Polyzêtas

Hoang Ngoc Minh, Université Lille 2,

1, Place Déliot, 59024 Lille Cédex, France.

Workshop QUANTUM and COMBINATORICS, Zakopane (Tatra Mountains, Poland), 12-14 November 2009. (joint work with G.H.E. Duchamp, Université Paris-Nord-LIPN)

Summary

1. Introduction.

2. Infinitesimal Braid relations and structures of polylogarithms,

2.1 From infinitesimal braid relations to KZ system to structure of

polylogarithms, 2.2 Polynomial relations among polyzêtas by computer.

3. Noncommutative generating series technology,

3.1 Double regularization to 0, 3.2 Results à la Abel and generalized Euler constants, 3.3 Regularization to γ, 3.4 Action of the differential Galois group of polylogarithms on

their asymptotic expansions. 3.5 Structure of polyzêtas. 3.6 Towards the transcendence of γ.

4. Coloured polyzêtas.

INFINITESIMAL BRAID RELATIONS

&

STRUCTURE OF POLYLOGARITHMS Infinitesimal braid relation & Knizhnik-Zamolodchikov syst.

In 1986, Drinfel’d introduced the differential system associated to the Lie algebra of pure braid groups T_{n} :

Ω_{n}(z_{1}, …, z_{n}) = ∑_{1≤i<j≤n} \frac{t_{i,j}}{2iπ} ω_{i,j}(z_{1}, …, z_{n}) = ∑_{1≤i<j≤n} \frac{t_{i,j}}{2iπ} \frac{d(z_{i} − z_{j})}{z_{i} − z_{j}}.

The flatness condition, dΩ_{n} − Ω_{n} ∧ Ω_{n} = 0, is equivalent to the infinitesimal braid relation generated by {t_{i,j}}_{1≤i,j≤n} :

t_{i,j} = 0  for  i = j,
t_{i,j} = t_{j,i}  for  i ≠ j,
[t_{i,j}, t_{i,k} + t_{j,k}] = 0 for distinct i, j, k,
[t_{i,j}, t_{k,l}] = 0 for distinct i, j, k, l.

Hence, the differential system KZ_{n}

dF (z_{1}, …, z_{n}) = Ω_{n}(z_{1}, …, z_{n})F (z_{1}, …, z_{n})

is completely integrable over {(z_{1}, …, z_{n}) ∈ ℂ^{n}|z_{i} ≠ z_{j} for i ≠ j}.

Examples

• T_{2} = {t_{1,2}}.

Ω_{2}(z) = \frac{t_{1,2}}{2iπ} \frac{d(z_{1} − z_{2})}{z_{1} − z_{2}}, F (z_{1}, z_{2}) = (z_{1} − z_{2})^{t_{1,2}/2iπ}.

• T_{3} = {t_{1,2}, t_{1,3}, t_{2,3}}, [t_{1,3}, t_{1,2} + t_{2,3}] = [t_{2,3}, t_{1,2} + t_{1,3}] = 0.

Ω_{3}(z) = \frac{1}{2iπ} [t_{1,2} \frac{d(z_{1} − z_{2})}{z_{1} − z_{2}} + t_{1,3} \frac{d(z_{1} − z_{3})}{z_{1} − z_{3}} + t_{2,3} \frac{d(z_{2} − z_{3})}{z_{2} − z_{3}}].
F (z_{1}, z_{2}, z_{3}) = G(\frac{z_{1} − z_{2}}{z_{1} − z_{3}})(z_{1} − z_{3})^{(t_{1,2}+t_{1,3}+t_{2,3})/2iπ},

where G satisfies the following differential equation

(DE) dG = (x_{0} ω_{0}(z) + x_{1} ω_{1})G,
with x_{0} := \frac{t_{1,2}}{2iπ} ,  ω_{0}(z) := \frac{dz}{z}
and x_{1} := −\frac{t_{2,3}}{2iπ} ,  ω_{1}(z) := \frac{dz}{1 − z}.

Iterated integral along a path and dilogarithms

Let ω_{0}(z) = dz/z et ω_{1}(z) = dz/(1 − z). The iterated integral over ω_{0}, ω_{1} associated to w = x_{i_{1}} ⋯ x_{i_{k}} ∈ X^{∗} is defined by

α_{z_{0}}^{z}(ε) = 1 and α_{z_{0}}^{z}(x_{i_{1}} … x_{i_{k}}) = ∫_{z_{0}}^{z} … ∫_{z_{0}}^{z_{k−1}} ω_{i_{1}}(z_{1}) … ω_{i_{k}}(z_{k}).

Example

α_{0}^{z}(x_{0}x_{1}) = ∫_{0}^{z} ∫_{0}^{s} ω_{0}(s)ω_{1}(t)
= ∫_{0}^{z} ∫_{0}^{s} \frac{ds}{s} \frac{dt}{1 − t}
= ∫_{0}^{z} \frac{ds}{s} ∫_{0}^{s} dt ∑_{k≥0}t^{k}
= ∑_{k≥1}∫_{0}^{z} ds \frac{s^{k−1}}{k}
= ∑_{k≥1} \frac{z^{k}}{k^{2}}
= Li_{2}(z).

