pith. sign in

arxiv: 2305.00789 · v2 · pith:Q5EWWNSSnew · submitted 2023-05-01 · 🧮 math.AG · math.KT· math.NT

A construction of the polylogarithm motive

Cl\'ement Dupont , Javier Fres\'an This is my paper

Pith reviewed 2026-05-24 08:31 UTC · model grok-4.3

classification 🧮 math.AG math.KTmath.NT
keywords polylogarithm motivemixed Tate motivesrelative cohomologyhypersurface complementaffine spaceKummer variationextension classHodge structures
0
0 comments X

The pith

The polylogarithm motive is the relative cohomology motive of the complement of the hypersurface {1 - z t1 ⋯ tn = 0} in A^n_S relative to the union of the hyperplanes {ti=0} and {ti=1}.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper supplies an explicit geometric model for the polylogarithm motive over the base S equal to the projective line minus the points 0, 1 and infinity. Classical polylogarithms determine a variation of mixed Hodge-Tate structures that extends a symmetric power of the Kummer variation by a trivial variation. Earlier results established the existence of a lift of this variation to mixed Tate motives by equating extension spaces in the motivic and Hodge settings. The construction realizes the motive directly as the relative cohomology of the indicated hypersurface complement in affine space. This turns an abstract existence statement into a concrete geometric object that can be examined by standard methods of algebraic geometry.

Core claim

The polylogarithm motive is constructed as the relative cohomology motive of the complement of the hypersurface {1−z t1⋯tn=0} in A^n_S relative to the union of the hyperplanes {ti=0} and {ti=1}. This explicit model realizes the known extension class in the category of mixed Tate motives over S.

What carries the argument

The relative cohomology motive of the pair (complement of the hypersurface {1 - z ∏ ti = 0} in affine n-space over S, union of the divisors ti = 0 and ti = 1).

Load-bearing premise

The category of mixed Tate motives over S exists and the comparison theorems correctly identify the extension spaces in the motivic and Hodge realizations.

What would settle it

A computation showing that the extension class realized by this relative cohomology motive fails to match the classical polylogarithm extension class in the Hodge realization would refute the identification.

Figures

Figures reproduced from arXiv: 2305.00789 by Cl\'ement Dupont, Javier Fres\'an.

Figure 1
Figure 1. Figure 1: The hypersurfaces A2 and B2 in the affine plane X2. rational coefficients over S as in the case where the base is a number field, explained by Levine [Lev93]. Inspired by the constructions of Wildeshaus and Huber–Wildeshaus in the Hodge and the ℓ-adic settings [Wil97, HW98], Ayoub [Ayo04] defined a polylogarithm motive as an ind-object of MT(S). The idea is to compute the extension group Ext1 Ind(MT(S)) (S… view at source ↗
Figure 2
Figure 2. Figure 2: In A2 , the blow-up of the origin separates the boundary of the integration simplex {0 ⩽ x1 ⩽ x2 ⩽ 1} from the pole divisor {(1 − zx1)x2 = 0}. By removing the strict transform of {x2 = 0} one recovers the geometry of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
read the original abstract

Classical polylogarithms give rise to a variation of mixed Hodge-Tate structures on the punctured projective line $S=\mathbb{P}^1\setminus \{0, 1, \infty\}$, which is an extension of the symmetric power of the Kummer variation by a trivial variation. By results of Beilinson-Deligne, Huber-Wildeshaus, and Ayoub, this polylogarithm variation has a lift to the category of mixed Tate motives over $S$, whose existence is proved by computing the corresponding space of extensions in both the motivic and the Hodge settings. In this paper, we construct the polylogarithm motive as an explicit relative cohomology motive, namely that of the complement of the hypersurface $\{1-zt_1\cdots t_n=0\}$ in affine space $\mathbb{A}^n_S$ relative to the union of the hyperplanes $\{t_i=0\}$ and $\{t_i=1\}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper constructs the polylogarithm motive explicitly as the relative cohomology motive of the complement of the hypersurface {1−z t1⋯tn=0} in A^n_S relative to the union of the hyperplanes {ti=0} and {ti=1}, where S=P^1∖{0,1,∞}. This is presented as realizing the extension of Sym^n of the Kummer variation by the trivial variation in the category of mixed Tate motives over S, using prior comparison results of Beilinson-Deligne, Huber-Wildeshaus, and Ayoub to equate the motivic and Hodge extension spaces.

