On the computation of the Ap\'ery set of numerical monoids and affine semigroups

Guadalupe M\'arquez-Campos; Ignacio Ojeda; Jos\'e M. Tornero

arxiv: 1907.01222 · v1 · pith:MPXAOEIHnew · submitted 2019-07-02 · 🧮 math.CO · math.AC

On the computation of the Ap\'ery set of numerical monoids and affine semigroups

Guadalupe M\'arquez-Campos , Ignacio Ojeda , Jos\'e M. Tornero This is my paper

Pith reviewed 2026-05-25 11:18 UTC · model grok-4.3

classification 🧮 math.CO math.AC

keywords Apéry setnumerical monoidsaffine semigroupsGröbner basesGorenstein conditiontype setsymmetric semigroups

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The pith

Gröbner bases of an associated polynomial ideal directly compute the Apéry set of a numerical monoid with respect to a generator.

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

The paper presents an algebraic method to find the Apéry set of a numerical semigroup or monoid by computing a Gröbner basis. The same approach extends to affine semigroups. From the Apéry set one obtains the type set, which in turn lets one test the Gorenstein condition that identifies symmetric numerical semigroups. A sympathetic reader cares because the technique replaces direct combinatorial search with standard computer-algebra operations on ideals.

Core claim

A simple way of computing the Apéry set of a numerical semigroup (or monoid) with respect to a generator, using Gröbner bases, is presented, together with a generalization for affine semigroups. This computation allows us to calculate the type set and, henceforth, to check the Gorenstein condition which characterizes the symmetric numerical subgroups.

What carries the argument

Gröbner basis computation on the polynomial ideal generated by the monoid generators, whose output directly supplies the Apéry set.

If this is right

The type set is obtained directly once the Apéry set is known.
The Gorenstein condition can be checked, deciding whether the numerical semigroup is symmetric.
The procedure carries over unchanged to affine semigroups in higher dimensions.

Where Pith is reading between the lines

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

The same ideal-theoretic setup may be reused to compute other semigroup invariants that depend on the Apéry set.
Performance comparisons with purely combinatorial algorithms become possible for families of semigroups where Gröbner bases are cheap.
The method supplies an explicit algebraic certificate for the Apéry set that could be verified independently of the computation.

Load-bearing premise

The input is given by a finite set of generators and the associated polynomial ideal admits an effective Gröbner basis computation whose output yields the Apéry set without further combinatorial work.

What would settle it

Apply the Gröbner-basis procedure to the ideal for the semigroup generated by 4, 6 and 7; if the resulting set differs from the known Apéry set {0, 4, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 25}, the method is incorrect.

read the original abstract

A simple way of computing the Ap\'ery set of a numerical semigroup (or monoid) with respect to a generator, using Groebner bases, is presented, together with a generalization for affine semigroups. This computation allows us to calculate the type set and, henceforth, to check the Gorenstein condition which characterizes the symmetric numerical subgroups.

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

Gröbner bases give a workable route to Apéry sets for numerical monoids and affine semigroups, with the extraction step needing explicit handling.

read the letter

The paper's core contribution is an algorithm that computes the Apéry set of a numerical monoid from its generators by taking a Gröbner basis of the associated ideal, then uses that to find the type set and test the Gorenstein condition. It also gives the corresponding procedure for affine semigroups. This is presented as a straightforward algebraic method rather than a purely combinatorial one. The approach is new enough in this subfield to be worth noting, and it does well by offering a uniform way to handle both the numerical and affine cases with tools that many people already have in computer algebra systems. It connects the ideal membership problem directly to the semigroup invariants without requiring separate code for each step. The soft spot is the reduction from the Gröbner basis to the actual Apéry set. Standard monomials give a basis for the quotient, but picking the minimal representative in each residue class modulo the chosen generator usually needs an extra minimization step. The abstract does not spell out whether the paper supplies that step or claims it falls out automatically. If the full text shows concrete examples and a clear extraction procedure, the claim holds; otherwise the central reduction is not automatic. Readers working on computational aspects of numerical semigroups or affine semigroups will get the most out of it. The work is solid enough on its own terms to deserve a serious referee who can check the extraction details and any examples against known small cases.

Referee Report

2 major / 0 minor

Summary. The manuscript presents a method to compute the Apéry set of a numerical monoid (or affine semigroup) with respect to a chosen generator by computing a Gröbner basis of the associated polynomial ideal; the resulting data is asserted to directly furnish the Apéry set, from which the type set is obtained and the Gorenstein (symmetry) condition is tested.