Polylogarithms, harmonic sums and polyzêtas

Classic cases (N ∈ ℕ_{+}, r > 0 and |z| < 1) :

Li_{r}(z) = α_{0}^{z}(x_{0}^{n−1}x_{1}) = ∑_{n≥1} \frac{z^{n}}{n^{r}} and H_{r}(N) = ∑_{n=1}^{N} \frac{1}{n^{r}} .

Generalization on multi-indices s = (s_{1}, …, s_{r}) :

α_{0}^{z}(x_{0}^{s_{1}−1}x_{1} … x_{0}^{s_{r}−1}x_{1}) = Li_{s}(z) = ∑_{n_{1}>...>n_{r}>0} \frac{z^{n_{1}}}{n_{1}^{s_{1}} … n_{r}^{s_{r}}} ,

P_{s}(z) = \frac{Li_{s}(z)}{1 − z} = ∑_{N≥0}H_{s}(N) z^{N}, where H_{s}(N) = ∑_{n_{1}>...>n_{r}=1}^{N} \frac{1}{n_{1}^{s_{1}} … n_{r}^{s_{r}}} .

If s_{1} > 1, by an Abel’s theorem, then one has

lim_{z→1}Li_{s}(z) = lim_{N→∞}H_{s}(N) = ζ(s) = ∑_{n_{1}>...>n_{r}>0}\frac{1}{n_{1}^{s_{1}} … n_{r}^{s_{r}}} else ?

Euler-Maclaurin sommation formula

∑_{n=1}^{N} \frac{1}{n} = log N + γ − ∑_{j=1}^{k−1} \frac{B_{j}}{j} \frac{1}{N^{j}} + O(\frac{1}{N^{k}} ),

∑_{n=1}^{N} \frac{1}{n^{r}} = ζ(r) − \frac{N^{1−r}}{(r − 1)} − ∑_{j=r}^{k−1} \frac{B_{j−r+1}}{j − r + 1}\binom{k − 1}{j − 1}\frac{1}{N^{j}} + O(\frac{1}{N^{k}} ).

Are γ and ζ(r) algebraically independent ? Let Z be the ℚ-algebra generated by convergent polyzêtas. Let Z′ be the ℚ[γ]-algebra generated by convergent polyzêtas. How to determine the asymptotic expansion and the constants associated to the divergents harmonic sums of the form

H_{\{1\}^{r}}(N) = ∑_{N≥n_{1}>...>n_{r}>0} \frac{1}{n_{1} … n_{r}} ?

H_{\{1\}^{k},\underbrace{s_{k+1}}_{>1},...,s_{r}}(N) = ∑_{N≥n_{1}>...>n_{r}>0}^{N} \frac{1}{n_{1} … n_{k}n_{k+1}^{s_{k+1}} … n_{r}^{s_{r}}} ?

Do there exists a generalization of γ ?

Encoding the multi-indices by words

Y = {y_{k}|k ∈ ℕ_{+}} (y_{1} < y_{2} < …) and X = {x_{0}, x_{1}} (x_{0} < x_{1}). Y^{∗} (resp. X^{∗}) : monoide generated by Y (resp. X ).

s = (s_{1}, …, s_{r}) ↔ w = y_{s_{1}} … y_{s_{r}} ↔ w = x_{0}^{s_{1}−1}x_{1} … x_{0}^{s_{r}−1}x_{1}.

u and v are convergent if s_{1} > 1. A word divergent is of the form

({1}^{k}, s_{k+1}, …, s_{r}) ↔ y_{1}^{k}y_{s_{k+1}} … y_{s_{r}} ↔ x_{1}^{k}x_{0}^{s_{k+1}−1}x_{1} … x_{0}^{s_{r}−1}x_{1}, for k ≥ 1.
Li_{w} : w ↦ Li_{w}(z) = ∑_{n_{1}>...>n_{r}>0} \frac{z^{n_{1}}}{n_{1}^{s_{1}} … n_{r}^{s_{r}}} ,
ζ_{w} : w ↦ ζ(w) = ∑_{n_{1}>...>n_{r}>0} \frac{1}{n_{1}^{s_{1}} … n_{r}^{s_{r}}} ,
H_{w} : w ↦ H_{w}(N) = ∑_{N≥n_{1}>...>n_{r}>0} \frac{1}{n_{1}^{s_{1}} … n_{r}^{s_{r}}} ,
P_{w} : w ↦ P_{w}(z) = ∑_{N≥0}H_{w}(N)z^{N} = \frac{Li_{w}(z)}{1 − z}.

Let Π_{X} : ℂ⟨⟨Y ⟩⟩ ⟶ ℂ⟨⟨X ⟩⟩ and Π_{Y} : ℂ⟨⟨X ⟩⟩ ⟶ ℂ⟨⟨Y ⟩⟩ denote the “change” of alphabets over ℂ⟨⟨X ⟩⟩ and ℂ⟨⟨Y ⟩⟩ respectively.

Structure of polylogarithms

Let C = ℂ[z, z^{−1}, (1 − z)^{−1}]

Theorem (HNM, van der Hoeven & Petitot, 1998)

Putting Li_{x_{0}}(z) = log z, Li : w ↦ Li_{w} becomes an isomorphism from (ℂ⟨X ⟩, � ) to (ℂ{Li_{w}}_{w∈X^{∗}}, .).

▶ Li_{w}, w ∈ X^{∗}, are C-linearly independent.

Then {Li_{w}}_{w∈X^{∗}} is universal Picard-Vessiot extension of fuchsian differential equations with three regular singularities.

