On generic double shuffle relations, localized multiple polylogarithms and algebraic functions

David Jarossay

REVIEW 4 major objections 5 minor 14 references

On generic double shuffle relations, localized multiple polylogarithms and algebraic functions

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper constructs explicit closed affine subvarieties Y^{(N)}_{0,n+3} of M_{0,n+3} equipped with algebraic functions that satisfy the generic double shuffle equations of multiple polylogarithms.

desk verdict The derivative computations are genuine, but the main theorem is a definition posed as a result—Y is defined by the equation it is supposed to solve, and nonemptiness is never established. read the letter →

arxiv 1908.01410 v1 pith:NHYBYCZM submitted 2019-08-04 math.NT

David Jarossay This is my paper · ORCID

classification math.NT MSC 11M3211G5514H10

keywords multiplepolylogarithmsgenericdoubleshufflerelationslocalizedalgebraicfunctionsmodulispaceM_{0n}zetavaluesiteratedintegralsChen'stheorem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Multiple polylogarithms are built from iterated integrals over the spaces M_{0,n+3} of configurations of distinct points on the projective line, and they satisfy two multiplication laws known as the generic double shuffle equations. This paper claims that those same equations can be solved by ordinary rational functions, after restricting to explicit subvarieties of M_{0,n+3}. For each positive integer N and each n at least 0, it constructs a closed affine subvariety $Y^{{(N)}}$_{0,n+3} and shows that on it the rational pieces extracted from N-th derivatives of multiple polylogarithms obey the same shuffle and quasi-shuffle products as the original functions. The interest is that relations normally visible only through transcendental functions are thus realized as purely algebraic identities, a step toward a geometric understanding of the algebraic relations among multiple zeta values.

What carries the argument

The load-bearing object is the rational fraction $I^{{rat}}$_N(a_0; a_1,\ldots,a_n; a_{n+1}), defined as the coefficient of the constant iterated integral in the expansion of \partial^N_{z_1}\cdots\partial^N_{z_n} applied to a multiple polylogarithm. The expansion is produced by a recursion that expresses each partial derivative as a \mathbb{Q}(z_1,\ldots,z_n)-linear combination of iterated integrals with simpler index sets, so localized multiple polylogarithms are built from the same data as the original functions. The transfer from polylogarithms to rational functions rests on the linear independence of iterated integrals over the ring of algebraic functions: once the generic double shuffle relations are differentiated, their rational parts must satisfy the same shuffle and quasi-shuffle laws. Finally, $Y^{{(N)}}$_{0,n+3} is the closed affine subvariety cut out by equating this derived rational law with the generic one.

What would settle it

For the smallest nontrivial case (N=1, two variables), the defining equations of $Y^{{(1)}}$_{0,5} are the explicit rational identities displayed in Example 3.3 (shuffle) and Example 3.6 (quasi-shuffle). Solving this system, for instance fixing z_0=0 and z_1=1 and searching for distinct a,b in $P^{1}$ minus {0,1,infinity}, would settle nonemptiness: one solution proves the variety is nonempty, while a Groebner basis computation reducing the system to a contradiction proves it is empty.

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Extended reading notes

Core claim

The paper's central claim is that there exists an infinite sequence of explicit closed affine subvarieties $Y^{{(N)}}$_{0,n+3} of M_{0,n+3}, parametrized by N in the positive integers and n in the nonnegative integers, each equipped with global algebraic functions that are solutions to the generic double shuffle equations. The functions are constructed by applying N partial derivatives in every coordinate to a multiple polylogarithm, expanding the result as a linear combination of iterated integrals with rational-function coefficients, and keeping the coefficient of the empty iterated integral; this coefficient is denoted $I^{{rat}}$_N. The classical theorem that iterated integrals are linearly independent over the ring of algebraic functions converts the double shuffle relations satisfied by multiple polylogarithms into equalities among these rational functions. The subvariety Y is then defined as the locus where this rational variant of the double shuffle relation agrees with the generic double shuffle relation, and on that locus $I^{{rat}}$_N gives the desired algebraic functions.