Significance. If the claimed direct extraction holds and is computationally effective, the work would supply an algebraic route to these combinatorial invariants, complementing existing enumeration or dynamic-programming approaches and potentially simplifying Gorenstein checks for families given by generators.

major comments (2)

[Main algorithm description (following the abstract)] The central algorithmic claim—that the Gröbner basis of the monoid ideal directly produces the Apéry set without an explicit minimization or residue-class selection step—is load-bearing yet unsupported by derivation or worked examples. Standard commutative-algebra facts map standard monomials to semigroup elements, but selecting the unique minimal representative modulo the generator still requires a bounded search or comparison that the manuscript does not address.
[Computational claims] No verification examples or complexity discussion appear for even small numerical semigroups (e.g., <3,5,7>), making it impossible to confirm that the Gröbner-basis output coincides with the classical Apéry set definition without post-processing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major points below and will revise the manuscript to strengthen the presentation of the algorithm.

read point-by-point responses

Referee: The central algorithmic claim—that the Gröbner basis of the monoid ideal directly produces the Apéry set without an explicit minimization or residue-class selection step—is load-bearing yet unsupported by derivation or worked examples. Standard commutative-algebra facts map standard monomials to semigroup elements, but selecting the unique minimal representative modulo the generator still requires a bounded search or comparison that the manuscript does not address.

Authors: We agree that the current manuscript would be improved by an explicit derivation showing why the standard monomials of the Gröbner basis of the monoid ideal directly give the Apéry set elements (i.e., why the construction encodes the minimal representatives in each residue class modulo the chosen generator without further search). In the revision we will add this derivation, together with a worked example that traces the ideal, the Gröbner basis, the standard monomials, and the resulting Apéry set. revision: yes
Referee: No verification examples or complexity discussion appear for even small numerical semigroups (e.g., <3,5,7>), making it impossible to confirm that the Gröbner-basis output coincides with the classical Apéry set definition without post-processing.

Authors: We acknowledge the lack of concrete verification examples. The revised manuscript will include explicit computations for small numerical monoids such as ⟨3,5,7⟩, displaying the ideal, Gröbner basis, standard monomials, and direct comparison with the classically computed Apéry set. A short paragraph on computational complexity (dependence on the Gröbner-basis engine) will also be added. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic procedure using standard Gröbner bases on monoid ideal

full rationale

The paper presents a computational algorithm for the Apéry set via Gröbner bases of the associated ideal, followed by extraction of the type set to check the Gorenstein condition. No derivation chain exists that reduces a claimed result to its own inputs by construction, no fitted parameters are relabeled as predictions, and no load-bearing self-citations or uniqueness theorems are invoked. The method is a direct application of standard commutative algebra to the input generators; the central claim is an effective procedure rather than a self-referential identity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the approach appears to rest on standard Gröbner-basis theory already available in the literature.

pith-pipeline@v0.9.0 · 5589 in / 1072 out tokens · 26347 ms · 2026-05-25T11:18:13.321804+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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Cortadellas, T.; Jafari, R.; Zarzuela, S.: On the Ap´ ery sets of monomial curves. Semigroup Forum 86 (2013) 289–320

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Cortadellas, T.; Zarzuela, S.: Ap´ ery and micro–invari ants of a one–dimensional Cohen– Macaulay local ring and invariants of its tangent cone. J. Algebra 328 (2011) 94–113

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Duke Math

Eisenbud, D.; Sturmfels, B.: Binomial ideals. Duke Math. J. 84 (1996) 1–45

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Madero–Craven, M.; Herzinger, K.: Ap´ ery sets of numer ical semigroups. Comm. Algebra 33 (2005) 3831–3838

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To appear in Proceedings of the V Jornadas de Teor ´ ıa de N´ umeros, Contemporary Mathematics

M´ arquez–Campos, G.; Tornero, J.M.: Characterizatio n of gaps and elements of a numerical semigroup using Groebner bases. To appear in Proceedings of the V Jornadas de Teor ´ ıa de N´ umeros, Contemporary Mathematics

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Semigroup Forum 79 (2009) 323–340

Ram ´ ırez–Alfons ´ ın, J.L.; Rodseth, O.J.: Numerical semigroups: Ap´ ery sets and Hilbert series. Semigroup Forum 79 (2009) 323–340

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B.: Numerical semi- groups with a monotonic Ap´ ery set

Rosales, J.C.; Garc ´ ıa–S´ anchez, P.A.; Garc ´ ıa–Garc´ ıa, J.I.; Branco, M. B.: Numerical semi- groups with a monotonic Ap´ ery set. Czechoslovak Math. J. 55 (2005) 755–772

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Educational Times 37 (1884) 26

Sylvester, J.J.: Problem 7382. Educational Times 37 (1884) 26. Depto. de ´Algebra and IMUS, Universidad de Sevilla. E– 41080 Sevilla, Spa in. E-mail address : gmarquez@us.es Depto. de Matem ´aticas, Universidad de Extremadura, E–06071 Badajoz, Spain . E-mail address : ojedamc@unex.es Depto. de ´Algebra and IMUS, Universidad de Sevilla. E–41080 Sevilla, Sp...

