplus x_0\mathbb{Q}\langle X\rangle x_1/\mathcal{R}_X$ and to $\mathbb{Q}1_{Y^*} \oplus (Y \setminus \{y_1\})\mathbb{Q}\langle Y\rangle/\mathcal{R}_Y$, is $\mathbb{Q}$-free and graded (see Theorem 4) and then the irreducible polyzas, belonging to $\mathcal{L}_{irr}^{\mathcal{X}, \infty}$ and being algebraic free, are transcendental numbers (see Corollary 3).

Up to weight 12, by these results, Conjecture 1 holds, i.e. $\mathcal{L}_{irr}^{\mathcal{X}, \le 12}$ is algebraically free (see Example 7 and see [29] for a short discussion, see also [21,34]$^{22}$). In particular, $\pi^2$ is algebraically free on $\{\zeta(2q+1)\}_{1 \le q \le 5}$ values. So is $\pi$ (see Corollary 4).

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<footnote>$^{21}$One can also base on (58)–(59) to get relations among $\{\zeta(w)\}_{w \in \text{CONV}}$, over fixed ring A.</footnote>

<footnote>$^{22}$All these implementations base on the “double shuffle relations” and provide linear relations.</footnote>
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