▶ Li_{l}, l ∈ LynX , are C-algebraically independent.

▶ ζ(l), l ∈ LynX \ {x_{0}, x_{1}}, are generators of the algebra Z.

More about structure of polylogarithms

Hoffman proved that ∀u, v ∈ Y^{∗}, H_{u}H_{v} = H_{u v}. Therefore, ∀u, v ∈ Y^{∗} \ y_{1}Y^{∗}, ζ(u)ζ(v) = ζ(u v).

P_{u}(z) ⊙ P_{v}(z) = ∑_{n≥0}H_{u}(n)H_{v}(n)z^{n} = ∑_{n≥0}H_{u v}(n)z^{n} = P_{u v}(z).

Theorem (HNM, 2003)

(ℂ{P_{w}}_{w∈Y^{∗}}, ⊙) ≅ (ℂ⟨Y ⟩, ) ≅ (ℂ{H_{w}}_{w∈Y^{∗}}, .).

▶ P_{w}, w ∈ Y^{∗}, are C-linearly independent.

Then H_{w}, w ∈ Y^{∗}, are linearly independent. ▶ P_{l}, l ∈ LynY , are C-algebraically independent.

Then H_{l}, l ∈ LynY , algebraically independent. ▶ ζ(l), l ∈ LynY \ {y_{1}}, are generators of the algebra Z.

Towards the structure of polyzêtas

Corollary

∀u, v ∈ X^{∗}, Li_{u} Li_{v} = Li_{u� v} ⇒ ∀u, v ∈ x_{0}X^{∗}x_{1}, ζ(u)ζ(v) = ζ(u� v).

Example

x_{0}x_{1}� x_{0}^{2}x_{1} = x_{0}x_{1}x_{0}^{2}x_{1} + 3x_{0}^{2}x_{1}x_{0}x_{1} + 6x_{0}^{3}x_{1}^{2},
Li_{2} Li_{3} = Li_{2,3} +3 Li_{3,2} +6 Li_{4,1},
ζ(2)ζ(3) = ζ(2, 3) + 3ζ(3, 2) + 6ζ(4, 1).

Corollary

∀u, v ∈ Y^{∗}, H_{u}H_{v} = H_{u v} ⇒ ∀u, v ∈ Y^{∗} \ y_{1}Y^{∗}, ζ(u)ζ(v) = ζ(u v).

Example

y_{2} y_{3} = y_{2}y_{3} + y_{3}y_{2} + y_{5},
P_{y_{2}} ⊙ P_{y_{3}} = P_{y_{2}y_{3}} + P_{y_{3}y_{2}} + P_{y_{5}},
H_{2}H_{3} = H_{2,3} + H_{3,2} + H_{5},
ζ(2)ζ(3) = ζ(2, 3) + ζ(3, 2) + ζ(5).

\begin{matrix}ζ(2)ζ(3) = ζ(2, 3) + 3ζ(3, 2) + 6ζ(4, 1) \\ ζ(2)ζ(3) = ζ(2, 3) + ζ(3, 2) + ζ(5) \end{matrix} \} ⇒ ζ(5) = 2ζ(3, 2)+6ζ(4, 1).

Polynomial relations among {ζ(l)}_{l∈LynX \{x_{0},x_{1}}}

ζ(2, 1) = ζ(3)
ζ(4) = \frac{2}{5} ζ(2)^{2}
ζ(3, 1) = \frac{1}{10} ζ(2)^{2}
ζ(2, 1, 1) = \frac{2}{5} ζ(2)^{2}
ζ(4, 1) = 2ζ(5) − ζ(2)ζ(3)
ζ(3, 2) = − \frac{11}{2} ζ(5) + 3ζ(2)ζ(3)
ζ(3, 1, 1) = 2ζ(5) − ζ(2)ζ(3)
ζ(2, 2, 1) = − \frac{11}{2} ζ(5) + 3ζ(2)ζ(3)
ζ(2, 1, 1, 1) = ζ(5)
ζ(6) = \frac{8}{35} ζ(2)^{3}
ζ(5, 1) = − \frac{1}{2} ζ(3)^{2} + \frac{6}{35} ζ(2)^{3}
ζ(4, 2) = ζ(3)^{2} − \frac{32}{105} ζ(2)^{3}
ζ(4, 1, 1) = −ζ(3)^{2} + \frac{23}{70} ζ(2)^{3}
ζ(3, 2, 1) = 3ζ(3)^{2} − \frac{29}{30} ζ(2)^{3}
ζ(3, 1, 2) = − \frac{3}{2} ζ(3)^{2} + \frac{53}{105} ζ(2)^{3}
ζ(3, 1, 1, 1) = − \frac{1}{2} ζ(3)^{2} + \frac{6}{35} ζ(2)^{3}
ζ(2, 2, 1, 1) = ζ(3)^{2} − \frac{32}{105} ζ(2)^{3}
ζ(2, 1, 1, 1, 1) = \frac{8}{35} ζ(2)^{3}

Irreducible polyzêtas by computer

in Axiom at weight 10 (avec Petitot), in Maple at weight 12 (with Bigotte, Jacob, Oussous, Petitot), in C++ at weight 16 (with El Wardi, Jacob, Oussous, Petitot).