Significance. If the identification is established, the explicit geometric model supplies a concrete object in the mixed Tate category whose extension class can be studied directly via relative cohomology, complementing the abstract existence proofs via extension-space computations. This could support explicit calculations of motivic polylogarithms and their realizations.

major comments (2)
  1. [Abstract] The central identification requires showing that the relative cohomology motive lies in the mixed Tate category over S and that its class in the motivic Ext^1 matches the polylogarithm generator; the abstract invokes the comparison theorems but supplies no derivation or verification steps for this particular geometric object.
  2. [Abstract] The construction is defined geometrically rather than via the extension space it is claimed to realize; without an explicit computation of the extension class (or a proof that the motive is mixed Tate), the claim that this object is the polylogarithm motive rests on the cited comparison isomorphisms holding for this specific case.
minor comments (1)
  1. Notation for the base S, the hypersurface, and the relative cohomology groups should be introduced with precise definitions early in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and the recommendation for major revision. The comments correctly note that the abstract relies on the cited comparison theorems without spelling out their application to this specific geometric object. Below we address each point. We agree that the abstract can be clarified and will make a partial revision to improve the exposition of the identification.

read point-by-point responses

  1. Referee: [Abstract] The central identification requires showing that the relative cohomology motive lies in the mixed Tate category over S and that its class in the motivic Ext^1 matches the polylogarithm generator; the abstract invokes the comparison theorems but supplies no derivation or verification steps for this particular geometric object.

    Authors: The comparison theorems of Beilinson-Deligne, Huber-Wildeshaus and Ayoub are stated for the category of mixed Tate motives over S and equate the relevant Ext^1 groups in the motivic and Hodge realizations. The geometric object is constructed as a relative cohomology motive whose weight filtration and graded pieces are manifestly mixed Tate (by the standard properties of relative cohomology with respect to the given hyperplanes and hypersurface). Because the polylogarithm generator is characterized precisely as the unique nontrivial class in the corresponding Ext^1, the identification follows directly once the object is shown to lie in the category; no separate computation of the class is required beyond the general isomorphism. We will revise the abstract to state this application of the theorems explicitly. revision: partial

  2. Referee: [Abstract] The construction is defined geometrically rather than via the extension space it is claimed to realize; without an explicit computation of the extension class (or a proof that the motive is mixed Tate), the claim that this object is the polylogarithm motive rests on the cited comparison isomorphisms holding for this specific case.

    Authors: The manuscript defines the object geometrically precisely to supply a concrete model inside the mixed Tate category whose extension class can then be studied by direct geometric means. The proof that it is mixed Tate is contained in the construction itself (the relative cohomology is filtered by the strata and the graded pieces are Tate by the Kummer and trivial variations). The comparison isomorphisms are functorial and apply to any object in the category over S; the paper invokes them in this standard way. An explicit cocycle-level computation of the class is not performed because the abstract characterization via Ext^1 already determines the generator uniquely. If the referee believes a direct cocycle computation would be valuable, we can discuss adding it, but it is not needed for the stated claim. revision: no

Circularity Check

0 steps flagged

Explicit geometric construction independent of extension-space definition

full rationale

The paper defines the polylogarithm motive directly as the relative cohomology motive of the complement of {1−z t1⋯tn=0} in A^n_S relative to the ti=0 and ti=1 hyperplanes. This geometric object is not defined in terms of the motivic Ext class it is claimed to realize. Identification with the classical polylogarithm variation rests on external comparison theorems of Beilinson-Deligne, Huber-Wildeshaus and Ayoub (cited for existence of the mixed Tate category and equality of extension spaces); these are not self-citations by the present authors and constitute independent support. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citation chains appear. The derivation therefore remains self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction relies on the existence of the category of mixed Tate motives and on the comparison isomorphisms between motivic and Hodge realizations that were established in the cited prior works; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The category of mixed Tate motives over S exists and the extension spaces computed by Beilinson-Deligne, Huber-Wildeshaus, and Ayoub correctly classify the polylogarithm variation.
    The abstract invokes these results to assert that the geometric object realizes the previously known motive.

