On the Chevalley-Bass number of a field

Florence Gillibert; Gabriele Ranieri; Jean Gillibert

arxiv: 2604.10802 · v1 · submitted 2026-04-12 · 🧮 math.NT

On the Chevalley-Bass number of a field

Jean Gillibert , Florence Gillibert , Gabriele Ranieri This is my paper

Pith reviewed 2026-05-10 15:14 UTC · model grok-4.3

classification 🧮 math.NT

keywords Chevalley-Bass numbercharacteristic zero fieldsmaximal abelian subextensionexponential Diophantine equationsnumber fieldsupper and lower boundscomputational algorithm

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The pith

Upper and lower bounds are given for the Chevalley-Bass number of any characteristic-zero field where the quantity is defined.

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

The paper establishes concrete upper and lower bounds that hold for the Chevalley-Bass number whenever the quantity exists for a field of characteristic zero. It supplies an explicit algorithm that returns the exact value once the maximal abelian subextension of the field is known. The same results yield a strictly smaller constant that appears in statements about solutions to exponential Diophantine equations. A reader cares because these bounds turn an abstract invariant into something that can be estimated or computed for concrete fields, directly tightening existence statements for integer solutions of certain exponential equations.

Core claim

We give upper and lower bounds on the Chevalley-Bass number of a field of characteristic zero, whenever this quantity is well-defined. We also describe an algorithm which computes the Chevalley-Bass number of a field, provided its maximal abelian subextension is known. As a primary application, we improve the value of a constant related to exponential diophantine equations.

What carries the argument

The Chevalley-Bass number together with the maximal abelian subextension, which together allow explicit bounds and a computable procedure for the invariant.

If this is right

The improved constant tightens the known thresholds in theorems on exponential Diophantine equations.
The algorithm produces the exact Chevalley-Bass number for any field whose maximal abelian subextension can be described explicitly.
Upper and lower bounds become available for every characteristic-zero field in which the number is defined.
Concrete estimates replace purely existential statements about the size of the invariant.

Where Pith is reading between the lines

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

The same bounding technique may extend to other arithmetic invariants that depend on abelian extensions.
Once the algorithm is implemented for standard number fields, it can be used to check conjectures about the growth of the Chevalley-Bass number with the degree.
The improved Diophantine constant could be fed into existing computer searches for solutions of exponential equations to rule out additional cases.

Load-bearing premise

The Chevalley-Bass number must be well-defined on the field and the maximal abelian subextension must already be known before the algorithm can be applied.

What would settle it

A concrete characteristic-zero field for which the computed upper or lower bound is violated by the actual Chevalley-Bass number, or for which the algorithm returns a value inconsistent with the definition once the maximal abelian subextension is supplied.

read the original abstract

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit bounds and a working algorithm for the Chevalley-Bass number when the maximal abelian subextension is known, plus a modest improvement to a constant in exponential Diophantine equations.

read the letter

The main point is that they supply upper and lower bounds on the Chevalley-Bass number for characteristic zero fields whenever it is defined, describe an algorithm to compute it once the maximal abelian subextension is known, and use the result to improve a constant tied to exponential Diophantine equations. That combination of bounds, algorithm, and application is what the paper actually delivers. The explicit bounds and the algorithm count as the concrete advance; in this corner of algebraic number theory, turning an invariant into something computable under a clear hypothesis is useful even if the hypothesis is not always easy to meet. The Diophantine application is a reasonable way to show why the number matters outside pure invariant theory. The paper stays focused and states its claims directly in the abstract, which is a plus. The main limitation is that the algorithm needs the maximal abelian subextension as input, so its reach depends on how often that data is already available or easy to obtain. The bounds are conditional on the number being well-defined, which is standard but narrows the scope. The improvement to the constant is presented as the primary payoff, yet without seeing the size of the gain it is hard to judge how much practical difference it makes. Nothing in the setup points to circular reasoning or unsupported steps. This is for number theorists who work with field invariants or who solve exponential Diophantine equations. A reader already interested in either topic could extract a usable tool or a slightly better constant. It deserves a serious referee because the contribution is grounded enough in its subfield to justify review time, even if the referee will probably ask for more detail on the algorithm's practical range and the exact improvement achieved.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to establish upper and lower bounds on the Chevalley-Bass number of a field of characteristic zero whenever the quantity is well-defined. It also describes an algorithm to compute the Chevalley-Bass number provided the maximal abelian subextension is known. The primary application is an improvement to a constant appearing in the theory of exponential Diophantine equations.

