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Structure of Polyzetas and Lyndon Words

Abstract

The effective construction of pairs of bases in duality for quasi-shuffle bialgebras and the infinite factorizations by Lyndon words of noncommutative generating series allow to prove that the algebra of polyzetas is graded by the weight and therefore lead to some consequences for their arithmetical nature.

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Notes

  1. Nielsen established several kind of identities for the double polyzetas and for polyzetas ζ(s,1,…,1) which are rediscovered recently by several authors studying, numerically or symbolically, the relations among general polyzetas given in (3).

    The polylogarithms \(\operatorname{Li}_{s,1,\ldots,1}(z)\) are also called a Nielsen polylogarithm or simply Nielsen function in [31, 36] and a link between these relations with the algebraic combinatorics on words and with transcendence basis of shuffle algebra was pointed out for the analysis of nonlinear differential equations with three singularities [30, 34, 35].

  2. They form a Gröbner basis of the ideal of polynomial relations among the convergent polyzetas and the ranking of this basis is based mainly on the order of Lyndon words [3, 37, 52]. For this case, this basis is also called Gröbner–Lyndon basis.

  3. The weight of

    • the composition \({\bf s}=(s_{1},\ldots,s_{r})\in ({\mathbb{N}}_{+})^{*}\) is the sum s 1+⋯+s r ,

    • the word \(u=x_{i_{1}}\cdots x_{i_{s}}\in\{x_{0},x_{1}\}^{*}\) is its length denoted usually by |u| and equals s,

    • the word \(v=y_{i_{1}}\cdots y_{i_{s}}\in(\{y_{k}\}_{k\ge 1})^{*}\) is denoted by (v) and equals i 1+⋯+i s .

  4. Note that Conjecture 1.2 amounts to way that the Hilbert–Poincaré series of the graded algebra is

    $$\begin{aligned} \sum_{n\ge0}d_n t^n =& \frac{1}{1-t^2-t^3}. \end{aligned}$$
  5. Recall that the combinatorial Hopf algebra of the shuffle product is contained in [48] and has been already exploited for studying dynamical systems [34, 38], polylogarithms [3941], polyzetas [37] and Drinfel’d associators [35]. We will give, in the future, an unified algebraic framework to treat all these shuffle products and their applications (see [19] for example).

  6. For the multidegree.

  7. I.e., a basis of polynomials indexed by their initial words which is a Lyndon word.

  8. Because \(\mathcal{H}\) is graded in finite dimensions by the multidegree.

  9. Recall that the duality preserves the (multi)-homogeneous degree and interchanges the triangularity of polynomials [48]. For that, one can construct the triangular matrices M and N admitting coefficients as the coefficients of the homogeneous of degree k triangular polynomials, \(\{{\mathrm{P}}_{w}\}_{w\in X^{*}}\) and \(\{S_{w}\}_{w\in X^{*}}\) in the basis \(\{w\}_{w\in X^{*}}\), respectively:

    $$\begin{aligned} M_{u,v}=\langle P_u\mid v \rangle \quad\mbox{and}\quad N_{v,u}=\langle S_u\mid v \rangle . \end{aligned}$$

    The triangular matrices M and N are unipotent and satisfy the identity N=(t M)−1.

  10. For any \(S\in{{\mathbb{Q}} \langle Y \rangle}\hat{\otimes}{{\mathbb{Q}} \langle Y \rangle}\), if \(\langle S\mid 1_{Y^{*}}\otimes1_{Y^{*}} \rangle =0\) then one defines

    $$\begin{aligned} \log(1_{Y^*}+S)=\sum_{n\ge1} \frac{(-1)^{n-1}}{n}S^n\quad\mathrm{and} \quad\exp(S)=\sum _{n\ge1}\frac{S^n}{n!} \end{aligned}$$

    and one has usual formulas log(exp(S))=S and \(\exp(\log (1_{Y^{*}}+S))=1_{Y^{*}}+S\).

  11. Such algebra isomorphism α certainly exists as shown by P (in Theorem 3) and H (in Theorem 4) below. But to be definite at this point, the reader can also take the alphabet duplication isomorphism

    $$\forall\bar{y}\in\bar{Y},\quad\bar{y}=\alpha(y) $$

    and use \(\{w\}_{w\in\bar{Y}^{*}}\) as a basis for \({\mathbb{Q}}\langle\bar{Y} \rangle \).