pith-pipeline@v0.9.0 · 5694 in / 1426 out tokens · 19503 ms · 2026-05-24T08:31:12.114780+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On a relation of a conjecture of Goncharov to the co-Lie algebra of Bloch-Kriz mixed Tate motives

    math.AG 2026-04 unverdicted novelty 6.0

    A possible linear map from Goncharov's B_n(F) to the co-Lie algebra of mixed Tate motives is considered via motivic polylogarithms, supported under Beilinson-Soulé K-group vanishing assumptions.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Arnol'd, The cohomology ring of the group of dyed braids, Mat.\ Zametki 5 (1969), 227--231

    V.\,I. Arnol'd, The cohomology ring of the group of dyed braids, Mat.\ Zametki 5 (1969), 227--231

    work page 1969

  2. [2]

    Ayoub, Motivic version of the classical polylogarithms, in: Polylogarithms, pp

    J. Ayoub, Motivic version of the classical polylogarithms, in: Polylogarithms, pp. 2563--2565, Oberwolfach Rep.\ 1 (2004), no. 4, doi:10.4171/owr/2004/48

    work page doi:10.4171/owr/2004/48 2004

  3. [3]

    I , Ast \'e risque 314 (2007)

    , Les six op \'e rations de G rothendieck et le formalisme des cycles \'e vanescents dans le monde motivique. I , Ast \'e risque 314 (2007)

    work page 2007

  4. [4]

    II , Ast \'e risque 315 (2007)

    , Les six op \'e rations de G rothendieck et le formalisme des cycles \'e vanescents dans le monde motivique. II , Ast \'e risque 315 (2007)

    work page 2007

  5. [5]

    Ball and T

    K. Ball and T. Rivoal, Irrationalit \'e d'une infinit \'e de valeurs de la fonction z\^eta aux entiers impairs , Invent.\ Math.\ 146 (2001), no. 1, 193--207, doi:10.1007/s002220100168

    work page doi:10.1007/s002220100168 2001

  6. [6]

    Barcelo, On the action of the symmetric group on the free L ie algebra and the partition lattice , J

    H. Barcelo, On the action of the symmetric group on the free L ie algebra and the partition lattice , J. Combin.\ Theory Ser. A 55 (1990), no. 1, 93--129, doi:10.1016/0097-3165(90)90050-7

    work page doi:10.1016/0097-3165(90)90050-7 1990

  7. [7]

    Beilinson, J

    A. Beilinson, J. Bernstein, P. Deligne, and O. Gabber, Faisceaux pervers, Ast \'e risque 100 (2018), doi:10.24033/ast.1042

    work page doi:10.24033/ast.1042 2018

  8. [8]

    Beilinson and P

    A. Beilinson and P. Deligne, Interpr \'e tation motivique de la conjecture de Z agier reliant polylogarithmes et r \'e gulateurs , in: Motives ( S eattle, WA , 1991), pp. 97--121, Proc.\ Sympos.\ Pure Math., vol. 55, Part 2, Amer.\ Math.\ Soc., Providence, RI, 1994, doi:10.1090/pspum/055.2/1265552

    work page doi:10.1090/pspum/055.2/1265552 1991

  9. [9]

    Beilinson and A

    A. Beilinson and A. Levin, The elliptic polylogarithm, in: Motives ( S eattle, WA , 1991), pp. 123--190, Proc.\ Sympos.\ Pure Math., vol. 55, P art 2, Amer.\ Math.\ Soc., Providence, RI, 1994, doi:10.1090/pspum/055.2/1265553

    work page doi:10.1090/pspum/055.2/1265553 1991

  10. [10]

    Brandt, The free L ie ring and L ie representations of the full linear group , Trans.\ Amer.\ Math.\ Soc.\ 56 (1944), 528--536, doi:10.2307/1990324

    A.\,J. Brandt, The free L ie ring and L ie representations of the full linear group , Trans.\ Amer.\ Math.\ Soc.\ 56 (1944), 528--536, doi:10.2307/1990324

    work page doi:10.2307/1990324 1944

  11. [11]

    Cisinski and F

    D.-C. Cisinski and F. D\' e glise, Triangulated categories of mixed motives, Springer Monogr.\ Math., Springer, Cham, 2019, doi:10.1007/978-3-030-33242-6

    work page doi:10.1007/978-3-030-33242-6 2019

  12. [12]

    Cohen, The homology of C _ n+1 -spaces, n 0 , in: The Homology of Iterated Loop Spaces, pp