{r \atop n}  1  2  3  4  5
2  ζ(2)
3  ζ(3)
5  ζ(5)
7  ζ(7)
8  ζ(6, 2)
9  ζ(9)
10  ζ(8, 2)
11 ζ(11)  ζ(8, 2, 1)
12  ζ(10, 2)  ζ(8, 2, 1, 1)
13 ζ(13)  ζ(9, 3, 1)
ζ10, 2, 1)
14  ζ(10, 4)  ζ(10, 2, 1, 1)
ζ(12, 2)
15 ζ(15)  ζ(11, 3, 1)  ζ(10, 2, 1, 1, 1)

ζ(12, 2, 1)

16  ζ(12, 4)  ζ(10, 4, 1, 1)
ζ(14, 2)  ζ(11, 3, 1, 1)

ζ(12, 2, 1, 1)

NONCOMMUTATIVE GENERATING SERIES TECHNOLOGY Noncommutative generating series

Definition

L(z) := ∑_{w∈X^{∗}} Li_{w}(z) w and H(N) := ∑_{w∈Y^{∗}} H_{w}(N) w.

Let LynX and l ∈ LynX (resp. {l̂}_{l∈LynY} and {l̂}_{l∈LynY}) be the transcendence basis of (ℂ⟨X ⟩, � ) (resp. (ℂ⟨Y ⟩, )) and its dual basis respectively. Then

Theorem

L and H are group-like and L(z) = e^{x_{1} log \frac{1}{1−z}} L_{reg}(z)e^{x_{0} log z} and H(N) = e^{H_{1}(N) y_{1}}H_{reg}(N),

where L_{reg}(z) = ∏_{{l∈LynX \atop l≠x_{0},x_{1}}}^{↘} e^{Li_{l}(z) l̂} and H_{reg}(N) = ∏_{{l∈LynY \atop l≠y_{1}}}^{↗} e^{H_{l}(N) l̂}.

Definition

Z� := L_{reg}(1) and Z := H_{reg}(∞).

Double regularization to 0

Proposition

Let ζ_{�} : ℂ⟨⟨X ⟩⟩ ⟶ ℂ be the shuffle algebra morphism defined by ▶ ζ_{�} (x_{0}) = ζ_{�} (x_{1}) = 0, ▶ for any r_{1} > 1, ζ_{�} (x_{0}^{r_{1}−1}x_{1} … x_{0}^{r_{k}−1}x_{1}) = ζ(r_{1}, …, r_{k}), ▶ for any u, v ∈ X^{∗}, ζ_{�} (u� v) = ζ_{�} (u)ζ_{�} (v).

Then ∑_{w∈X^{∗}} ζ_{�} (w) w = Z� .

Proposition

Let ζ : ℂ⟨⟨Y ⟩⟩ ⟶ ℂ be the algebra morphism defined by ▶ ζ (y_{1}) = 0, ▶ for any r_{1} > 1, ζ (y_{r_{1}} … y_{r_{k}}) = ζ(r_{1}, …, r_{k}), ▶ for any u, v ∈ Y^{∗}, ζ_{�} (u v) = ζ (u)ζ (v).

Then ∑_{w∈Y^{∗}} ζ (w) w = Z .

Results à la Abel

Theorem (HNM, 2005)

lim_{z→1}e^{y_{1} log \frac{1}{1−z}} Π_{Y}L(z) = lim_{N→∞}[∑_{k≥0}H_{y_{1}^{k}}(N)(−y_{1})^{k}]H(N) = Π_{Y}Z� .

⇒ H(N){ \atop N→∞} exp[− ∑_{k≥1}H_{y_{k}}(N)\frac{(−y_{1})^{k}}{k} ]Π_{Y}Z� .

Theorem (Costermans, Enjalbert & HNM, 2005)

There exists algorithmically computable coefficients b_{i} ∈ Z′, the ℚ-algebra generated by Z and by γ, κ_{i} ∈ ℕ and η_{i} ∈ ℤ such that ∀w ∈ Y^{∗}, H_{w}(N){ \atop N→∞} ∑_{i=0}^{+∞}b_{i}N^{η_{i}} log^{κ_{i}}(N).

Example

H_{4,2}(N) = ζ(4, 2) − \frac{π^{2}}{6} \frac{1}{N^{3}} + (\frac{π^{2}}{12} + \frac{1}{4}) \frac{1}{N^{4}} − (\frac{π^{2}}{18} + \frac{2}{5}) \frac{1}{N^{5}} + O (\frac{1}{N^{6}}) ,
H_{1,4}(N) = \frac{π^{4}}{90} ln (N) + \frac{π^{4}}{90} γ − ζ (4, 1) − ζ (5)
+ \frac{π^{4}}{180} \frac{1}{N} − \frac{π^{4}}{1080} \frac{1}{N^{2}} + \frac{1}{9} \frac{1}{N^{3}} + (\frac{π^{4}}{10800} − \frac{1}{24}) \frac{1}{N^{4}} + O (\frac{1}{N^{5}}) .

Generalized Euler constants

Theorem (HNM, 2005)

For any k ≥ 0 and for any w ∈ Y^{∗} \ {y_{1}}, let γ_{y_{1}^{k} w} be the constant associated to H_{y_{1}^{k} w}. Let G := ∑_{w∈Y^{∗}} γ_{w} w.

