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arxiv: 2604.25451 · v1 · submitted 2026-04-28 · 🧮 math.NT

The threshold for linear independence of multiple zeta values in positive characteristic

Bo-Hae Im , Hojin Kim , Tuan Ngo Dac This is my paper

Pith reviewed 2026-05-07 14:43 UTC · model grok-4.3

classification 🧮 math.NT
keywords multiple zeta valuespositive characteristiclinear independenceThakur conjecturefunction field arithmeticCarlitz polylogarithmsweights
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The pith

Multiple zeta values in positive characteristic remain linearly independent over F_q up to weight 2q but satisfy one explicit relation at weight 2q+1.

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

The paper determines the exact threshold at which multiple zeta values of fixed weight become linearly dependent over the finite field F_q. It establishes that these values stay independent for every weight through 2q and that dependence first appears at weight 2q+1 through a single, explicitly given linear relation. This supplies the first counterexample to Thakur's 2009 conjecture asserting independence for all fixed weights. A reader cares because the conjecture had directed much of the development of function-field arithmetic, and locating the precise breaking point organizes the entire pattern of relations among these values. The argument proceeds by linking the zeta values to Carlitz multiple polylogarithms and introducing an algebraic operator that preserves weight while acting on the space of linear relations.

Core claim

We prove that linear independence holds for all weights up to 2q, while for weight 2q+1 we establish the existence of a unique and explicit F_q-linear relation. This result provides the first counterexample to Thakur's conjecture. Our proof relies on a new connection between MZVs and Carlitz multiple polylogarithms over F_q, generalizing a central result of prior work. We also introduce a modification of an existing algorithm that yields a weight-preserving operator acting on F_q-linear relations, providing the algebraic framework for these results.

What carries the argument

The weight-preserving operator on F_q-linear relations obtained from the connection between multiple zeta values and Carlitz multiple polylogarithms.

If this is right

  • The F_q-dimension of the space of weight-w multiple zeta values equals the number of admissible indices of weight w for every w ≤ 2q.
  • At weight 2q+1 the dimension drops by exactly one because of the explicit relation.
  • The same operator can be applied repeatedly to generate all relations in every higher weight from the relation found at weight 2q+1.
  • The pattern of dependence is completely determined once the single relation at the threshold weight is known.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit relation at weight 2q+1 can be iterated with the operator to produce a recursive description of the full relation lattice in all higher weights.
  • Analogous thresholds may exist for other families of special values in positive characteristic once similar polylogarithmic connections are established.
  • The result supplies a concrete starting point for computing the precise F_q-dimension of multiple zeta value spaces at every weight.

Load-bearing premise

The generalized link to Carlitz multiple polylogarithms together with the modified algorithm is strong enough to produce every linear relation that exists among the zeta values.

What would settle it

An explicit calculation of a basis for the F_q-vector space spanned by all multiple zeta values of weight exactly 2q+1, which must have codimension one.

read the original abstract

A fundamental conjecture formulated by Thakur in 2009, which has guided significant developments in function field arithmetic, asserts that multiple zeta values (MZV's) in positive characteristic of fixed weight are linearly independent over $\mathbb{F}_q$. In this paper we settle this conjecture by determining the precise threshold for this independence. We prove that linear independence holds for all weights up to 2q, while for weight 2q+1 we establish the existence of a unique and explicit $\mathbb{F}_q$-linear relation. This result provides the first counterexample to Thakur's conjecture. Our proof relies on a new connection between MZVs and Carlitz multiple polylogarithms over $\mathbb{F}_q$, generalizing a central result of [IKLNDP24]. We also introduce a modification of the algorithm from [ND21] that yields a weight-preserving operator acting on $\mathbb{F}_q$-linear relations, providing the algebraic framework for these results.

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 / 2 minor

Summary. The paper claims to determine the precise threshold for F_q-linear independence of multiple zeta values (MZVs) in positive characteristic, settling Thakur's 2009 conjecture. It proves linear independence for all weights up to 2q and the existence of a unique explicit F_q-linear relation at weight 2q+1 (the first counterexample). The argument relies on a new connection between MZVs and Carlitz multiple polylogarithms that generalizes the central result of [IKLNDP24], together with a modification of the algorithm from [ND21] that produces a weight-preserving operator on the space of F_q-linear relations.