    F.\,R. Cohen, The homology of C _ n+1 -spaces, n 0 , in: The Homology of Iterated Loop Spaces, pp. 207--351, Lecture Notes in Math., vol. 533, Springer, Berlin, Heidelberg, 1976, doi:10.1007/BFb0080467

    work page doi:10.1007/bfb0080467 1976

  13. [13]

    Pure Appl.\ Algebra 100 (1995), no

    , On configuration spaces, their homology, and L ie algebras , J. Pure Appl.\ Algebra 100 (1995), no. 1-3, 19--42, doi:10.1016/0022-4049(95)00054-Z

    work page doi:10.1016/0022-4049(95)00054-z 1995

  14. [14]

    Deligne, Unpublished letter to Spencer Bloch, April 3, 1984

    P. Deligne, Unpublished letter to Spencer Bloch, April 3, 1984

    work page 1984

  15. [15]

    79--297, Math.\ Sci.\ Res.\ Inst.\ Publ., vol

    , Le groupe fondamental de la droite projective moins trois points, in: Galois groups over Q (Berkeley, CA, 1987), pp. 79--297, Math.\ Sci.\ Res.\ Inst.\ Publ., vol. 16, Springer-Verlag, New York, 1989, doi:10.1007/978-1-4613-9649-9_3

    work page doi:10.1007/978-1-4613-9649-9_3 1987

  16. [16]

    , Unpublished letter to Alexander Beilinson, February 16, 2001

    work page 2001

  17. [17]

    Deligne and A.\,B

    P. Deligne and A.\,B. Goncharov, Groupes fondamentaux motiviques de T ate mixte , Ann.\ Sci.\ \'E cole Norm.\ Sup.\ (4) 38 (2005), no. 1, 1--56, doi:10.1016/j.ansens.2004.11.001

    work page doi:10.1016/j.ansens.2004.11.001 2005

  18. [18]

    Dupont, Odd zeta motive and linear forms in odd zeta values, (with a joint appendix with D

    C. Dupont, Odd zeta motive and linear forms in odd zeta values, (with a joint appendix with D. Zagier), Compos.\ Math.\ 154 (2018), no. 2, 342--379, doi:10.1112/S0010437X17007588

    work page doi:10.1112/s0010437x17007588 2018

  19. [19]

    , Progr \'e s r \'e cents sur la conjecture de Z agier et le programme de G oncharov [d'apr\' e s G oncharov, R udenko, G angl, ] , in: S\' e min.\ Bourbaki. Vol. 2019/2021. Expos\'es 1166--1180 , Exp.\ No. 1176, pp. 295–343, Ast\'erisque 430 (2021), doi:10.24033/ast.1165

    work page doi:10.24033/ast.1165 2019

  20. [20]

    Dupont and D

    C. Dupont and D. Juteau, The localization spectral sequence in the motivic setting, Algebr.\ Geom.\ Topol.\ 24 (2024), no. 3, 1431--1466, doi:10.2140/agt.2024.24.1431

    work page doi:10.2140/agt.2024.24.1431 2024

  21. [21]

    Tate motives and the fundamental group

    H. Esnault and M. Levine, T ate motives and the fundamental group , preprint arXiv:0708.4034 (2007)

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  22. [22]

    Getzler, Resolving mixed H odge modules on configuration spaces , Duke Math

    E. Getzler, Resolving mixed H odge modules on configuration spaces , Duke Math. J.\ 96 (1999), no. 1, 175--203, doi:10.1215/S0012-7094-99-09605-9

    work page doi:10.1215/s0012-7094-99-09605-9 1999

  23. [23]

    Griffiths and J

    P. Griffiths and J. Harris, Principles of algebraic geometry (reprint of the 1978 original), Wiley Classics Lib., John Wiley & Sons, Inc., New York, 1994, doi:10.1002/9781118032527

    work page doi:10.1002/9781118032527 1978

  24. [24]

    u rich, 1994), pp. 374--387, Birkh \

    A.\,B. Goncharov, Polylogarithms in arithmetic and geometry, in: Proceedings of the I nternational C ongress of M athematicians, V ol.\ 1, 2 ( Z \"u rich, 1994), pp. 374--387, Birkh \"a user Verlag, Basel, 1995, doi:10.1007/978-3-0348-9078-6_31

    work page doi:10.1007/978-3-0348-9078-6_31 1994

  25. [25]

    , Multiple polylogarithms and mixed T ate motives , preprint arXiv:0103059 (2001)

    work page 2001

  26. [26]