Load-bearing premise

The construction defines $Y^{{(N)}}$_{0,n+3} as the locus where two explicitly written rational double-shuffle identities coincide, but the paper does not prove that this locus contains any point; if the defining equations are inconsistent, the claimed algebraic functions exist on an empty set.

Editorial extensions

If this is right

For every pair (N,n) the construction yields an explicit closed affine subvariety Y^{(N)}_{0,n+3} of M_{0,n+3}; these form infinitely many such subvarieties, one for each depth n and each order N of derivation.
On each such subvariety, the rational fractions I^{rat}_N satisfy the same shuffle and quasi-shuffle product laws as the original multiple polylogarithms, so the generic double shuffle equations are realized as identities among rational functions rather than among transcendental functions.
Because the derivation operators apply to any algebraic relation among multiple polylogarithms, the same mechanism transfers other algebraic relations, not only double shuffle relations, to algebraic functions.
The classical double shuffle relations for multiple zeta values are limits of the generic ones; the paper states that limits of the restricted equations and the resulting periods will be treated in subsequent work.
On the p-adic side, the paper positions this construction as a step toward a variant of the motivic Galois theory of multiple polylogarithms in which algebraic functions play the special role that multiple harmonic values play in the p-adic setting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the subvarieties are nonempty, they give rational solutions to the generic double shuffle equations in families parametrized by moduli space, so the solutions are geometric objects depending on the configuration of points rather than isolated formal series.
Nonemptiness can be checked computationally for the first few cases: the defining equations of Y are explicit rational equations, so elimination theory can decide whether a solution exists without any new idea.
The same localization-and-derivation transfer could be applied to other functional equations of multiple polylogarithms, for instance duality or homography transformations, yielding analogous algebraic solutions if the double shuffle case works.
A positive answer would provide a new source of periods attached to algebraic functions on moduli spaces of curves, potentially connecting the algebraic relations of multiple zeta values to algebraic geometry more directly.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: the derivative calculations are real, but the main theorem is a definition wearing a theorem's clothes. Y^(N) is defined (Definition 4.3) as the subvariety where the variant double shuffle relations switch over to the generic ones, so on that subvariety the equations are satisfied by construction. Nonemptiness is never addressed, and the promised explicit equations are pushed to a later version.

What's worth saving here is the computational core. Proposition 2.2 gives a recursive formula for higher partial derivatives of multiple polylogarithms, with coefficients that are rational functions, and the examples in weights 2 and 3 are explicit enough to check by hand. Proposition 3.2 and 3.5 use this to derive "localized" versions of the shuffle and quasi-shuffle relations, and Example 3.3 shows a concrete identity among rational functions that looks exactly like the shuffle relation. That is a genuinely new way of producing algebraic identities from derivatives of MPLs, and it could be useful for the p-adic multiple zeta value program the author has been developing.

The problem is the arrow from those identities to an existence theorem. Proposition 4.2 claims that the rational parts I^rat_N satisfy a variant of the generic double shuffle relations. A proof is sketched in one line, citing Chen's linear independence theorem. That might be enough if everything were laid out, but the variant relations themselves are not stated in closed form, and the text says more explicit versions will come later. Then Definition 4.3 sets Y to be the zero locus of the difference between variant and generic. The theorem in the introduction then asserts that Y exists and is equipped with algebraic functions solving generic double shuffle. That assertion is true by definition if Y is nonempty—but nonemptiness is exactly what is missing. Nothing in the paper shows that the equations defining Y are consistent, or that the rational fractions remain regular on it.

The citation pattern is fine; the heavy self-citation is normal for a single-author program, and the external references (Chen, Deligne, Goncharov, IHKS) are the standard ones. The math that is actually written is plausible and the computations are detailed. The gap is not a small technicality; it is the central claim.

Who gets value from this: someone working on p-adic multiple zeta values or Goncharov multiple polylogarithms, mainly for the derivative formulas. As a standalone paper, it is not ready. My recommendation: do not accept it for peer review as is. Send it back with a request to prove that Y is nonempty, or to give at least one nontrivial explicit example, and to write out the defining equations. If the author can do that, the result would be worth a serious referee.

