Short Salem polynomials

Chris Smyth; James McKee

arxiv: 2605.19013 · v1 · pith:7YFWDG4Fnew · submitted 2026-05-18 · 🧮 math.NT

Short Salem polynomials

James McKee , Chris Smyth This is my paper

Pith reviewed 2026-05-20 07:47 UTC · model grok-4.3

classification 🧮 math.NT

keywords Salem polynomialsSalem numbersLehmer's conjectureinfinite familiesclassificationreciprocal polynomialsPisot numbersminimal polynomials

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The pith

All Salem polynomials of length 5 are completely classified while for length 6 all but finitely many lie in one of 12 infinite families with 126 exceptions under Lehmer's conjecture.

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

The paper classifies all Salem polynomials of length 5. For length 6 it shows that all but finitely many belong to one of 12 infinite families. Subject to Lehmer's conjecture the exceptions total 126. A table of short polynomials is given for known Salem numbers below the smallest Pisot number.

Core claim

We give a complete classification of all Salem polynomials of length 5. For length 6 we show that all but finitely many Salem polynomials lie in one of 12 infinite families, and subject to Lehmer's Conjecture we give a complete list of the 126 exceptions. We provide a table of short polynomials for all known Salem numbers below the smallest Pisot number.

What carries the argument

Classification of Salem polynomials by length, with reduction of length-6 cases to twelve infinite families plus a finite list of exceptions.

If this is right

Every Salem polynomial of length 5 appears in an explicit list.
Length-6 Salem polynomials fall into the 12 families or the 126 exceptions.
The table links short polynomials to all known small Salem numbers.
The result for length 5 requires no extra assumptions.
The length-6 result becomes complete once Lehmer's conjecture is assumed.

Where Pith is reading between the lines

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

Analogous infinite families may exist for Salem polynomials of length greater than 6.
The listed exceptions could be checked directly to test Lehmer's conjecture in this setting.
The classification provides a foundation for studying the distribution of Salem numbers by their polynomial lengths.
Short polynomials may correspond to Salem numbers with small Mahler measure.

Load-bearing premise

Lehmer's conjecture must hold so that the 126 exceptions exhaust all cases outside the twelve families for length 6.

What would settle it

A Salem polynomial of length 6 not belonging to any of the twelve families and not among the 126 exceptions would show the classification is incomplete.

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

McKee and Smyth classify all length-5 Salem polynomials and reduce length 6 to twelve families plus 126 conditional exceptions.

read the letter

The main thing to know is that this paper gives a complete classification of Salem polynomials of length 5 and shows that length-6 ones fall into one of twelve explicit infinite families except for 126 cases, provided Lehmer's conjecture holds. They also include a table of short polynomials tied to known Salem numbers below the smallest Pisot number. This is concrete work that supplies usable lists and parametric forms rather than just existence statements. The explicit families for length 6 and the full length-5 list look like genuine additions that do not reduce to earlier published results on Salem numbers. The table is a practical compilation that connects the classification to existing data in the field. People searching for small Mahler measures can pull the lists directly for checks or bounds. The argument stays within standard definitions and does not create circular dependencies inside the paper itself. The finiteness claim for the length-6 exceptions stands independently of the conjecture, which keeps the core structure intact. The main limitation is the conditional status of the 126 exceptions. If Lehmer's conjecture turns out false, additional polynomials outside the families could exist, though the paper states this limitation clearly. Without the full proofs it is hard to confirm that every case was covered in the enumeration, but the abstract and stress-test note give no sign of an internal gap that would break the stated claims. This paper is for algebraic number theorists working on Salem numbers, Mahler measures, and related classification questions. Readers who need explicit data or families for computation or further bounds would get direct value from it. It has enough specificity and engagement with the literature to deserve a serious referee rather than a desk reject. I would send it out for peer review.

Referee Report

1 major / 3 minor

Summary. The manuscript claims a complete classification of all Salem polynomials of length 5. For length 6 it proves that all but finitely many such polynomials belong to one of 12 explicit infinite families and, subject to Lehmer's conjecture, supplies an explicit list of the 126 exceptions. A table of short polynomials for known Salem numbers below the smallest Pisot number is also included.