  12. Due to the fact this Hopf algebra is co-commutative and \({\mathbb{N}}\)-graded, then by the theorem of CQMM, \({{\mathbb{Q}} \langle Y \rangle}\simeq \mathcal{U}(\mathcal{P})\).

  13. The duality preserves the homogeneous degree and interchanges the triangularity of polynomials (see Note 9 for the same construction of the triangular matrices of coefficients).

  14. The duality preserves the homogeneous degree and interchanges the triangularity of polynomials (see Note 9 for the same construction of the triangular matrices of coefficients).

  15. This result is an analogue of a theorem by Radford (see [48]). Thus the bases \({\mathcal{L}\mathit{yn}}Y\) and \(\{\varSigma _{l}\}_{l\in{\mathcal{L}\mathit{yn}}Y}\) belong to the class of Radford bases, i.e., the class of transcendence bases, of the quasi-shuffle algebra, as well as the bases \({\mathcal{L}\mathit{yn}}X\) and \(\{P_{l}\}_{l\in{\mathcal {L}\mathit{yn}}X}\) belong to the class of Radford bases of the shuffle algebra.

  16. z 0z is a path of the domain \(\varOmega=\widetilde{{\mathbb{C}}-\{0,1\}}\) (see below) fixed once for all.

  17. Here, π Y is extended, term by term, to a series.

  18. Note that the words x 1 x 0 x 1 and y 1 y 2 are divergent.

  19. Here, the identities ζ(2,1,1,1)=ζ(5),ζ(3,1,1)=ζ(4,1),ζ(2,2,1)=ζ(3,2) can be obtained by duality in the meaning of [25].

  20. As previously, the identities ζ(2,1,1,1,1)=ζ(6),ζ(3,1,1,1)=ζ(5,1),ζ(2,2,1,1)=ζ(4,2) can be obtained also by duality.

  21. Up to weight 10, these polynomial relations are verified numerically by Blondel using the computer program EZface (http://oldweb.cecm.sfu.ca/projects/EZFace/).

  22. Let \({\mathcal{A}}\) be the alphabet {y 2,y 3} with y 2>y 3. For any \(w\in{\mathcal{A}}^{*}\) of weight 16, it is verified, by El Wardi [52], ζ(w) is polynomial on these irreducible polyzetas and, conversely, any polyzeta admits a decomposition on the family of generators \(\{\zeta (w)\}_{w\in{\mathcal{A}}^{*}}\) by using the facts

    • except y 2,y 3 being of weight 2,3, respectively,, any Lyndon words \(y_{s_{1}}\cdots y_{s_{n}}\) belong to \(y_{2}{\mathcal{A}}^{*}y_{3}\) and are of weight s 1+⋯+s n ≥5 meaning there are no Lyndon words of weight 1,4 or 6 (i.e., i 1=i 4=i 6=0),

    • the numbers of words and Lyndon words of length n over \({\mathcal{A}}\) are, respectively, 2n and 2(n) satisfying, via a Witt’s formula,

      $$\begin{aligned} \ell_2(n)=\frac{1}{n}\sum_{d|n} \mu(d)2^{n/d}\le2^n, \end{aligned}$$

      where μ denotes the Möbius’ function (see [50]).

    The existence of this family was conjectured firstly by Hoffman and proved by Brown [6].

  23. Table [52] offers i 1=0, i 2=1, i 3=1, i 4=0, i 5=1, i 6=0, i 7=1, i 8=1, i 9=1, i 10=1, i 11=2, i 12=2, i 13=3, i 14=3, i 15=4, i 16=5.

  24. Hence, in the power series , only convergent polyzetas (Definition 5) appear. This series corresponds to the expected Drinfel’d associator Φ KZ [39] (see Theorem 10 to perform the other associators).

  25. Since the coefficient of z N in the ordinary Taylor expansion of \({\mathrm{P}}_{y_{1}^{k}}\) is \(\mathrm{H}_{y_{1}^{k}}(N)\).