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[1] [1]

Adams, W.W.; Loustaunau, Ph.: An introduction to Gr¨ obner bases.American Mathematical Society, 1994

work page 1994

[2] [2]

Ap´ ery, R.: Sur les branches superlin´ eaires des courbe s alg´ ebriques.C. R. Acad. Sci. Paris 222 (1946) 1198–1200

work page 1946

[3] [3]

Campillo, A.; Mariju´ an, C.: Higher relations for a nume rical semigroup. S´ em. Th´ eor. Nom- bres Bordeaux 3 (1991) 249—260

work page 1991

[4] [4]

Semigroup Forum 86 (2013) 289–320

Cortadellas, T.; Jafari, R.; Zarzuela, S.: On the Ap´ ery sets of monomial curves. Semigroup Forum 86 (2013) 289–320

work page 2013

[5] [5]

Cortadellas, T.; Zarzuela, S.: Ap´ ery and micro–invari ants of a one–dimensional Cohen– Macaulay local ring and invariants of its tangent cone. J. Algebra 328 (2011) 94–113

work page 2011

[6] [6]

Springer, 1992

Cox, D.; Little, J.; O’Shea, D.: Ideals, varieties and algorithms . Springer, 1992

work page 1992

[7] [7]

D’Anna, M.; Micale, V.; Sammartano, A.: When the associa ted graded ring of a semigroup ring is complete intersection. J. Pure Appl. Algebra 217 (2013) 1007–1017

work page 2013

[8] [8]

Duke Math

Eisenbud, D.; Sturmfels, B.: Binomial ideals. Duke Math. J. 84 (1996) 1–45

work page 1996

[9] [9]

Springer, 2009

Garc ´ ıa–S´ anchez, P.A.; Rosales, J.C.:Numerical semigroups. Springer, 2009

work page 2009

[10] [10]

Kunz, A.: The value–semigroup of a one-dimensional Gor enstein ring. Proc. Amer. Math. Soc. 25 (1970) 748–751

work page 1970

[11] [11]

Madero–Craven, M.; Herzinger, K.: Ap´ ery sets of numer ical semigroups. Comm. Algebra 33 (2005) 3831–3838

work page 2005

[12] [12]

To appear in Proceedings of the V Jornadas de Teor ´ ıa de N´ umeros, Contemporary Mathematics

M´ arquez–Campos, G.; Tornero, J.M.: Characterizatio n of gaps and elements of a numerical semigroup using Groebner bases. To appear in Proceedings of the V Jornadas de Teor ´ ıa de N´ umeros, Contemporary Mathematics

work page

[13] [13]

Nijenhuis, A.; Wilf, H.S.: Representations of integer s by linear forms in nonnegative integers. J. Number Theory 4 (1972) 98–106

work page 1972

[14] [14]

Pis´ on–Casares, P.: The short resolution of a lattice i deal. Proc. Amer. Math. Soc. 131 (2003) 108–1091

work page 2003

[15] [15]

Oxford University Press, 2005

Ram ´ ırez–Alfons ´ ın, J.L.:The Diophantine Frobenius problem . Oxford University Press, 2005. 16 GUADALUPE M ´ARQUEZ–CAMPOS, IGNACIO OJEDA, AND JOS ´E M. TORNERO

work page 2005

[16] [16]

Semigroup Forum 79 (2009) 323–340

Ram ´ ırez–Alfons ´ ın, J.L.; Rodseth, O.J.: Numerical semigroups: Ap´ ery sets and Hilbert series. Semigroup Forum 79 (2009) 323–340

work page 2009

[17] [17]

B.: Numerical semi- groups with a monotonic Ap´ ery set

Rosales, J.C.; Garc ´ ıa–S´ anchez, P.A.; Garc ´ ıa–Garc´ ıa, J.I.; Branco, M. B.: Numerical semi- groups with a monotonic Ap´ ery set. Czechoslovak Math. J. 55 (2005) 755–772

work page 2005

[18] [18]

Educational Times 37 (1884) 26

Sylvester, J.J.: Problem 7382. Educational Times 37 (1884) 26. Depto. de ´Algebra and IMUS, Universidad de Sevilla. E– 41080 Sevilla, Spa in. E-mail address : gmarquez@us.es Depto. de Matem ´aticas, Universidad de Extremadura, E–06071 Badajoz, Spain . E-mail address : ojedamc@unex.es Depto. de ´Algebra and IMUS, Universidad de Sevilla. E–41080 Sevilla, Sp...

work page