Then G is group-like and G = exp[γy_{1} − ∑_{k≥2}ζ(k)\frac{(−y_{1})^{k}}{k} ]Π_{Y}Z� . Let b_{n,k}(t_{1}, …, t_{k}) be the Bell polynomials. By specializing at t_{1} = γ and for l ≥ 2, t_{l} = (−1)^{l−1}(l − 1)!ζ(l) and by using the identity, for any u ∈ X^{∗}, x_{1}^{k}x_{0}u = ∑_{l=0}^{k}x_{1}^{l}� (x_{0}[(−x_{1})^{k−l}� u]), we get

Corollary

γ_{y_{1}^{k} w} = ∑_{i=0}^{k} \frac{ζ(x_{0}[(−x_{1})^{k−i}� Π_{X}w])}{i!}[∑_{j=1}^{i}b_{i,j}(γ, −ζ(2), 2ζ(3), …)]. In

particular, γ_{y_{1}^{k}} = ∑_{s_{1},...,s_{k}>0 s_{1}+...+ks_{k}=k} \frac{(−1)^{k}}{s_{1}! … s_{k}!}(−γ)^{s_{1}}(−\frac{ζ(2)}{2})^{s_{2}} … (−\frac{ζ(k)}{k})^{s_{k}}.

Regularization to γ

γ_{1,1} = [γ^{2} − ζ(2)]/2,
γ_{1,1,1} = [γ^{3} − 3ζ(2)γ + 2ζ(3)]/6,
γ_{1,1,1,1} = [80ζ(3)γ − 60ζ(2)γ^{2} + 6ζ(2)^{2} + 10γ^{4}]/240,
γ_{1,7} = ζ(7)γ + ζ(3)ζ(5) − 54ζ(2)^{4}/175,
γ_{1,1,6} = 4ζ(2)^{3}γ^{2}/35 + [ζ(2)ζ(5) + 2ζ(3)ζ(2)^{2}/5 − 4ζ(7)]γ
+ ζ(6, 2) + 19ζ(2)^{4}/35 + ζ(2)ζ(3)^{2}/2 − 4ζ(3)ζ(5),
γ_{1,1,1,5} = \frac{3ζ(6, 2)}{4} − \frac{14ζ(3)ζ(5)}{3} + \frac{3ζ(2)ζ(3)^{2}}{4} + \frac{809ζ(2)^{4}}{1400} + \frac{ζ(5)γ^{3}}{6}
+ [\frac{ζ(3)^{2}}{4} − \frac{ζ(2)^{3}}{5} ]γ^{2} − [2ζ(7) − \frac{3ζ(2)ζ(5)}{2} + \frac{ζ(3)ζ(2)^{2}}{10} ]γ.

Theorem

γ_{•} realizes the morphism from (ℂ⟨⟨Y ⟩⟩, ) to (ℂ, .) verifying ▶ for any word u, v ∈ Y^{∗}, γ_{u v} = γ_{u}γ_{v}, ▶ for any convergent word w ∈ Y^{∗} − y_{1}Y^{∗}, γ_{w} = ζ(w), ▶ γ_{y_{1}} = γ.

Then G = e^{γy_{1}}Z .

The meaning of the double regularization to 0

The constant γ_{y_{1}} = γ is obtained as the finite part of the asymptotic expansion of H_{1}(n) in the scale {n^{a} log^{b}(n)}_{a∈ℤ,b∈ℕ}.

In the same way, since for any n ∈ ℕ, n and H_{1}(n) are algebraically independent then {n^{a}H_{1}^{b}(n)}_{a∈ℤ,b∈ℕ} constitutes a new scale for asymptotic expansions.

Let C_{1} = ℚ ⊕ x_{0}ℚ⟨X ⟩x_{1} and C_{2} = ℚ ⊕ (Y \ {y_{1}})ℚ⟨Y ⟩. By the Radford theorem and its generalization over Y (due to Malvenuto & Reutenauer), one has respectively

(ℚ⟨X ⟩, � ) ≅ ℚ[LynX ] = C_{1}[x_{0}, x_{1}],
(ℚ⟨Y ⟩, ) ≅ ℚ[LynY ] = C_{2}[y_{1}].

Thus, ζ_{�} (x_{1}) = 0 and ζ (y_{1}) = 0 can be interpreted as the finite part of the asymptotic expansions of Li_{1} and H_{1} in the scales {(1 − z)^{a} log(1 − z)^{b}}_{a∈ℤ,b∈ℕ} and {n^{a}H_{1}^{b}(n)}_{a∈ℤ,b∈ℕ} respectively.

Differential Galois group of polylogarithms

LI_{C} is the smallest algebra containing C closed by derivation, by integration w.r.t. ω_{0} and ω_{1}. It is the C-modulus generated by {Li_{w}}_{w∈X^{∗}}.

Let σ ∈ Gal(LI_{C}). Then ∑_{w∈X^{∗}} σ Li_{w} w = ∏_{l∈Lyn}^{↘}e^{σ LiŠ_{l} S_{l}}.

Since dσ Li_{x_{i}} = σd Li_{x_{i}} = ω_{i} then σ Li_{x_{i}} = Li_{x_{i}} +c_{x_{i}}.

More generally, σ LiŠ_{l} = ∫ ω_{x_{i}} \frac{σ Lii_{1} Š_{l_{1}}}{i_{1}!} ⋯ \frac{σ Lii_{k} Š_{l_{k}}}{i_{k}!} + cŠ_{l}.

