A convexity proof of Pohst's inequality

Scott Duke Kominers

arxiv: 2606.08665 · v1 · pith:D2Z4LH5Ynew · submitted 2026-06-07 · 🧮 math.NT · math.CO

A convexity proof of Pohst's inequality

Scott Duke Kominers This is my paper

Pith reviewed 2026-06-27 17:54 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords Pohst's inequalityRaposo refinementsigned inequalityconvexity proofanalytic proofalgebraic number theorylattice bounds

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The pith

Convexity of key quantities yields a short analytic proof of Raposo's signed refinement of Pohst's inequality.

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

The paper supplies an analytic proof that Raposo's signed version of Pohst's inequality holds. It proceeds by verifying that the quantities in the refinement obey convexity conditions that permit direct application of standard analytic inequalities. The resulting argument is shorter than prior ones and replaces algebraic machinery with convexity. A reader would care because the refinement sharpens a classical bound that appears in the study of algebraic integers and lattices.

Core claim

Raposo's signed refinement of Pohst's inequality follows at once from the convexity of the functions that appear when the inequality is written in signed form, supplying an analytic derivation that avoids the longer algebraic arguments previously used.

What carries the argument

Convexity of the quantities in the signed refinement, which licenses the direct analytic comparison that establishes the inequality.

If this is right

The signed refinement holds for the class of objects to which Pohst's inequality applies.
Verification of the inequality can be performed by checking convexity rather than by direct algebraic manipulation.
The same convexity properties may be reusable for other refinements that share the same functional form.

Where Pith is reading between the lines

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

The method could be tested on similar inequalities whose statements involve sign choices or absolute values.
If the convexity can be established uniformly, the proof technique might extend to higher-dimensional or non-commutative analogues without new algebraic input.

Load-bearing premise

The quantities appearing in the signed refinement satisfy the convexity properties needed for the analytic argument to go through.

What would settle it

An explicit numerical instance in which the signed refinement fails, or in which one of the relevant quantities is shown to be non-convex, would refute the argument.

read the original abstract

We give a short analytic proof of Raposo's signed refinement of Pohst's inequality.

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

Kominers gives a short convexity-based analytic proof of Raposo's already-known signed refinement of Pohst's inequality.

read the letter

Kominers has written a short note that gives an analytic proof, based on convexity, of the signed refinement of Pohst's inequality that Raposo previously established. That's the core of it.

The new element is the proof technique itself. Using convexity properties to derive the inequality analytically is distinct from prior approaches, and the paper presents it as a straightforward argument. This could appeal to readers who prefer analytic methods over more algebraic ones in this area of number theory.

It does well in keeping the argument short and focused, as the abstract promises. If the convexity holds for the quantities involved, this provides a clean way to see the result.

The main uncertainty is whether the convexity properties actually apply in the way needed. The abstract doesn't include the steps, so it's not possible to verify if there are any gaps in showing that the relevant functions or quantities are convex, or if additional conditions are required. That matches the reader's note on the weakest assumption.

Overall, this is a modest but honest piece of work that fills in a proof for an existing statement. It's aimed at specialists in algebraic number theory who work with inequalities like Pohst's, perhaps in the context of discriminant bounds or related problems. A reader looking for alternative proofs or convexity applications in number theory would get some value from it.

It deserves a serious referee because the claim is precise and the method is new enough to warrant checking, even though the underlying inequality isn't novel. I'd recommend sending it to peer review rather than desk rejecting it.

Referee Report

1 major / 0 minor

Summary. The manuscript asserts that it provides a short analytic proof of Raposo's signed refinement of Pohst's inequality based on convexity properties of the relevant quantities.

Significance. If the convexity argument is valid and free of hidden assumptions, a short analytic proof would be a modest but useful contribution to algebraic number theory by simplifying access to Raposo's refinement of Pohst's inequality. The manuscript supplies no machine-checked proofs, code, or explicit parameter-free derivations to strengthen this assessment.

major comments (1)

The provided manuscript text consists solely of the title and the one-sentence abstract; no derivation, convexity verification, or explicit argument is supplied. This prevents any check that the quantities in the signed refinement satisfy the convexity properties required for the analytic argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The observation that the submitted version contained only the title and abstract is accurate, and we will address this by including the full convexity-based derivation in the revision.

read point-by-point responses

Referee: The provided manuscript text consists solely of the title and the one-sentence abstract; no derivation, convexity verification, or explicit argument is supplied. This prevents any check that the quantities in the signed refinement satisfy the convexity properties required for the analytic argument.

