Characterization of Gaps and Elements of a Numerical Semigroup Using Groebner Bases

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

arxiv: 1907.01217 · v1 · pith:KMBFNZ4Tnew · submitted 2019-07-02 · 🧮 math.CO · math.AC

Characterization of Gaps and Elements of a Numerical Semigroup Using Groebner Bases

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

Pith reviewed 2026-05-25 11:20 UTC · model grok-4.3

classification 🧮 math.CO math.AC

keywords numerical semigroupsGroebner basesgapselementspolynomial idealsideal membershipcharacterization

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The pith

Numerical semigroups have their elements and gaps characterized using Groebner bases of associated polynomial ideals.

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

The paper develops a systematic use of Groebner bases to handle problems in numerical semigroups. It translates questions about semigroup membership and gaps into ideal membership questions in a polynomial ring. This yields a characterization of the elements of the semigroup expressed in terms of the Groebner bases. The approach also allows uniform proofs of some existing results on numerical semigroups. A reader would care because it turns combinatorial questions into decidable algebraic computations.

Core claim

The elements of a numerical semigroup can be characterized in terms of Groebner bases of certain ideals constructed from the semigroup generators, and this same translation systematically resolves related problems such as identifying gaps.

What carries the argument

Groebner bases of polynomial ideals associated to the numerical semigroup, which decide membership questions and identify the gaps.

If this is right

An integer belongs to the semigroup precisely when it satisfies a condition derived from the Groebner basis of the ideal.
The gaps of the semigroup correspond to specific monomials not reducible via the leading ideal.
Some known results on numerical semigroups receive uniform proofs through the same ideal-theoretic setup.
The method provides an algorithmic procedure for computing the set of elements and gaps.

Where Pith is reading between the lines

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

The same translation technique could be tested on other classes of semigroups to see if analogous characterizations emerge.
It suggests direct links between combinatorial number theory and effective methods in computational algebra.
One could implement the construction for small generators and compare the output gaps against the known list to check consistency.

Load-bearing premise

Problems about numerical semigroups can be translated into polynomial ideal membership questions in a way that Groebner bases systematically resolve.

What would settle it

A concrete numerical semigroup where the Groebner basis of the associated ideal fails to correctly identify an element or gap according to the stated characterization.

read the original abstract

This article is partly a survey and partly a research paper. It tackles the use of Groebner bases for addressing problems of numerical semigroups, which is a topic that has been around for some years, but it does it in a systematic way which enables us to prove some results and a hopefully interesting characterization of the elements of a semigroup in terms of Groebner bases.

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 a systematic Groebner basis setup for numerical semigroups that yields a clean characterization of elements and gaps, but most of the material is a reorganization of known ideas.

read the letter

This paper takes the existing link between numerical semigroups and Groebner bases and applies it in a more organized fashion. The result is a characterization that reads the elements and gaps straight off the basis of the associated ideal, plus some proofs that follow from the same construction. The systematic translation is the useful part: once the ideal is set up, standard basis computations handle membership and gap detection without case-by-case arguments. That approach is reliable and could save time for anyone doing explicit calculations in this area. The authors are upfront that much of the work is a survey of techniques that have existed for years, so the new piece is mainly the uniform presentation and the derived characterization. The soft spot is the limited novelty. The characterization looks like a direct consequence of the translation rather than a result that opens new questions or handles cases the old methods could not reach. No evidence is given here of faster algorithms, new examples that were previously intractable, or extensions beyond the usual cofinite case. The paper stays inside the standard algebraic setup for numerical semigroups and does not claim broader impact. It is written for specialists already working on numerical semigroups who want a single reference for the Groebner basis method. A reader outside that niche will not find much to take away. The work is honest and the claims are modest, so it deserves a serious referee who can check the details of the ideal construction and the characterization. I would send it for review in a combinatorial algebra journal.

Referee Report

0 major / 2 minor

Summary. The manuscript is partly a survey and partly a research contribution on the systematic application of Gröbner bases to problems in numerical semigroup theory. It claims that this approach enables proofs of certain results and yields a characterization of the gaps and elements of a numerical semigroup in terms of Gröbner bases.

Significance. If the claimed characterization and systematic translation to ideal-membership problems hold, the work would supply a concrete computational bridge between commutative algebra and numerical semigroup theory, potentially allowing machine-assisted proofs and new algorithms for membership and gap computation. The survey component could also consolidate existing scattered applications of Gröbner bases in the area.

minor comments (2)

The abstract asserts that the systematic method 'enables us to prove some results and a hopefully interesting characterization' but supplies neither an explicit statement of the characterization nor any illustrative equation, example, or reference to a specific theorem number. Adding a concise statement of the main characterization (e.g., in terms of leading terms or standard monomials) would allow readers to assess the claim immediately.
Because the manuscript is described as partly a survey, a brief comparison table or section that distinguishes the new systematic results from previously known Gröbner-basis applications to numerical semigroups would clarify the incremental contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript arXiv:1907.01217. The report correctly identifies the work as partly a survey and partly a research contribution that applies Gröbner bases systematically to numerical semigroup problems, including a characterization of gaps and elements. No specific major comments appear in the provided report, so we offer no point-by-point responses below. We remain ready to supply further details, examples, or clarifications on the claimed characterization if the editor requests them.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper describes a systematic translation of numerical semigroup problems (gaps, membership) into polynomial ideal membership questions solved via Groebner bases, yielding a characterization of elements. No equations, self-definitional constructions, fitted parameters renamed as predictions, or load-bearing self-citations are present in the provided abstract or description. The central claim rests on an algebraic reformulation whose validity is independent of the result itself and can be verified externally via standard Groebner basis algorithms. This is a standard non-circular research contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities; all ledger entries are therefore empty.