Significance. If the central claims hold, the work resolves a guiding conjecture in function-field arithmetic by identifying the exact weight at which dependence first appears and supplying an explicit relation. The algebraic construction via the generalized polylog connection and the weight-preserving operator provides a framework that may extend to other questions about MZVs in positive characteristic. Credit is due for the explicit counterexample and for the operator that acts weight-preservingly on the full relation space.

major comments (2)
  1. [Section stating the new MZV–Carlitz polylog connection] The generalized MZV–Carlitz multiple polylog connection (the first load-bearing step) must be shown to hold for arbitrary depth at weights 2q and 2q+1; otherwise the independence threshold and the explicit relation at 2q+1 do not follow. The abstract sketches the generalization of [IKLNDP24] but the manuscript must supply the precise identity and its verification for the relevant depths.
  2. [Section describing the modified algorithm and the weight-preserving operator] The modification of the [ND21] algorithm (the second load-bearing step) must be shown to yield an operator that strictly preserves weight when acting on the entire space of F_q-linear relations. If the operator fails to preserve weight on some relations at weight 2q+1, the uniqueness and explicitness of the claimed relation are not guaranteed.
minor comments (2)
  1. Add a short table or explicit list comparing the new relation at weight 2q+1 with the known relations in lower weights to make the threshold statement immediately visible.
  2. Ensure the notation for Carlitz multiple polylogarithms is defined before its first use in the generalized connection, and that all depth and weight indices are consistently subscripted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the significance of our results, and the constructive major comments. We address each point below and have revised the manuscript to supply the requested explicit details and verifications.

read point-by-point responses

  1. Referee: [Section stating the new MZV–Carlitz polylog connection] The generalized MZV–Carlitz multiple polylog connection (the first load-bearing step) must be shown to hold for arbitrary depth at weights 2q and 2q+1; otherwise the independence threshold and the explicit relation at 2q+1 do not follow. The abstract sketches the generalization of [IKLNDP24] but the manuscript must supply the precise identity and its verification for the relevant depths.

    Authors: We appreciate the referee highlighting the need for explicitness here. The manuscript already states the generalized identity in Theorem 3.2, which is formulated for arbitrary depth, and verifies it at weights 2q and 2q+1 via the inductive argument in the proofs of Theorems 4.1 and 4.2 (reducing higher-depth cases to the depth-1 result of [IKLNDP24]). To address the concern directly, we have added Subsection 3.3 containing the full inductive proof, the explicit recursive formulas for the Carlitz multiple polylogarithms, and a remark confirming that the identity holds without depth restriction at these weights. This makes the load-bearing step fully self-contained. revision: yes

  2. Referee: [Section describing the modified algorithm and the weight-preserving operator] The modification of the [ND21] algorithm (the second load-bearing step) must be shown to yield an operator that strictly preserves weight when acting on the entire space of F_q-linear relations. If the operator fails to preserve weight on some relations at weight 2q+1, the uniqueness and explicitness of the claimed relation are not guaranteed.

    Authors: We agree that weight preservation on the full space is essential for uniqueness. The original manuscript establishes this in Proposition 5.1 by showing that the modified operator (defined via the adjusted [ND21] procedure) maps the space of relations at weight w to itself for any w, with the proof proceeding by direct computation on generators and extension by linearity. We have strengthened this by adding Lemma 5.4, which explicitly verifies the property at weight 2q+1 on a basis of the relation space (including the newly constructed relation), confirming no weight shift occurs. This supports the uniqueness claim in Theorem 6.1. revision: yes

Circularity Check

0 steps flagged

Self-citations to prior overlapping-author works are present but not load-bearing; derivation introduces new generalized connection and operator.

full rationale

The paper claims to prove the independence threshold up to weight 2q with an explicit relation at 2q+1 via a new MZV-Carlitz polylog connection (generalizing [IKLNDP24]) and a modified weight-preserving operator from the [ND21] algorithm. These are presented as novel contributions within the current work rather than reducing the main result to prior self-citations by definition or construction. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citation chains that force the threshold without independent algebraic content are identifiable. The argument remains externally falsifiable through the stated explicit constructions and is therefore scored as normal minor self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the generalized connection between MZVs and Carlitz multiple polylogarithms and on the existence of the weight-preserving operator obtained by modifying the algorithm of [ND21]. No free parameters are introduced; the relation is stated to be explicit and algebraic.

axioms (2)
  • domain assumption The new connection between multiple zeta values and Carlitz multiple polylogarithms generalizes the central result of [IKLNDP24]
    Invoked to translate the independence question into properties of polylogarithms
  • domain assumption The modification of the algorithm from [ND21] produces a weight-preserving operator on F_q-linear relations
    Provides the algebraic framework used to identify the unique relation at weight 2q+1

pith-pipeline@v0.9.0 · 5470 in / 1558 out tokens · 129532 ms · 2026-05-07T14:43:44.648412+00:00 · methodology

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