  26. Here, the coefficient \(\langle B(y_{1})\mid y_{1}^{k} \rangle \) corresponds to the Euler–Mac Laurin constant associated to \(\langle\mathrm{Const}(N)\mid y_{1}^{k} \rangle \) [24], i.e., the finite part of its asymptotic expansion in the scale of comparison \(\{n^{a}\log^{b}(n)\}_{a\in{\mathbb{Z}},b\in{\mathbb{N}}}\).

  27. In fact, these constants \(\{\gamma_{w}\}_{w \in Y^{*}}\) are the Euler–Mac Laurin constants associated to \(\{\mathrm{H}_{w}\}_{w \in Y^{*}}\) [24]. They are then a generalization of the Euler γ constant.

  28. For example, \(A={\mathbb{Q}}(2\mathrm{}i\pi)\).

  29. See Theorem 7.

  30. The noncommutative generating series Z γ does not correspond to the mould of regularized polyzetas leading to the formal polyzetas (see [5, 18, 20, 28]).

  31. The power series B′(y 1) corresponds in fact to the mould Mono in [18] and to the \(\varPhi_{\rm corr}\) in [20] (see also [5, 7]). While the power series B(y 1) corresponds indeed to the Gamma Euler function with its famous product expansion arising the constant γ,

    $$\begin{aligned} B(y_1)=\varGamma(y_1+1),\quad \frac{1}{\varGamma(y_1+1)}=e^{\gamma y_1} \prod_{n\ge1} \biggl(1+\frac{y_1}{n} \biggr)e^{-\gamma/n}. \end{aligned}$$
  32. This explains a useful idea of the regularization making “+∞−∞=0”.

  33. Note that in the left sums, there are regularized polyzetas and in the right products, all polyzetas are convergent. The couple of characters and corresponds to the pair of regularization morphisms sending and to the finite parts of the asymptotic expansions of \(\operatorname{Li}_{w}(z)\) and H w (n) in the scales of comparison \(\{(1-z)^{a}\log^{b}(1-z)\}_{a\in{\mathbb{Z}},b\in{\mathbb{N}}}\) and \(\{n^{a}\mathrm{H}_{1}^{b}(n)\}_{a\in{\mathbb{Z}},b\in{\mathbb{N}}}\), respectively, i.e., to 0 [42].

  34. In the first identity, involve the “0-regularized” polyzetas while the last one indicates how to get the finite parts of the asymptotic expansions of H w (n) in \(\{n^{a}\log _{1}^{b}(n)\}_{a\in{\mathbb{Z}},b\in{\mathbb{N}}}\), i.e., the separateγ-regularized” polyzetas, for the quasi-shuffle product.

  35. One can, of course, decompose the lower triangular and homogeneous polynomials \(\{\pi _{Y}P_{l}-\varPi_{\pi_{Y}l}\}_{l\in{\mathcal{L}\mathit{yn}}X-X}\) (which are still lower triangular and homogeneous) in PBW-Lyndon basis.

  36. Or equivalently, identifying the coefficients of \(\{\varPi_{l}\}_{l\in{\mathcal {L}\mathit{yn}}Y-y_{1}}\) in Corollary 8(1) involved the regularized polyzetas while only convergent polyzetas arise in Corollary 8(2).

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Acknowledgements

I would like to acknowledge the influence of the lectures of Pierre Cartier at the Groupe de travail “Polylogarithmes et nombres zêta multiples” and the fruitful discussions with Michel Waldschmidt on Corollaries 13 and 17. More particularly, let me thank Gérard H.E. Duchamp for his advice and kindness to improve this writing.

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A Pierre Cartier, pour ses 80 ans.

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Minh, H.N. Structure of Polyzetas and Lyndon Words. Viet J Math 41, 409–450 (2013). https://doi.org/10.1007/s10013-013-0047-x

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Keywords

  • Algebraic computation
  • Combinatorial Hopf algebra
  • Drinfel’d associators
  • Free Lie algebra
  • Noncommutative symbolic computation
  • Polylogarithm
  • Polyzeta
  • Renormalization
  • Regularization
  • Special functions
  • Transcendence basis

Mathematics Subject Classification (2000)

  • 05E
  • 11M32
  • 37F25
  • 68W30