pith-pipeline@v0.9.0 · 5588 in / 928 out tokens · 21335 ms · 2026-05-25T11:20:34.464886+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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Sammartano, A.: Numerical semigroups with large embed ding dimension satisfy Wilf’s con- jecture. Semigroup Forum 85 (2012) 439–447

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W ang, X.; Yau, S.S.T.: On the GLY conjecture of upper est imate of positive integral points in real right-angled simplices. J. Number Theory 122 (2007) 184–210

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[16]

Wilf, H.S.: A circle–of–lights algorithm for the money changing problem. Amer. Math. Monthly 85 (1978) 562–565

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[17]

Xu, Y.J.; Yau, S.S.T.: A sharp estimate of number of inte gral points in a tetrahedron. J. Reine Angew. Math. 423 (1992) 199–219

work page 1992
[18]

Xu, Y.J.; Yau, S.S.T.: Durfee conjecture and coordinat e free characterization of homogeneous singularities. J. Diﬀerential Geom. 37 (1993) 375–396

work page 1993
[19]

Xu, Y.J.; Yau, S.S.T.: A sharp estimate of number of inte gral points in a 4–dimensional tetrahedra. J. Reine Angew. Math. 473 (1996) 1–23

work page 1996
[20]

Yau, S.S.T.; Zhang, L.: An upper estimate of integral po ints in real simplices with an appli- cation to singularity theory. Math. Res. Lett. 13 (2006) 911–921. Departamento de ´Algebra, Universidad de Sevilla. P.O. 1160. 41080 Sevilla, Sp ain. E-mail address : gmarquez@us.es Departamento de ´Algebra, Universidad de Sevilla. P.O. 1160. 41080 Sevilla, Sp a...

work page 2006

[1] [1]

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

work page 1994

[2] [2]

Semigroup Forum 76 (2008) 379–384

Bras-Amor´ os, M.: Fibonacci–like behavior of the numbe r of numerical semigroups of a given genus. Semigroup Forum 76 (2008) 379–384

work page 2008

[3] [3]

Duke Math

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

work page 1996

[4] [4]

Springer, 2009

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

work page 2009

[5] [5]

Manuscripta Math

Herzog, J.: Generators and relations of abelian semigro ups and semigroup rings. Manuscripta Math. 3 (1970) 175–193

work page 1970

[6] [6]

Kaplan, N.: Counting numerical semigroups by genus and s ome cases of a question of Wilf. J. Pure Appl. Algebra 216 (2012) 1016–1032

work page 2012

[7] [7]

Lin, K.P.; Yau, S.T.: Analysis of sharp polynomial upper estimate of number of positive integral points in 4–dimensional tetrahedra. J. Reine Angew. Math. 547 (2002) 191–205

work page 2002

[8] [8]

Lin,K.P.; Yau, S.T.: Analysis of sharp polynomial upper estimate of number of positive integral points in 5–dimensional tetrahedra. J. Number Theory 93 (2002) 207–234

work page 2002

[9] [9]

Lin, K.P.; Yau, S.S.T.: Counting the number of integral p oints in general n–dimensional tetrahedra and Bernoulli polynomials. Canad. Math. Bull. 24 (2003) 229–241

work page 2003

[10] [10]

Combinatorica 16 (1996) 143–147

Ram ´ ırez–Alfons ´ ın, J.L.: Complexity of the Frobeniu s problem. Combinatorica 16 (1996) 143–147

work page 1996

[11] [11]

Oxford University Press, 2005

Ram ´ ırez–Alfons ´ ın, J.L.:The Diophantine Frobenius problem . Oxford University Press, 2005

work page 2005

[12] [12]

Semigroup Forum 85 (2012) 439–447

Sammartano, A.: Numerical semigroups with large embed ding dimension satisfy Wilf’s con- jecture. Semigroup Forum 85 (2012) 439–447

work page 2012

[13] [13]

Educational Times 37 (1884) 26

Sylvester, J.J.: Problem 7382. Educational Times 37 (1884) 26

work page

[14] [14]

Sylvester, J.J.: On the partition of numbers. Quart. J. Pure Appl. Math. 1 (1857) 141–152

work page

[15] [15]

W ang, X.; Yau, S.S.T.: On the GLY conjecture of upper est imate of positive integral points in real right-angled simplices. J. Number Theory 122 (2007) 184–210

work page 2007

[16] [16]

Wilf, H.S.: A circle–of–lights algorithm for the money changing problem. Amer. Math. Monthly 85 (1978) 562–565

work page 1978

[17] [17]

Xu, Y.J.; Yau, S.S.T.: A sharp estimate of number of inte gral points in a tetrahedron. J. Reine Angew. Math. 423 (1992) 199–219

work page 1992

[18] [18]

Xu, Y.J.; Yau, S.S.T.: Durfee conjecture and coordinat e free characterization of homogeneous singularities. J. Diﬀerential Geom. 37 (1993) 375–396

work page 1993

[19] [19]

Xu, Y.J.; Yau, S.S.T.: A sharp estimate of number of inte gral points in a 4–dimensional tetrahedra. J. Reine Angew. Math. 473 (1996) 1–23

work page 1996

[20] [20]

Yau, S.S.T.; Zhang, L.: An upper estimate of integral po ints in real simplices with an appli- cation to singularity theory. Math. Res. Lett. 13 (2006) 911–921. Departamento de ´Algebra, Universidad de Sevilla. P.O. 1160. 41080 Sevilla, Sp ain. E-mail address : gmarquez@us.es Departamento de ´Algebra, Universidad de Sevilla. P.O. 1160. 41080 Sevilla, Sp a...

work page 2006