Referee Report

4 major / 5 minor

Summary. The paper proposes to construct, for each N and n, closed affine subvarieties Y^{(N)}_{0,n+3} of M_{0,n+3} together with global algebraic functions that satisfy the generic double shuffle relations of multiple polylogarithms. The strategy is to apply iterated partial derivatives to the generic double shuffle relations, collect the purely rational terms via localized multiple polylogarithms, and then define the subvariety Y as the locus where the resulting 'variant' double shuffle relations coincide with the generic ones. The central existence theorem is stated in Section 0.4, and the construction is given in Section 4.

Significance. If the central theorem were established, the paper would supply a new class of algebraic functions satisfying the same shuffle and quasi-shuffle relations as multiple polylogarithms, with potential applications to p-adic multiple zeta values. The paper also contains some useful explicit formulas for derivatives of multiple polylogarithms (Section 2.2 and Examples 2.4–2.6), and it identifies an interesting transfer principle. However, the main existence theorem is not proved in the manuscript: the decisive step is essentially a definition presented as a conclusion, and the paper itself defers explicit equations and further details to later versions. As written, the central claim is unsupported.

major comments (4)

[Section 4.3 (Definition 4.3) and Section 0.4] The argument is circular at the central point. Definition 4.3 defines Y^{(N)}_{L+3} as the subvariety cut out by 'the equation saying that the variant of the generic double shuffle relation satisfied by I^rat_N is equal to the generic double shuffle relation.' The immediately following paragraph then concludes that 'on this subvariety, one has explicit rational fractions over Q which satisfy the generic double shuffle relations.' This conclusion is a restatement of the definition, not a theorem. The manuscript never proves that this locus is nonempty, that it has positive dimension, or that it is an infinite sequence of varieties as claimed in Section 0.4. Without nonemptiness, the theorem's assertion 'equipped with global algebraic functions' is vacuous.
[Section 4.2 (Proposition 4.2)] Proposition 4.2 is not proved. The proof consists of one sentence citing Chen's theorem and asserting that multiple polylogarithms are linearly independent over algebraic functions on M_{0,L+3}. The 'variant of the generic double shuffle relations' is never stated in general; only low-weight examples are given in Section 3.3 and Section 3.4. Moreover, the linear independence statement over algebraic functions on the moduli space requires a precise formulation and justification; Chen's theorem concerns iterated integrals on a path space, and the passage to algebraic functions on M_{0,L+3} is not supplied. Thus the proposition cannot support the construction that follows.
[Section 3.2 and Section 0.4] The manuscript explicitly defers the main technical content. Section 3.2 states: 'We will add in the next versions more explicit versions of these equations as well as similar results about the homographical transformations,' and Section 0.4 states: 'More details and consequences of the result will appear in the next version of this text.' The central theorem promises 'explicit closed affine subvarieties' and 'global algebraic functions,' but the promised explicit equations are not present. The reader cannot verify the claimed existence or explicitness from the submitted text.
[Section 4.3 (Definition 4.3)] Definition 4.3 is not mathematically well-posed as stated. The generic double shuffle equations are a family of polynomial equations, and the phrase 'the equation saying that the variant ... is equal to the generic double shuffle relation' does not identify a finite list of explicit polynomials cutting out Y. Moreover, even if the locus were defined, one must check that the rational fractions I^rat_N are regular on it, i.e., that the denominators occurring in the coefficients do not vanish identically on Y; this regularity is asserted in the theorem but never demonstrated.

minor comments (5)

[Section 0.1, Eq. (3)] There is a typo: 'quasi-shuﬄe procuct' should be 'quasi-shuﬄe product'.
[Section 0.3] The word 'varians' should be 'variants', and the sentence about multiple harmonic values contains several typographical errors (e.g., 'followng', 'terminology multiple harmonic values c omes').
[Section 1.2] The phrase 'We will use the term localization to takl about a settng' contains typos; it should read 'to talk about a setting'.
[Throughout] The notation for the dimension of the moduli space is inconsistent: the theorem in Section 0.4 uses Y^{(N)}_{0,n+3} and M_{0,n+3}, while Definition 4.3 and the surrounding text use Y^{(N)}_{L+3} and M_{0,L+3}. The relation between the variables n and L should be clarified.
[Section 3.2] The sentence 'For simplicitly, we restrict to the case...' contains a typo; it should be 'For simplicity'.