Authors: We agree with this assessment. The submitted file was incomplete and contained only the abstract. The revised manuscript will include the explicit analytic argument establishing the required convexity of the relevant functions in Raposo's signed refinement, allowing direct verification of the properties used. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a short analytic proof of an existing inequality (Raposo's signed refinement of Pohst's) via convexity properties. No derivation steps, equations, or citations are supplied that reduce the claimed result to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The argument is therefore self-contained as a standard convexity-based proof and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the precise free parameters, axioms, and entities cannot be enumerated. The proof is described as relying on convexity, which is a standard tool in real analysis.

axioms (1)

standard math Standard properties of convex functions on the reals
The title and abstract indicate that the proof proceeds by establishing or invoking convexity of the relevant functions.

pith-pipeline@v0.9.1-grok · 5511 in / 969 out tokens · 23285 ms · 2026-06-27T17:54:56.309962+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references

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Francesco Battistoni and Giuseppe Molteni,Generalization of a Pohst’s inequality, J. Number Theory228(2021), 73–86

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Eduardo Friedman,Analytic formulas for the regulator of a number field, Invent. Math.98(1989), no. 3, 599–622

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Number Theory198(2019), 381–385

Eduardo Friedman and Gabriel Ramirez-Raposo,Filling the gap in the table of small- est regulators up to degree7, J. Number Theory198(2019), 381–385

2019
[8]

Number Theory9(1977), 459–492

Michael Pohst,Regulatorabsch¨ atzungen f¨ ur total reelle algebraische Zahlk¨ orper, J. Number Theory9(1977), 459–492

1977
[9]

Gabriel Raposo,A proof of Pohst’s inequality, Funct. Approx. Comment. Math.70 (2024), no. 1, 71–83

2024
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,A refinement of Pohst’s inequality, J. Th´ eor. Nombres Bordeaux37(2025), no. 3, 747–773

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Harvard Business School; Department of Economics and Center of Mathe- matical Sciences and Applications, Harvard University; and a16z crypto Email address:kominers@fas.harvard.edu

Robert Remak, ¨Uber Gr¨ ossenbeziehungen zwischen Diskriminante und Regulator eines algebraischen Zahlk¨ orpers, Compositio Math.10(1952), 245–285. Harvard Business School; Department of Economics and Center of Mathe- matical Sciences and Applications, Harvard University; and a16z crypto Email address:kominers@fas.harvard.edu

1952

[1] [1]

Number Theory167(2016), 232–258

Sergio Astudillo, Francisco Diaz y Diaz, and Eduardo Friedman,Sharp lower bounds for regulators of small-degree number fields, J. Number Theory167(2016), 232–258

2016

[2] [2]

Unione Mat

Francesco Battistoni,A conjectural improvement for inequalities related to regulators of number fields, Boll. Unione Mat. Ital.14(2021), no. 4, 609–627

2021

[3] [3]

Number Theory228(2021), 73–86

Francesco Battistoni and Giuseppe Molteni,Generalization of a Pohst’s inequality, J. Number Theory228(2021), 73–86

2021

[4] [4]

Comp.94(2025), no

,Generalized Pohst inequality and small regulators, Math. Comp.94(2025), no. 351, 475–504

2025

[5] [5]

M. J. Bertin,Sur une conjecture de Pohst, Acta Arith.74(1996), no. 4, 347–349

1996

[6] [6]

Math.98(1989), no

Eduardo Friedman,Analytic formulas for the regulator of a number field, Invent. Math.98(1989), no. 3, 599–622

1989

[7] [7]

Number Theory198(2019), 381–385

Eduardo Friedman and Gabriel Ramirez-Raposo,Filling the gap in the table of small- est regulators up to degree7, J. Number Theory198(2019), 381–385

2019

[8] [8]

Number Theory9(1977), 459–492

Michael Pohst,Regulatorabsch¨ atzungen f¨ ur total reelle algebraische Zahlk¨ orper, J. Number Theory9(1977), 459–492

1977

[9] [9]

Gabriel Raposo,A proof of Pohst’s inequality, Funct. Approx. Comment. Math.70 (2024), no. 1, 71–83

2024

[10] [10]

,A refinement of Pohst’s inequality, J. Th´ eor. Nombres Bordeaux37(2025), no. 3, 747–773

2025

[11] [11]

Harvard Business School; Department of Economics and Center of Mathe- matical Sciences and Applications, Harvard University; and a16z crypto Email address:kominers@fas.harvard.edu

Robert Remak, ¨Uber Gr¨ ossenbeziehungen zwischen Diskriminante und Regulator eines algebraischen Zahlk¨ orpers, Compositio Math.10(1952), 245–285. Harvard Business School; Department of Economics and Center of Mathe- matical Sciences and Applications, Harvard University; and a16z crypto Email address:kominers@fas.harvard.edu

1952