Consequently, ∑_{w∈X^{∗}} σ Li_{w} w = L ∏_{l∈Lyn}^{↘}e^{cŠ_{l}S_{l}} = Le^{C_{σ}}.

The action of σ ∈ Gal(LI_{C}) over {Li_{w}}_{w∈X^{∗}} is equivalent to the action of e^{C_{σ}} ∈ Gal(DE) over the exponential solution L. So,

Theorem (HNM, 2003)

Gal(LI_{C}) ≅ Gal(DE) = {e^{C} | C ∈ Lie_{ℂ}⟨⟨X ⟩⟩}.

Action of Gal(DE) on the asymptotic expansions

Theorem (Group of associators theorem)

For any commutative ℚ-algebra A, let Φ ∈ A⟨⟨X ⟩⟩ and Ψ ∈ A⟨⟨Y ⟩⟩ be group-like elements such that Ψ = B(y_{1})Π_{Y}Φ. There exists an unique C ∈ Lie_{A}⟨⟨X ⟩⟩ such that Φ = Z� e^{C} and Ψ = GΠ_{Y}e^{C}.

If C ∈ Lie_{A}⟨⟨X ⟩⟩ then L′ = Le^{C} is group-like and e^{C} ∈ Gal(DE). Let H′(N) be the n.c.g.s. of the Taylor coefficients, belonging the harmonic algebra, of {(1 − z)^{−1}L′_{w}(z)}_{w∈Y^{∗}}. Then H′(N) is group-like.

\frac{L′(1 − ε)}{ε}{ \atop ε→0^{+}}e^{−(1+x_{1}) log ε}Φ_{KZ}e^{C} ⇒ H′(N){ \atop N→∞}H(N)Π_{Y}e^{C}.

Let κ_{w} be the constant part of H′_{w}(N). Then,

∑_{w∈Y^{∗}} κ_{w} w = Ψ_{KZ}Π_{Y}e^{C}, or eqivalenty Π_{X} ∑_{w∈Y^{∗}} κ_{w} w = B^{−1}(x_{1})Φ_{KZ}e^{C}.

We put then Ψ := Z� Π_{Y}e^{C} and Φ := GΠ_{Y}e^{C}.

Examples (action of the monodromy group)

For t ∈]0, 1[, the monodromies around 0, 1 of L are given respectively by (p = 2iπ)

M_{0}L(t) = L(t)e^{p𝔪_{0}} and M_{1}L(t) = L(t)Φ_{KZ}^{−1}e^{−px_{1}}Φ_{KZ}
= L(t)e^{p𝔪_{1}},
where 𝔪_{0} = x_{0} and 𝔪_{1} = ∏_{l∈Lyn,l≠x_{0},x_{1}}^{↘} e^{−ζ(Š_{l}) ad_{S_{l}}}(−x_{1}).

▶ If C = p𝔪_{0} then Φ = Z� e^{px_{0}} and

Ψ = exp[γy_{1} − ∑_{k≥2}ζ(k)\frac{(−y_{1})^{k}}{k} ]Π_{Y}Z� = G.

▶ If C = p𝔪_{1} then Φ = e^{−px_{1}}Z� and

Ψ = exp[(\underbrace{γ − p}_{=T})y_{1} − ∑_{k≥2}ζ(k)\frac{(−y_{1})^{k}}{k} ]Π_{Y}Z� = e^{−py_{1}}G.

Hence, the monodromies could not allow neither to introduce the factor e^{γx_{1}} on the left of Z� , neither to eliminate the left factor e^{γy_{1}} in G (by putting T = 0, for example).

Polynomial relations among generators of polyzêtas

{ζ_{�} (l)}_{l∈LynX} and {ζ (l)}_{l∈LynY} are also generators of the algebras Z and Z′. One also gets

Theorem

∏_{l∈LynX ,l≠x_{0},x_{1}}^{↘} e^{ζ(l) l̂} = e∑_{k≥2}ζ(k)\frac{(−x_{1})^{k}}{k} Π_{X} ∏_{l∈LynY ,l≠y_{1}}^{↗} e^{ζ(l) l̂}.

Since ∀l ∈ LynY ⟺ Π_{X}l ∈ LynX \ {x_{0}} then identifying the local coordinates, in the Lyndon-PBW basis, we get polynomial relations among these generators which are algebraically independent on γ.

Corollary

For any l ∈ LynY \ {y_{1}}, let P_{l} be the decomposition of Π_{X}l̂ in the Lyndon-PBW basis, over X , and let P̌_{l} be its dual. Then Π_{X}l − P̌_{l} ∈ ker ζ. Moreover, Π_{X}λ − P̌_{λ} is homogenous of degree |λ| and if Π_{X}l = P̌_{l} then ζ(l) is irreducible.

Structure of polyzêtas

Theorem (Structure of polyzêtas)

The ℚ-algebra generated by convergent polyzêtas, Z, is isomorphic to the graded algebra (ℚ ⊕ (Y − y_{1})ℚ⟨Y ⟩/ ker ζ, ).

Proof.

Since ker ζ is an ideal generated by the homogenous polynomials then the quotient ℚ ⊕ (Y − y_{1})ℚ⟨Y ⟩/ ker ζ is graded.

Corollary

The ℚ-algebra of polyzêtas is freely generated by irreducible polyzêtas.

Proof.

