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arxiv: 2605.17756 · v1 · pith:KRIGQ7ZAnew · submitted 2026-05-18 · 🧮 math.NT

Linear independence of periods related to polylogarithms

Makoto Kawashima This is my paper

Pith reviewed 2026-05-20 01:30 UTC · model grok-4.3

classification 🧮 math.NT
keywords multiple polylogarithmslinear independencealgebraic number fieldsPadé approximantsperiodstranscendence theorypolylogarithm values
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The pith

This paper provides the first criteria for the linear independence of multiple polylogarithm values over algebraic number fields.

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

The author seeks to prove specific criteria that ensure certain multiple polylogarithm values and their products are linearly independent when considered over algebraic number fields. If these criteria hold, it would mean that there are no unexpected algebraic relations among these important constants, helping to clarify the structure of periods in number theory. The focus is on values at distinct points, which are particularly relevant for understanding products of such functions. The proof relies on constructing special approximants that control the errors in a way that reveals independence.

Core claim

By explicitly constructing Padé-type approximants tailored for multiple polylogarithms, the paper derives error terms and relations that establish linear independence criteria for the values over algebraic number fields, yielding new results specifically for products of polylogarithms evaluated at distinct points.

What carries the argument

Explicit construction of Padé-type approximants tailored for multiple polylogarithms, which produce the necessary error terms to imply the independence.

If this is right

  • Linear independence holds for products of polylogarithms at distinct points over algebraic number fields.
  • These criteria represent the first such general results for multiple polylogarithm values in this context.
  • New understanding of the algebraic relations among polylogarithmic periods is obtained through these independence statements.

Where Pith is reading between the lines

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

  • Specializing the criteria to lower weight cases might connect to known results on zeta values and their products.
  • The Padé approximant method could potentially be applied to other classes of periods, such as those from elliptic integrals.
  • These independence results may have implications for the expected dimensions of spaces of multiple zeta values in conjectural frameworks.

Load-bearing premise

The Padé-type approximants constructed specifically for multiple polylogarithms have error terms or functional equations that are strong enough to imply the linear independence criteria over algebraic number fields.

What would settle it

Discovering a nontrivial linear relation with algebraic coefficients among a set of multiple polylogarithm values or their products at distinct points that the criteria claim should be independent would falsify the result.

read the original abstract

This paper provides the first criteria for the linear independence of multiple polylogarithm values over algebraic number fields. In particular, we derive novel results regarding the linear independence of products of polylogarithms at distinct points over an algebraic number field. Our approach is based on the explicit construction of Pad\'{e}-type approximants tailored for multiple polylogarithms.

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 / 2 minor

Summary. The paper claims to establish the first criteria for the linear independence of multiple polylogarithm values over algebraic number fields. It derives novel results on the linear independence of products of polylogarithms at distinct points, relying on an explicit construction of Padé-type approximants tailored to multiple polylogarithms whose error terms or functional equations are asserted to yield the independence statements.

Significance. If the explicit Padé-type construction and the resulting error estimates are correct, the work would supply new, concrete criteria for linear independence among periods attached to multiple polylogarithms over number fields. Such results are central to the arithmetic study of these transcendental numbers; an explicit approximant method that avoids reduction to previously fitted quantities would constitute a genuine technical advance.

minor comments (2)
  1. The abstract asserts that the approximants are 'tailored for multiple polylogarithms'; the introduction or §2 should contain a precise statement of the functional equations or error bounds that convert the approximants into independence criteria, with explicit reference to the relevant theorem numbers.
  2. Notation for the multiple polylogarithms (e.g., the precise indexing of the weight and depth parameters) should be fixed at the first appearance and used consistently in all subsequent statements of the independence results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. We appreciate the recognition that an explicit Padé-type construction for multiple polylogarithms would represent a technical advance if the error estimates hold. No specific major comments were raised in the report, so we address the overall evaluation below and will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation rests on explicit construction

full rationale

The paper states that its criteria for linear independence of multiple polylogarithm values follow from an explicit construction of Padé-type approximants tailored to multiple polylogarithms, with the error terms or functional equations of those approximants stated to be sufficient to imply the independence results over algebraic number fields. No load-bearing step is shown to reduce by definition to a fitted parameter, a self-cited uniqueness theorem, or a renaming of prior results; the central claim is presented as arising directly from the properties of the newly constructed approximants rather than from any circular dependence on the target independence statements themselves. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

With only the abstract available, the ledger is limited to the core modeling choice described: the assumption that the constructed Padé-type approximants behave as needed for the independence argument. No free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption The Padé-type approximants constructed for multiple polylogarithms satisfy the analytic or algebraic properties required to deduce linear independence over algebraic number fields.
    The entire approach rests on this property of the approximants; if it fails, the criteria do not follow.

pith-pipeline@v0.9.0 · 5571 in / 1289 out tokens · 39018 ms · 2026-05-20T01:30:04.713643+00:00 · methodology

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Reference graph

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