    Goncharov and Yu.\,I

    A.\,B. Goncharov and Yu.\,I. Manin, Multiple -motives and moduli spaces M _ 0,n , Compos.\ Math.\ 140 (2004), no. 1, 1--14, doi:10.1112/S0010437X03000125

    work page doi:10.1112/s0010437x03000125 2004

  27. [27]

    Hain, Classical polylogarithms, in: Motives ( S eattle, WA , 1991), pp

    R.\,M. Hain, Classical polylogarithms, in: Motives ( S eattle, WA , 1991), pp. 3--42, Proc.\ Sympos.\ Pure Math., vol. 55, Part 2, Amer.\ Math.\ Soc., Providence, RI, 1994, doi:10.1090/pspum/055.2/1265551

    work page doi:10.1090/pspum/055.2/1265551 1991

  28. [28]

    Hanlon, The fixed-point partition lattices, Pacific J

    P. Hanlon, The fixed-point partition lattices, Pacific J. Math.\ 96 (1981), no. 2, 319--341, doi:10.2140/pjm.1981.96.319

    work page doi:10.2140/pjm.1981.96.319 1981

  29. [29]

    Hanlon and M

    P. Hanlon and M. Wachs, On L ie k -algebras , Adv.\ Math.\ 113 (1995), no. 2, 206--236, doi:10.1006/aima.1995.1038

    work page doi:10.1006/aima.1995.1038 1995

  30. [30]

    Hatcher, Algebraic topology, Cambridge Univ.\ Press, Cambridge, 2002

    A. Hatcher, Algebraic topology, Cambridge Univ.\ Press, Cambridge, 2002

    work page 2002

  31. [31]

    Huber and G

    A. Huber and G. Kings, Polylogarithm for families of commutative group schemes, J. Algebraic Geom.\ 27 (2018), no. 3, 449--495, doi:10.1090/jag/717

    work page doi:10.1090/jag/717 2018

  32. [32]

    Huber and J

    A. Huber and J. Wildeshaus, Classical motivic polylogarithm according to B eilinson and D eligne , Doc.\ Math.\ 3 (1998), 27--133, doi:10.4171/dm/37-5

    work page doi:10.4171/dm/37-5 1998

  33. [33]

    Ivorra and S

    F. Ivorra and S. Morel, The four operations on perverse motives , J. Eur.\ Math.\ Soc.\ (JEMS) 26 (2024), no. 11, 4191--4272, doi:10.4171/jems/1402

    work page doi:10.4171/jems/1402 2024

  34. [34]

    Joyal, Foncteurs analytiques et esp \'e ces de structures , in: Combinatoire \'e num \'e rative (Montreal, Que., 1985/Quebec, Que., 1985), pp

    A. Joyal, Foncteurs analytiques et esp \'e ces de structures , in: Combinatoire \'e num \'e rative (Montreal, Que., 1985/Quebec, Que., 1985), pp. 126--159, Lecture Notes in Math., vol. 1234, Springer-Verlag, Berlin, 1986, doi:10.1007/BFb0072514

    work page doi:10.1007/bfb0072514 1985

  35. [35]

    Jin and E

    F. Jin and E. Yang, K\" u nneth formulas for motives and additivity of traces , Adv.\ Math.\ 376 (2021), Paper No. 107446, doi:10.1016/j.aim.2020.107446

    work page doi:10.1016/j.aim.2020.107446 2021

  36. [36]

    Klyachko, Lie elements in a tensor algebra, Sibirsk.\ Mat.\ Z

    A.\,A. Klyachko, Lie elements in a tensor algebra, Sibirsk.\ Mat.\ Z . 15 (1974), 1296--1304, 1430

    work page 1974

  37. [37]

    Levine, Tate motives and the vanishing conjectures for algebraic K -theory , in: Algebraic K -theory and algebraic topology ( L ake L ouise, AB , 1991), pp

    M. Levine, Tate motives and the vanishing conjectures for algebraic K -theory , in: Algebraic K -theory and algebraic topology ( L ake L ouise, AB , 1991), pp. 167--188, NATO Adv.\ Sci.\ Inst.\ Ser. C: Math.\ Phys.\ Sci., vol. 407, Kluwer Acad. Publ.\ Group, Dordrecht, 1993, doi:10.1007/978-94-017-0695-7_7

    work page doi:10.1007/978-94-017-0695-7_7 1991

  38. [38]