Circularity Check

1 steps flagged · score 8.0 of 10

Definition 4.3 defines Y as the locus where the variant double shuffle relation equals the generic one, so the theorem's assertion that Y carries rational functions solving the generic double shuffle equations is true by construction; nonemptiness, regularity, and explicitness remain unproved.

self definitional [Section 4.3, Definition 4.3; cf. Section 0.4, Theorem]
"Let Y (N ) L+3 be the subvariety of the affine variety M0,L+3 defined by the equation saying that the variant of the generic double shuffle relation satisfied by I rat N (a0; a1, . . . , am; am+1) is equal to the generic double shuffle relation. By the previous proposition, on this subvariety, one has explicit rational fractions over Q which satisfy the generic double shuffle relations."

The subvariety Y is defined in Definition 4.3 by the very equation that the theorem needs to prove: Y is the locus where the variant double shuffle relation, which Proposition 4.2 asserts holds identically for the rational terms I^rat_N, is equal to the generic double shuffle relation. Once Y is defined this way, the statement that on Y the rational fractions satisfy the generic double shuffle equations is an immediate unpacking of the definition, not an independent existence result.

full rationale

The main load-bearing step is self-definitional. The theorem in Section 0.4 is introduced by saying that one restricts to a subvariety defined by the equation that the derived generic double shuffle equals the generic double shuffle; Definition 4.3 then defines Y^(N)_{L+3} exactly as the subvariety where the variant of the generic double shuffle satisfied by I^rat_N equals the generic double shuffle. With Proposition 4.2 asserting that I^rat_N satisfies the variant relation, the conclusion that the rational fractions satisfy the generic double shuffle relations on Y is a formal consequence of the defining equation of Y, i.e. it is forced by construction. The remaining substantive issues, nonemptiness of Y, regularity and globality of I^rat_N on Y, and actual explicit equations, are not proved in the text and are explicitly deferred to later versions. This warrants a high circularity score, because the paper's central existence claim is not independently established beyond its own definition. The self-citations to the author's earlier p-adic papers are motivational and not load-bearing for this theorem, and the appeal to Chen's theorem is to an external result, so those do not contribute additional circularity. The score is set at 8 rather than 10 because the rational functions I^rat_N and the derivative formalism are genuine constructions; however, the theorem's conclusion about those functions is made true by the definition of Y rather than by derivation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical or algebro-geometric entities beyond the subvarieties Y itself, which are defined to be the solution sets of the very equations under study. The main axioms are standard background in the multiple polylogarithm and fundamental groupoid literature, but the linear independence statement from Chen's theorem is used without precise hypotheses, and the consistency of the defining equations of Y is assumed without proof.

assumptions (4)

domain assumption Multiple polylogarithms satisfy the generic double shuffle relations (7) and (8).
Used throughout Section 3 to derive the algebraic variants; taken from Goncharov's theory [G].
domain assumption Chen's theorem gives linear independence of multiple polylogarithms over the ring of algebraic functions on M_{0,L+3}.
Invoked in the proof of Proposition 4.2 to isolate the rational term; the precise ring and hypotheses are not stated.
domain assumption The localization of the de Rham fundamental groupoid with formal inverses of integration operators is a valid construction.
Section 1.2 introduces formal inverses to interpret partial derivatives; relies on Deligne's pro-unipotent groupoid [D].
standard math M_{0,L+3} is an affine variety and the defining equation of Y is algebraic.
Used implicitly in Definition 4.3 to call Y an affine subvariety.