Significance. If the stated classifications are correct, the work supplies concrete infinite families and a conditional finite list that organize the landscape of short Salem polynomials, directly supporting computational and theoretical study of small Mahler measures. The explicit families constitute a structural result independent of the conjecture, while the table provides immediate reference data for known examples.

major comments (1)

[Length-6 classification section] The completeness claim for the 126 length-6 exceptions is explicitly conditional on Lehmer's conjecture; the manuscript should verify that the bounding argument used to produce the finite list (e.g., via Mahler-measure estimates or root-location constraints) is stated with the precise dependence on the conjecture so that the finiteness statement remains unconditional.

minor comments (3)

[Introduction] Define the term 'length' for Salem polynomials at first use; the abstract and introduction should make clear whether it refers to the number of non-zero coefficients, the degree, or another invariant.
[Table of short polynomials] In the table of known Salem numbers, add explicit citations or references for each listed Salem number so that the data can be independently verified.
[Length-6 families] Ensure that the 12 infinite families for length 6 are presented with explicit generating polynomials or recurrence relations so that membership in a family can be checked algorithmically.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the helpful suggestion for improving the clarity of the length-6 results. We agree that making the dependence on Lehmer's conjecture explicit will strengthen the manuscript and will revise accordingly.

read point-by-point responses

Referee: [Length-6 classification section] The completeness claim for the 126 length-6 exceptions is explicitly conditional on Lehmer's conjecture; the manuscript should verify that the bounding argument used to produce the finite list (e.g., via Mahler-measure estimates or root-location constraints) is stated with the precise dependence on the conjecture so that the finiteness statement remains unconditional.

Authors: We thank the referee for this observation. The finiteness of the exceptions outside the twelve families follows from an unconditional upper bound on the Mahler measure together with root-location constraints that do not invoke Lehmer's conjecture; this part of the argument therefore remains unconditional. The explicit enumeration yielding the list of 126 exceptions is obtained by exhaustive search within that bound and is complete only under Lehmer's conjecture. In the revised manuscript we will add a precise statement in the length-6 classification section that separates the unconditional finiteness result from the conditional completeness of the finite list, thereby making the dependence on the conjecture fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper classifies Salem polynomials of length 5 completely and shows that length-6 cases fall into 12 explicit infinite families (with 126 exceptions listed conditionally on the external Lehmer conjecture). No derivation step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the argument uses standard definitions of Salem numbers together with algebraic enumeration and finiteness arguments that remain independent of the conjecture. The conditional statement on exceptions is presented transparently rather than smuggled in as an internal result. This is a normal non-circular outcome for a classification paper relying on external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The classification rests on the standard definition of Salem numbers as reciprocal algebraic integers with the required conjugate distribution and on the external Lehmer conjecture for the length-6 completeness statement. No free parameters or new entities are introduced.

axioms (1)

domain assumption Salem numbers are algebraic integers >1 whose conjugates lie inside or on the unit circle with at least one on the circle, and whose minimal polynomials are reciprocal.
This is the foundational definition used for the entire classification.

pith-pipeline@v0.9.0 · 5563 in / 1257 out tokens · 65098 ms · 2026-05-20T07:47:23.175279+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2. There are 12 infinite families of length-6 Salem polynomials, of the shape Pn(z)=z^n P(z)+ε P^*(z), where P(z)∈{z−2,z^2−z−1,...}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

D. W. Boyd, Small Salem numbers. Duke Math. J.44(1977), no. 2, 315-328

work page 1977
[2]

, Pisot and Salem numbers in intervals of the real line. Math. Comp.32(1978), no. 144, 1244-1260

work page 1978
[3]

, Speculations concerning the range of Mahler’s measure, Canad. Math. Bull.24(1981), no. 4, 453–469

work page 1981
[4]

Chinburg, On the arithmetic of two constructions of Salem numbers, J

T. Chinburg, On the arithmetic of two constructions of Salem numbers, J. Reine Angew. Math.348 (1984), 166–179

work page 1984
[5]

Dobrowolski, Mahler’s measure of a polynomial in function of the number of its coefficients

E. Dobrowolski, Mahler’s measure of a polynomial in function of the number of its coefficients. Canad. Math. Bull.34(2), 186–195 (1991)

work page 1991
[6]

Acta Arith

, Mahler’s measure of a polynomial in terms of the number of its monomials. Acta Arith. 123(3) (2006), 201–231

work page 2006
[7]

El-Serafy, J

S. El-Serafy, J. F. McKee, Small Mahler measures with bounds on the house and shortness. Canad. Math. Bull.68, no. 2 (2025), 603–619

work page 2025
[8]

Ghate, E

E. Ghate, E. Hironaka, The arithmetic and geometry of Salem numbers, Bull. Amer. Math. Soc. (N.S.)38(2001), no. 3, 293–314

work page 2001
[9]

D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2)34(1933), no. 3, 461-479

work page 1933
[10]

J. F. McKee, C. J. Smyth, Around the unit circle. Universitext, Springer, Cham, 2021

work page 2021
[11]

M. J. Mossinghoff, http://wayback.cecm.sfu.ca/∼mjm/Lehmer/lists/SalemList.html [Currently of- fline]

work page
[12]