For any λ ∈ LynY , if λ = P̌_{λ} then one gets the conclusion else Π_{X}λ − P̌_{λ} ∈ ker ζ. Since P̌_{λ} ∈ ℚ[LynX ] then P̌_{λ} is polynomial on Lyndon words of degree ≤ |λ|. For each Lyndon word does appear in this decomposition of P̌_{λ}, after applying Π_{Y}, the same process goes on untill having irreducible polyzêtas.

Towards the transcendence of γ over Z

By considering the commutative indeterminates t_{1}, t_{2}, …, then let A = ℚ[t_{1}, t_{2}, …].

Lemma

For any Φ ∈ {Φ_{KZ}e^{C}|C ∈ Lie_{A}⟨⟨X ⟩⟩}, one get

Ψ = B(y_{1})Π_{Y}Φ ⟺ Ψ′ = B′(y_{1})Π_{Y}Φ.

Theorem

For all Φ ∈ {Φ_{KZ}e^{C}|C ∈ Lie_{ℚ}⟨⟨X ⟩⟩}, the identies Ψ = B(y_{1})Π_{Y}Φ yield all polynomial relations among convergent polyzêtas. Moreover, these relations are algebraically independent on γ. In other words, under the hypothesis γ ∉ ℚ, the constant γ does not verify any polynomial of coefficients in Z.

Corollary

If γ ∉ ℚ then it is transcendental over the ℚ-algebra generated by convergent polyzêtas.

COLOURED POLYLOGARITHMS, HARMONIC SUMS AND POLYZETAS Sommes harmoniques et polylogarithmes colorés

Let

\binom{b}{s} = \binom{b_{1}, …, b_{r}}{s_{1}, …, s_{r}} ,

where b_{1}, .., b_{r} ∈ {1, q, …, q^{n−1}} and q = e^{2iπ/n}.

H_{{b \atop s}}(N) = ∑_{N≥n_{1}>...>n_{r}>0} \frac{b_{1}^{n_{1}} ⋯ b_{r}^{n_{r}}}{n_{1}^{s_{1}} … n_{r}^{s_{r}}} ,
Li_{{b \atop s}}(z) = ∑_{n_{1}>···>n_{r}>0} \frac{b_{1}^{n_{1}} ⋯ b_{r}^{n_{r}}}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}} z^{n_{1}}.

If \binom{b_{1}}{s_{1}} ≠ \binom{1}{1} then

lim_{N→∞}H_{{b \atop s}}(N) = lim_{z→1}Li_{{b \atop s}}(z) = ∑_{n_{1}>...>n_{r}>0} \frac{b_{1}^{n_{1}} ⋯ b_{r}^{n_{r}}}{n_{1}^{s_{1}} … n_{r}^{s_{r}}} = ζ \binom{b}{s} .

P_{{b \atop s}}(z) = ∑_{n≥0}H_{{b \atop s}}(n)z^{n} = \frac{Li_{{b \atop s}}(z)}{1 − z}.

Coloured alphabets

Let us consider the alphabets X = {x_{0}, x_{q}, …, x_{q^{n}}} and Y = \{y_{{b \atop k}}\}_{{b∈\{1,...,q^{n−1}\} \atop k∈ℕ_{+}}}.

\binom{b_{1}, …, b_{r}}{s_{1}, …, s_{r}} ↔ y_{{b_{1} \atop s_{1}}} … y_{{br \atop sr}} ↔ x_{0}^{s_{1}−1}x_{τ_{1}} ⋯ x_{0}^{s_{r}−1}x_{τ_{r}},

where j = 1, .., r, τ_{j} = ∏_{i=1}^{j}b_{i}.

Example

\binom{−1,1,−1}{3,4, 5} ↔ x_{0}^{2}x_{−1}x_{0}^{3}x_{−1}x_{0}^{4}x_{1}.

Let Z be the ℚ-algebra generated by convergent coloured polyzêtas. Let Z′ be the ℚ[γ]-algebra generated by convergent coloured polyzêtas.

Harmonic algebra

For u, v ∈ Y^{∗}, we have H_{u}H_{v} = H_{u⋆v}, where u ⋆ v is defined by ε ⋆ u = u ⋆ ε = u and (y_{{b \atop i}}u) ⋆ (y_{{c \atop j}} v) = y_{{b \atop i}}(u ⋆ y_{{c \atop j}} v) + y_{{c \atop j}} (y_{{b \atop i}}u ⋆ v) + y_{{bc \atop i+j}} (u ⋆ v). By consequent, P_{u} ⊙ P_{v} = P_{u⋆v}.