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Pith. "Pith review of On generic double shuffle relations, localized multiple polylogarithms and algebraic functions." pith.science (2026). https://pith.science/paper/NHYBYCZM

@misc{pith2026190801410,
  author       = {Pith},
  title        = {Pith review of: On generic double shuffle relations, localized multiple polylogarithms and algebraic functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NHYBYCZM}},
  note         = {Machine review of arXiv:1908.01410}
}

abstract

We define subvarieties of $\mathcal{M}_{0,n}$ equipped with algebraic functions that are solutions to the generic double shuffle equations satisfied by multiple polylogarithms on $\mathcal{M}_{0,n}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages

[1]

K. T. Chen - Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), 831-879

work page 1977
[2]

P.Deligne, Le groupe fondamental de la droite projective moins trois points, Galois Groups over Q (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ. 16, Springer-Verlag, New York, 1989

work page 1987
[3]

Deligne, A.B

P. Deligne, A.B. Goncharov, Groupes fondamentaux motiviques de Tate mixtes, Ann. Sci. Ecole Norm. Sup. 38.1 , 2005, pp. 1-56

work page 2005
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Algebraic differential formulas for the shuffle, stuffle and duality relations of iterated integrals

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work page Pith review arXiv
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Ihara, M

K. Ihara, M. Kaneko, D. Zagier - Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006) 307-338

work page 2006
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A bound on the norm of overconvergent $p$-adic multiple polylogarithms

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work page Pith review arXiv
[8]

Jarossay - Pro-unipotent harmonic actions and a computation of p -adic cyclotomic multiple zeta values, arXiv:1501.04893 (submitted)

D. Jarossay - Pro-unipotent harmonic actions and a computation of p -adic cyclotomic multiple zeta values, arXiv:1501.04893 (submitted)

arXiv

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[9]

Jarossay - Pro-unipotent harmonic actions and a dynamical method for the computation of p -adic cyclotomic multiple zeta values, arXiv:1610.09107 (submitted)

D. Jarossay - Pro-unipotent harmonic actions and a dynamical method for the computation of p -adic cyclotomic multiple zeta values, arXiv:1610.09107 (submitted)

arXiv
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Jarossay - Adjoint cyclotomic multiple zeta values and multiple harmonic values, arXiv:1412.5099 (submitted)

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Jarossay - The adjoint quasi-shuffle relation of p -adic cyclotomic multiple zeta values recovered by explicit formulas, arXiv:1601.01158

D. Jarossay - The adjoint quasi-shuffle relation of p -adic cyclotomic multiple zeta values recovered by explicit formulas, arXiv:1601.01158

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Jarossay - Cyclotomic multiple harmonic values regarded as periods, arXiv:1601.01159

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Jarossay - p -adic cyclotomic multiple zeta values at roots of unity of order non prime to p , arXiv:1708.08009

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arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

K. T. Chen - Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), 831-879

work page 1977

[2] [2]

P.Deligne, Le groupe fondamental de la droite projective moins trois points, Galois Groups over Q (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ. 16, Springer-Verlag, New York, 1989

work page 1987

[3] [3]

Deligne, A.B

P. Deligne, A.B. Goncharov, Groupes fondamentaux motiviques de Tate mixtes, Ann. Sci. Ecole Norm. Sup. 38.1 , 2005, pp. 1-56

work page 2005

[4] [4]

A.B.Goncharov - Multiple polylogarithms and mixed Tate motives, arXiv:0103059v4

work page

[5] [5]

Algebraic differential formulas for the shuffle, stuffle and duality relations of iterated integrals

M. Hirose, S. Sato - Algebraic differential formulas for the shuffle, stuffle and duality relations of iterated integrals, arXiv:1801.03165

work page Pith review arXiv

[6] [6]

Ihara, M

K. Ihara, M. Kaneko, D. Zagier - Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006) 307-338

work page 2006

[7] [7]

A bound on the norm of overconvergent $p$-adic multiple polylogarithms

D. Jarossay - A bound on the norm of overconvergent p -adic multiple polylogarithms, arXiv:1503.08756, to appear in J. Number Theory

work page Pith review arXiv

[8] [8]

Jarossay - Pro-unipotent harmonic actions and a computation of p -adic cyclotomic multiple zeta values, arXiv:1501.04893 (submitted)

D. Jarossay - Pro-unipotent harmonic actions and a computation of p -adic cyclotomic multiple zeta values, arXiv:1501.04893 (submitted)

arXiv