Significance. If the bounds, algorithm, and application are correct, the work supplies concrete tools for computing a field invariant in characteristic zero and improves a Diophantine constant, which could be useful for both theoretical investigations and explicit computations in algebraic number theory.

minor comments (1)

The abstract and introduction should include a brief, self-contained definition or reference for the Chevalley-Bass number to aid readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for recognizing the potential utility of the bounds, algorithm, and Diophantine application if correct. The recommendation is listed as uncertain, yet the report contains no specific major comments or points of concern. We remain confident that the stated results are correct as presented and would be pleased to provide any additional clarifications or details the referee may require.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states conditional upper and lower bounds on the Chevalley-Bass number for fields of characteristic zero (when well-defined) and an algorithm that explicitly requires the maximal abelian subextension as known input. No load-bearing derivation step reduces by construction to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain. The application to exponential Diophantine equations is presented as a downstream consequence without evidence of internal reduction to the paper's own inputs. The derivation chain remains self-contained against external number-theoretic constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the prior definition of the Chevalley-Bass number and standard assumptions in algebraic number theory about field extensions.

axioms (2)

domain assumption The Chevalley-Bass number is well-defined for fields of characteristic zero under consideration
The abstract qualifies the bounds with 'whenever this quantity is well-defined'
domain assumption Knowledge of the maximal abelian subextension allows computation via the algorithm
The algorithm is conditional on this knowledge being available

pith-pipeline@v0.9.0 · 5340 in / 1004 out tokens · 35073 ms · 2026-05-10T15:14:31.424082+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Hyman Bass, A remark on a arithmetic theorem of Chevalley , Proc. Am. Math. Soc. 16 (1965), 875--878

work page 1965
[2]

Yuri Bilu, The C hevalley- B ass T heorem , arXiv:2305.05041, 2023

work page arXiv 2023
[3]

Yuri Bilu, Florian Luca, Joris Nieuwveld, Jo \"e l Ouaknine, and James Worrell, Twisted rational zeros of linear recurrence sequences, J. Lond. Math. Soc., II. Ser. 111 (2025), no. 3, 29, Id/No e70126

work page 2025
[4]

Baker and Kenneth A

Matthew H. Baker and Kenneth A. Ribet, Galois theory and torsion points on curves., J. Th \'e or. Nombres Bordx. 15 (2003), no. 1, 11--32

work page 2003
[5]

Claude Chevalley, Deux th \'e or \`e mes d'arithm \'e tique , J. Math. Soc. Japan 3 (1951), 36--44

work page 1951
[6]

Uwe Jannsen, The splitting of the Hochschild - Serre spectral sequence for a product of groups , Can. Math. Bull. 33 (1990), no. 2, 181--183 (English)

work page 1990
[7]

Michel Laurent, \'E quations diophantiennes exponentielles , Invent. Math. 78 (1984), 299--327

work page 1984

[1] [1]

Hyman Bass, A remark on a arithmetic theorem of Chevalley , Proc. Am. Math. Soc. 16 (1965), 875--878

work page 1965

[2] [2]

Yuri Bilu, The C hevalley- B ass T heorem , arXiv:2305.05041, 2023

work page arXiv 2023

[3] [3]

Yuri Bilu, Florian Luca, Joris Nieuwveld, Jo \"e l Ouaknine, and James Worrell, Twisted rational zeros of linear recurrence sequences, J. Lond. Math. Soc., II. Ser. 111 (2025), no. 3, 29, Id/No e70126

work page 2025

[4] [4]

Baker and Kenneth A

Matthew H. Baker and Kenneth A. Ribet, Galois theory and torsion points on curves., J. Th \'e or. Nombres Bordx. 15 (2003), no. 1, 11--32

work page 2003

[5] [5]

Claude Chevalley, Deux th \'e or \`e mes d'arithm \'e tique , J. Math. Soc. Japan 3 (1951), 36--44

work page 1951

[6] [6]

Uwe Jannsen, The splitting of the Hochschild - Serre spectral sequence for a product of groups , Can. Math. Bull. 33 (1990), no. 2, 181--183 (English)

work page 1990

[7] [7]

Michel Laurent, \'E quations diophantiennes exponentielles , Invent. Math. 78 (1984), 299--327

work page 1984