, Polynomials with small Mahler measure. Math. Comp.67(1998), no. 224, 1697-1705, S11- S14

work page 1998
[13]

Sac- ´Ep´ ee, Salem numbers less than 49/37

J.-M. Sac- ´Ep´ ee, Salem numbers less than 49/37. Rocky Mountain Journal of Mathematics (in press). Also arXiv 2409.11159v3, hal-04700271v3

work page arXiv
[14]

Salem, Power series with integral coefficients

R. Salem, Power series with integral coefficients. Duke Math. J.12(1945), 153–172

work page 1945
[15]

, Algebraic numbers and Fourier analysis. D. C. Heath and Company, Boston, (1963)

work page 1963
[16]

C. L. Siegel, Algebraic integers whose conjugates lie in the unit circle. Duke Math. J.11(1944), 597–602

work page 1944
[17]

C. J. Smyth, Salem numbers of negative trace. Math. Comp.69(2000), no. 230, 827–838

work page 2000
[18]

, Seventy years of Salem numbers. Bull. Lond. Math. Soc.47(2015), no. 3, 379–395. Royal Holloway, University of London, Egham Hill, Egham, Surrey TW20 0EX, UK Email address:james.mckee@rhul.ac.uk University of Edinburgh, Edinburgh EH9 3FD, Scotland, UK Email address:c.smyth@ed.ac.uk

work page 2015

[1] [1]

D. W. Boyd, Small Salem numbers. Duke Math. J.44(1977), no. 2, 315-328

work page 1977

[2] [2]

, Pisot and Salem numbers in intervals of the real line. Math. Comp.32(1978), no. 144, 1244-1260

work page 1978

[3] [3]

, Speculations concerning the range of Mahler’s measure, Canad. Math. Bull.24(1981), no. 4, 453–469

work page 1981

[4] [4]

Chinburg, On the arithmetic of two constructions of Salem numbers, J

T. Chinburg, On the arithmetic of two constructions of Salem numbers, J. Reine Angew. Math.348 (1984), 166–179

work page 1984

[5] [5]

Dobrowolski, Mahler’s measure of a polynomial in function of the number of its coefficients

E. Dobrowolski, Mahler’s measure of a polynomial in function of the number of its coefficients. Canad. Math. Bull.34(2), 186–195 (1991)

work page 1991

[6] [6]

Acta Arith

, Mahler’s measure of a polynomial in terms of the number of its monomials. Acta Arith. 123(3) (2006), 201–231

work page 2006

[7] [7]

El-Serafy, J

S. El-Serafy, J. F. McKee, Small Mahler measures with bounds on the house and shortness. Canad. Math. Bull.68, no. 2 (2025), 603–619

work page 2025

[8] [8]

Ghate, E

E. Ghate, E. Hironaka, The arithmetic and geometry of Salem numbers, Bull. Amer. Math. Soc. (N.S.)38(2001), no. 3, 293–314

work page 2001

[9] [9]

D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2)34(1933), no. 3, 461-479

work page 1933

[10] [10]

J. F. McKee, C. J. Smyth, Around the unit circle. Universitext, Springer, Cham, 2021

work page 2021

[11] [11]

M. J. Mossinghoff, http://wayback.cecm.sfu.ca/∼mjm/Lehmer/lists/SalemList.html [Currently of- fline]

work page

[12] [12]

, Polynomials with small Mahler measure. Math. Comp.67(1998), no. 224, 1697-1705, S11- S14

work page 1998

[13] [13]

Sac- ´Ep´ ee, Salem numbers less than 49/37

J.-M. Sac- ´Ep´ ee, Salem numbers less than 49/37. Rocky Mountain Journal of Mathematics (in press). Also arXiv 2409.11159v3, hal-04700271v3

work page arXiv

[14] [14]

Salem, Power series with integral coefficients

R. Salem, Power series with integral coefficients. Duke Math. J.12(1945), 153–172

work page 1945

[15] [15]

, Algebraic numbers and Fourier analysis. D. C. Heath and Company, Boston, (1963)

work page 1963

[16] [16]

C. L. Siegel, Algebraic integers whose conjugates lie in the unit circle. Duke Math. J.11(1944), 597–602

work page 1944

[17] [17]

C. J. Smyth, Salem numbers of negative trace. Math. Comp.69(2000), no. 230, 827–838

work page 2000

[18] [18]

, Seventy years of Salem numbers. Bull. Lond. Math. Soc.47(2015), no. 3, 379–395. Royal Holloway, University of London, Egham Hill, Egham, Surrey TW20 0EX, UK Email address:james.mckee@rhul.ac.uk University of Edinburgh, Edinburgh EH9 3FD, Scotland, UK Email address:c.smyth@ed.ac.uk

work page 2015