    , Motivic tubular neighborhoods, Doc.\ Math.\ 12 (2007), 71--146, doi:10.4171/dm/221

    work page doi:10.4171/dm/221 2007

  39. [39]

    Orlik and L

    P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent.\ Math.\ 56 (1980), no. 2, 167--189, doi:10.1007/BF01392549

    work page doi:10.1007/bf01392549 1980

  40. [40]

    Petersen, A spectral sequence for stratified spaces and configuration spaces of points, Geom.\ Topol.\ 21 (2017), no

    D. Petersen, A spectral sequence for stratified spaces and configuration spaces of points, Geom.\ Topol.\ 21 (2017), no. 4, 2527--2555, doi:10.2140/gt.2017.21.2527

    work page doi:10.2140/gt.2017.21.2527 2017

  41. [41]

    Ramakrishnan, On the monodromy of higher logarithms, Proc.\ Amer.\ Math.\ Soc.\ 85 (1982), no

    D. Ramakrishnan, On the monodromy of higher logarithms, Proc.\ Amer.\ Math.\ Soc.\ 85 (1982), no. 4, 596--599, doi:10.2307/2044073

    work page doi:10.2307/2044073 1982

  42. [42]

    183--310, Contemp.\ Math., vol

    , Regulators, algebraic cycles, and values of L -functions , in: Algebraic K -theory and algebraic number theory (Honolulu, HI, 1987), pp. 183--310, Contemp.\ Math., vol. 83, Amer.\ Math.\ Soc., Provindence, RI, 1989, doi:10.1090/conm/083/991982

    work page doi:10.1090/conm/083/991982 1987

  43. [43]

    Rivoal, La fonction z\^eta de R iemann prend une infinit \'e de valeurs irrationnelles aux entiers impairs , C

    T. Rivoal, La fonction z\^eta de R iemann prend une infinit \'e de valeurs irrationnelles aux entiers impairs , C. R. Acad.\ Sci.\ Paris S \'e r. I Math.\ 331 (2000), no. 4, 267--270, doi:10.1016/S0764-4442(00)01624-4

    work page doi:10.1016/s0764-4442(00)01624-4 2000

  44. [44]

    Stanley, Some aspects of groups acting on finite posets, J

    R.\,P. Stanley, Some aspects of groups acting on finite posets, J. Combin.\ Theory Ser. A 32 (1982), no. 2, 132--161, doi:10.1016/0097-3165(82)90017-6

    work page doi:10.1016/0097-3165(82)90017-6 1982

  45. [45]

    Voevodsky, Triangulated categories of motives over a field, in: Cycles, transfers, and motivic homology theories, pp

    V. Voevodsky, Triangulated categories of motives over a field, in: Cycles, transfers, and motivic homology theories, pp. 188--238, Ann.\ of Math.\ Stud., vol. 143, Princeton Univ.\ Press, Princeton, NJ, 2000, doi:10.1515/9781400837120.188

    work page doi:10.1515/9781400837120.188 2000

  46. [46]

    Wang, Moduli spaces and multiple polylogarithm motives, Adv.\ Math.\ 206 (2006), no

    Q. Wang, Moduli spaces and multiple polylogarithm motives, Adv.\ Math.\ 206 (2006), no. 2, 329--357, doi:10.1016/j.aim.2005.09.002

    work page doi:10.1016/j.aim.2005.09.002 2006

  47. [47]

    Wildeshaus, Realizations of polylogarithms, Lecture Notes in Math., vol

    J. Wildeshaus, Realizations of polylogarithms, Lecture Notes in Math., vol. 1650, Springer-Verlag, Berlin, 1997, doi:10.1007/BFb0093051

    work page doi:10.1007/bfb0093051 1997

  48. [48]

    Zagier, Polylogarithms, D edekind zeta functions and the algebraic K -theory of fields , in: Arithmetic algebraic geometry ( T exel, 1989), pp

    D. Zagier, Polylogarithms, D edekind zeta functions and the algebraic K -theory of fields , in: Arithmetic algebraic geometry ( T exel, 1989), pp. 391--430, Progr.\ Math., vol. 89, Birkh \"a user Boston, Inc., Boston, MA, 1991, doi:10.1007/978-1-4612-0457-2_19

    work page doi:10.1007/978-1-4612-0457-2_19 1989