Une nouvelle minoration pour la trace des entiers alg\'ebriques totalement positifs

V.Flammang

arxiv: 1907.09407 · v1 · pith:IYDYKTI4new · submitted 2019-07-22 · 🧮 math.NT

Une nouvelle minoration pour la trace des entiers alg\'ebriques totalement positifs

V.Flammang This is my paper

Pith reviewed 2026-05-24 17:47 UTC · model grok-4.3

classification 🧮 math.NT

keywords totally positive algebraic integersabsolute tracelower boundsrecursive algorithmreciprocal integers

0 comments

The pith

A slight variant in a recursive algorithm improves lower bounds for the absolute trace of totally positive algebraic integers.

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

The paper shows that modifying the application of an existing recursive algorithm produces stronger lower bounds on the absolute trace of totally positive algebraic integers. It also connects the absolute trace of such an integer to the absolute trace of a totally positive reciprocal integer. A reader would care because these refinements help determine the smallest possible traces for algebraic integers of given degree.

Core claim

A slight variant in the use of the recursive algorithm leads to improved known lower bounds for the absolute trace of a totally positive algebraic integer. This approach also links the absolute trace of a totally positive algebraic integer and the absolute trace of a totally positive reciprocal integer.

What carries the argument

The recursive algorithm for computing lower bounds on traces, applied with a slight variant in its usage.

If this is right

Improved lower bounds are obtained for the absolute trace of totally positive algebraic integers.
The absolute traces of totally positive algebraic integers and totally positive reciprocal integers are linked.
The variant method can be used to compute bounds in additional cases.

Where Pith is reading between the lines

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

The link between the two classes of integers may allow bounds computed in one setting to inform the other.
The same variant approach could be tested on related minimization problems involving algebraic integers.

Load-bearing premise

The recursive algorithm remains valid when applied with the described slight variant.

What would settle it

A totally positive algebraic integer whose absolute trace falls below the new lower bound obtained from the variant would falsify the claimed improvement.

read the original abstract

We explain how a slight variant in the use of our recursive algorithm leads to improve the known lower bounds for the absolute trace of a totally positive algebraic integer. We also link the absolute trace of a totally positive algebraic integer and the absolute trace of a totally positive reciprocal integer.

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

A minor tweak to an existing recursive algorithm is claimed to give slightly better explicit lower bounds on traces of totally positive algebraic integers, plus a link to the reciprocal case, but the gain looks incremental and the supporting details are missing.

read the letter

The main point is that this short note says a slight variant in applying their prior recursive algorithm produces improved lower bounds on the absolute trace of totally positive algebraic integers, and it also connects that quantity to the trace for totally positive reciprocal integers. The work stays inside an established computational approach rather than introducing a new framework or proof technique. If the variant is valid and the numerical gain is real, it could supply tighter constants for people who use these bounds in Diophantine problems or effective versions of results on algebraic integers. That kind of concrete sharpening is the right target for this subfield. The paper earns credit for keeping the focus on explicit, usable bounds instead of asymptotics and for noting the reciprocal link, which might allow cross-comparisons that were not obvious before. The central claim does not contain internal contradictions or obvious circular steps based on what is stated. The soft spot is the complete absence of any description of the variant, the actual new bound value, or sample calculations. Without those, it is impossible to judge whether the improvement is meaningful, whether the change introduces new conditions, or whether the numbers check out. The abstract alone supplies no data or derivation to inspect. This paper is written for the small group of researchers who already follow the sequence of papers on trace bounds for algebraic integers. A reader outside that niche will find little to use. I would not bring it to a reading group. It does not look strong enough to justify sending to peer review; the advance is too narrow and the evidence too thin to occupy referees' time.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that a slight variant in the application of an existing recursive algorithm produces improved explicit lower bounds on the absolute trace of totally positive algebraic integers, and that this variant also yields a relation linking the absolute trace in the totally positive case to the absolute trace in the totally positive reciprocal case.

Significance. If the claimed improvement is verified by explicit computation or proof and the bounds are strictly better than those in the literature, the result would constitute a modest but concrete advance in the study of minimal traces of algebraic integers. The additional link to the reciprocal case could facilitate comparisons between the two settings.

minor comments (2)

[Introduction] The abstract refers to 'our recursive algorithm' without a citation or brief recap of the prior work; adding a reference or short description in the introduction would improve accessibility.
The title is in French while the abstract is in English; the manuscript should state its primary language clearly and ensure consistent usage throughout.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. No specific major comments are listed in the report, but we address the referee summary below for completeness.

read point-by-point responses

Referee: The manuscript claims that a slight variant in the application of an existing recursive algorithm produces improved explicit lower bounds on the absolute trace of totally positive algebraic integers, and that this variant also yields a relation linking the absolute trace in the totally positive case to the absolute trace in the totally positive reciprocal case.

Authors: The manuscript provides explicit computations verifying the improved lower bounds obtained via the variant of the recursive algorithm, which are strictly better than those previously known. The relation to the reciprocal case is derived directly from the same recursive construction and is stated explicitly. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claim rests on applying a described slight variant to an existing recursive algorithm to obtain improved explicit lower bounds, together with a linking relation between absolute traces. No equations, fitted parameters, or derivations are presented that reduce by construction to the inputs themselves. The modification is presented as an independent change in application rather than a redefinition or statistical forcing of the target quantity. Self-reference to the author's prior algorithm is normal and does not constitute load-bearing circularity when the variant itself supplies the improvement.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no information sufficient to identify free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5555 in / 901 out tokens · 42338 ms · 2026-05-24T17:47:46.430453+00:00 · methodology

Une nouvelle minoration pour la trace des entiers alg\'ebriques totalement positifs

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)