Example

y_{{−1 \atop 2}} ⋆ y_{{−1 \atop 3}} = y_{{−1 \atop 2}} y_{{−1 \atop 3}} + y_{{−1 \atop 3}} y_{{−1 \atop 2}} + y_{{1 \atop 5}} ,
H_{{−1 \atop 2}} H_{{−1 \atop 3}} = H_{{−1,−1 \atop 2,3}} + H_{{−1,−1 \atop 3,2}} + H_{{1 \atop 5}} ,
P_{{−1 \atop 2}} ⊙ P_{{−1 \atop 3}} = P_{{−1,−1 \atop 2,3}} + P_{{−1,−1 \atop 3,2}} + P_{{1 \atop 5}} ,
y_{{−1 \atop 2}} y_{{−1 \atop 5}} ⋆ y_{{1 \atop 4}} = y_{{−1 \atop 2}} (y_{{−1 \atop 5}} ⋆ y_{{1 \atop 4}} ) + y_{{1 \atop 4}} (y_{{−1 \atop 2}} y_{{−1 \atop 5}} ⋆ ε) + y_{{−1 \atop 6}} (y_{{−1 \atop 5}} ⋆ ε)
= y_{{−1 \atop 2}} (y_{{−1 \atop 5}} y_{{1 \atop 4}} + y_{{1 \atop 4}} y_{{−1 \atop 5}} + y_{{−1 \atop 9}} ) + y_{{1 \atop 4}} y_{{−1 \atop 2}} y_{{−1 \atop 5}} + y_{{−1 \atop 6}} y_{{−1 \atop 5}}
= y_{{−1 \atop 2}} y_{{−1 \atop 5}} y_{{1 \atop 4}} + y_{{−1 \atop 2}} y_{{1 \atop 4}} y_{{−1 \atop 5}} + y_{{1 \atop 4}} y_{{−1 \atop 2}} y_{{−1 \atop 5}} + y_{{−1 \atop 2}} y_{{−1 \atop 9}} + y_{{−1 \atop 6}} y_{{−1 \atop 5}} ,
H_{{−1,−1 \atop 2,5}} H_{{1 \atop 4}} = H_{{−1,−1,1 \atop 2,5,4}} + H_{{−1,1,−1 \atop 2,4,5}} + H_{{1,−1,−1 \atop 4,2,5}} + H_{{−1,−1 \atop 2,9}} + H_{{−1,−1 \atop 6,5}} ,
P_{{−1,−1 \atop 2,5}} ⊙ P_{{1 \atop 4}} = P_{{−1,−1,1 \atop 2,5,4}} + P_{{−1,1,−1 \atop 2,4,5}} + P_{{1,−1,−1 \atop 4,2,5}} + P_{{−1,−1 \atop 2,9}} + P_{{−1,−1 \atop 6,5}} .

Structure theorems

Let O = {0} ∪ {q^{n} = 1} and let C = ℂ[z, {a_{i}(z)}_{i∈O}] where a_{0}(z) = z^{−1} and, for ρ ∈ O \ {0}, a_{ρ}(z) = ρ(1 − ρz)^{−1}.

Theorem (Bigotte, Jacob, Oussous, Petitot, 2000)

(C{Li_{w}}_{w∈X^{∗}}, .) ≃ (C[Li_{LynX}], .).

Proposition

(ℂ{P_{w}}_{w∈Y^{∗}}, ⊙) ≃ (ℂ{H_{w}}_{w∈Y^{∗}}, .) ≃ (ℂ⟨Y ⟩, ⋆).

Theorem (HNM, 2003)

(ℂ{P_{w}}_{w∈Y^{∗}}, ⊙) ≃ (ℂ[P_{LynY}], ⊙), (ℂ{H_{w}}_{w∈Y^{∗}}, .) ≃ (ℂ[H_{LynY}], .).

Distribution relation

Proposition

Let q be a m-th root of unity. Let s and i be two compositions. Assume that 0 ≤ i_{1} < … < i_{r} ≤ m − 1. Then,

ζ\binom{s}{q^{i}} = \frac{1}{m^{r}} ∑_{b_{1},...,b_{r}=1}^{m} ∑_{j_{1},...,j_{r}=0}^{m−1} \frac{q^{∑_{l=1}^{r}(i_{1}+...+i_{l})b_{l}}}{q^{∑_{l=1}^{r}(j_{1}+...+j_{l})b_{l}}} ζ\binom{s}{q^{j}},

or equivalently

∑_{b_{1},...,b_{r}=1}^{m} ∑_{j_{1}≠i_{1},...,j_{r} ≠i_{r}^{j_{1},...,jr=0}}^{m−1} \frac{∏_{l=1}^{r} q^{∑{r \atop l=1}}(i_{1} + … + i_{l})b_{l}}{q^{∑{r \atop l=1}}(j_{1} + … + j_{l})b_{l}} ζ\binom{s}{q^{j}} = 0.

Asymptotic expansions

Theorem (HNM, 2005)

(C{P_{w}}_{w∈Y^{∗}}, ⊙) ≃ (C[P_{Lyn(Y )}], ⊙) and for any g ∈ C{P_{w}}_{w∈Y^{∗}}, there exists algorithmically computable coefficients b_{i} ∈ ℂ, the ℚ-algebra generated by Z and by γ, κ_{i} ∈ ℕ and η_{i} ∈ ℤ such that

g(z) ∼ ∑_{j=0}^{+∞}c_{j}(1 − z)^{α_{j}}P_{y_{1}^{β_{j}}}(z) for z → 1.

By conssequence, there exists algorithmically computable coefficients b_{i} ∈ ℂ, η_{i} ∈ ℤ and κ_{i} ∈ ℕ such that

[z^{n}]g(z) ∼ ∑_{i=0}^{+∞}b_{i}n^{η_{i}} log^{κ_{i}}(n) for n → ∞.

Corollary

There exists algorithmically computable coefficients b_{i} ∈ Z′, η_{i} ∈ ℤ and κ_{i} ∈ ℕ such that

∀w ∈ Y^{∗}, H_{w}(N) ∼ ∑_{i=0}^{+∞}b_{i}N^{η_{i}} log^{κ_{i}}(N) for N → +∞.

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