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arxiv: 2308.05825 · v1 · pith:YA65YGGYnew · submitted 2023-08-10 · 🧮 math.NT

A v-adic variant of Anderson-Brownawell-Papanikolas linear independence criterion and its application

Yen-Tsung Chen This is my paper

Pith reviewed 2026-05-24 07:30 UTC · model grok-4.3

classification 🧮 math.NT
keywords v-adic Carlitz multiple polylogarithmslinear independenceAnderson-Brownawell-Papanikolas criterionfunction fieldsmultiple zeta valuesFurusho-Yamashita conjecture
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The pith

When v has degree one, all algebraic-closure linear relations among v-adic Carlitz multiple polylogarithms arise from base-field relations of the same weight.

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

The paper proves a linear-independence result for v-adic Carlitz multiple polylogarithms evaluated at algebraic points in function fields. When the finite place v has degree one, every linear dependence over the algebraic closure of the base field is shown to follow from an existing linear dependence over the base field itself, and only among values of identical weight. This criterion is then applied to obtain the function-field analogue of Furusho-Yamashita's conjecture on the relations satisfied by v-adic multiple zeta values, again restricted to degree-one places. A reader cares because the result supplies a precise description of the algebraic dependencies among these special values, mirroring known phenomena in characteristic-zero number theory but now inside positive-characteristic function fields.

Core claim

The paper establishes that, for any finite place v of degree one, every linear relation over the algebraic closure among the v-adic Carlitz multiple polylogarithms at algebraic points is a consequence of a linear relation over the base field k among values of the same weight. The proof proceeds by developing a v-adic variant of the Anderson-Brownawell-Papanikolas criterion adapted to this setting. As a direct corollary, the function-field version of Furusho-Yamashita's conjecture holds for the v-adic multiple zeta values whenever the place v has degree one.

What carries the argument

The v-adic variant of the Anderson-Brownawell-Papanikolas linear independence criterion, which isolates the precise linear relations among the polylogarithm values that descend from the base field.

If this is right

  • The function-field analogue of Furusho-Yamashita's conjecture holds for v-adic multiple zeta values at every degree-one place.
  • All ̄k-linear relations among the values are generated by the k-linear relations of equal weight.
  • The same-weight restriction on the base-field relations is part of the precise description of the dependence module.

Where Pith is reading between the lines

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

  • The degree-one restriction suggests that higher-degree places may require additional correction terms or a modified statement.
  • The result supplies a concrete test for verifying whether a given collection of v-adic multiple polylogarithm values is linearly independent over the algebraic closure.
  • It opens the possibility of using the criterion to bound the dimension of the span of these values in explicit low-weight cases.

Load-bearing premise

The finite place v must have degree exactly one; the statement is not claimed for places of higher degree.

What would settle it

An explicit linear dependence over the algebraic closure, for some degree-one place v, among v-adic Carlitz multiple polylogarithms at algebraic points that cannot be explained by any base-field linear relation of matching weight.

read the original abstract

Let $\overline{k}$ be a fixed algebraic closure of $k$. When the finite place $v$ is of degree one, we show that all $\overline{k}$-linear relations among $v$-adic Carlitz multiple polylogarithms at algebraic points arise from $k$-linear relations among these values of the same weight. As an application, we establish a function field analogue of Furusho-Yamashita's conjecture for $v$-adic multiple zeta values whenever the degree of the place $v$ is one.

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

0 major / 0 minor

Summary. The manuscript establishes a v-adic variant of the Anderson-Brownawell-Papanikolas linear independence criterion. When the finite place v has degree one, it proves that every overline{k}-linear relation among v-adic Carlitz multiple polylogarithms evaluated at algebraic points is induced by a k-linear relation among values of the same weight. The result is applied to obtain a function-field analogue of the Furusho-Yamashita conjecture for v-adic multiple zeta values under the same degree-one hypothesis on v.

Significance. The criterion supplies a precise description of the linear dependence relations in the v-adic setting, directly extending the classical Anderson-Brownawell-Papanikolas framework to a new context. The explicit restriction to places of degree one is stated clearly and the resulting statement on multiple zeta values constitutes a concrete, falsifiable advance in the arithmetic of function fields.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept. The report correctly summarizes the main result on the v-adic Anderson-Brownawell-Papanikolas criterion under the degree-one hypothesis on v and its application to a function-field analogue of the Furusho-Yamashita conjecture.

Circularity Check

0 steps flagged

No significant circularity; derivation extends prior criterion without self-referential reduction

full rationale

The paper states a v-adic variant of the Anderson-Brownawell-Papanikolas criterion, explicitly restricted to degree-one places v, and applies it to obtain a function-field analogue of the Furusho-Yamashita conjecture under the same hypothesis. The abstract and description indicate the result is derived from the existing ABP criterion in the literature rather than by fitting parameters to the target relations or by self-citation chains that bear the load. No equations or steps are shown reducing the claimed linear-independence statement to a definition or fit of the same quantities. The degree-one restriction is stated as a hypothesis rather than smuggled in. This is a standard extension of prior work with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted beyond the standard setup of function-field arithmetic.

pith-pipeline@v0.9.0 · 5612 in / 1220 out tokens · 42377 ms · 2026-05-24T07:30:31.220569+00:00 · methodology

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Reference graph

Works this paper leans on

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