On numerical semigroups with embedding dimension four

Kazimierz Chomicz

arxiv: 2604.25653 · v2 · pith:3VY7AUU5new · submitted 2026-04-28 · 🧮 math.NT

On numerical semigroups with embedding dimension four

Kazimierz Chomicz This is my paper

Pith reviewed 2026-05-07 14:52 UTC · model grok-4.3

classification 🧮 math.NT

keywords numerical semigroupsembedding dimensionApéry setFrobenius numbergenusBetti elementscatenary degreeminimal presentation

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The pith

A geometric procedure determines the Apéry set for any numerical semigroup with embedding dimension four.

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

The paper develops a geometric procedure to compute the Apéry set for numerical semigroups with exactly four minimal generators. This set encodes the smallest representatives of the semigroup in each residue class modulo its multiplicity, which in turn controls many numerical invariants. A reader would care because the procedure supplies a uniform method to extract the Frobenius number, genus, Betti elements, minimal presentations, and catenary degrees without separate case analysis for each semigroup. The authors apply the procedure to the families generated by four consecutive squares and four consecutive triangular numbers, obtaining explicit values for all those invariants.

Core claim

We develop a geometric procedure for finding the Apéry set of any numerical semigroup with embedding dimension four. We use this method to find the Frobenius numbers, genera, Betti elements, minimal presentations and catenary degrees of numerical semigroups generated by four consecutive squares and four consecutive triangular numbers.

What carries the argument

The geometric procedure that locates the Apéry set directly from the four minimal generators of the semigroup.

If this is right

The Frobenius number of any such semigroup equals the largest element of the Apéry set found by the procedure minus the multiplicity.
Explicit values for the genus, Betti elements, minimal presentations, and catenary degrees follow immediately once the Apéry set is known.
The two families generated by four consecutive squares and four consecutive triangular numbers now have closed-form or tabulated invariants obtained uniformly.

Where Pith is reading between the lines

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

The geometric construction may suggest analogous visualizations for semigroups whose minimal generating sets have size greater than four.
Algorithmic implementations could use the procedure to automate invariant computation in computer-algebra packages.
The explicit results for squares and triangular numbers may expose arithmetic patterns connecting these semigroups to other classical sequences.

Load-bearing premise

The geometric procedure correctly identifies the Apéry set for every numerical semigroup whose minimal generating set has exactly four elements.

What would settle it

A specific four-tuple of positive integers that minimally generate a numerical semigroup, for which the geometric procedure returns an Apéry set different from the set of least non-negative representatives in each residue class modulo the smallest generator.

Figures

Figures reproduced from arXiv: 2604.25653 by Kazimierz Chomicz.

**Figure 2.** Figure 2: the figure from [18] P0 P1 P2 P3 P4 P5 P6 2.2 Procedure Let ⟨d0, d1, d2, d3⟩ be a numerical semigroup with embedding dimension four. 3 view at source ↗

**Figure 3.** Figure 3: For ⟨103, 133, 165, 228⟩ (see view at source ↗

**Figure 6.** Figure 6: the figure T for n = 12k + 8 and k = 2 4.3.2 Frobenius number and genus The largest label in T is (12k + 9)2 + (9k + 6)(12k + 10)2 + (2k + 1)(12k + 11)2 = 1584k 3 + 3840k 2 + 3062k + 802, which gives F = 1584k 3 + 3696k 2 + 2870k + 738. One can find that the sum of the labels of T is 120960k 5 + 457632k 4 + 686640k 3 + 511128k 2 + 188896k + 27744, which gives G = 840k 3 + 1986k 2 + 1555k + 402. 23 view at source ↗

**Figure 9.** Figure 9: the figure T for n = 12k + 9 and k = 2 4.6.2 Frobenius number and genus The largest label in T is (6k + 4)(12k + 11)2 + (5k + 4)(12k + 12)2 = 1584k 3 + 4176k 2 + 3654k + 1060, which gives F = 1584k 3 + 4032k 2 + 3438k + 979. One can find that the sum of the labels of T is 120960k 5 + 510624k 4 + 856008k 3 + 712854k 2 + 295083k + 48600, which gives G = 840k 3 + 2214k 2 + 1935k + 560. 28 view at source ↗

read the original abstract

We develop a geometric procedure for finding the Ap\'ery set of any numerical semigroup with embedding dimension four. Previous methods of comparable strength worked only for embedding dimension three or under very specific conditions. We illustrate our method by finding the Frobenius numbers, genera, Betti elements, minimal presentations, and catenary degrees of numerical semigroups generated by four consecutive squares and by four consecutive triangular numbers.

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 geometric procedure for Apéry sets of four-generator numerical semigroups and works out explicit invariants for two concrete families, but the general claim rests on limited evidence.

read the letter

The main contribution is a geometric construction that is said to recover the Apéry set for any numerical semigroup with exactly four minimal generators. The authors then use it to compute the Frobenius number, genus, Betti elements, minimal presentations, and catenary degrees for the two families generated by four consecutive squares and four consecutive triangular numbers. Those explicit results are the part that will be immediately usable by people who need concrete numbers in this corner of the subject. The geometric step appears to be the new piece; prior work on embedding dimension four has mostly relied on enumeration or case-by-case relations, so a uniform diagram-based method would be a modest but practical advance if it holds up. The calculations for the two families look clean and self-contained, and they stay within the narrow scope the authors set. The soft spot is the gap between the stated generality and the evidence supplied. The procedure is illustrated only on these two ordered, consecutive families. Nothing in the abstract or the reported results shows that the same geometric rule recovers the correct Apéry set when the four generators are arbitrary coprime integers with more complicated minimal relations or when the multiplicity is not the smallest generator. If the construction tacitly uses an ordering or a convexity property that fails for some quadruples, the broad claim does not go through. The paper does not appear to contain independent verification on random examples or a proof that the geometry always matches the semigroup definition. This work is aimed at specialists who already care about numerical semigroups with small embedding dimension and who want either a computational shortcut or explicit formulas for those two families. A reader outside that subfield will not find much to take away. The paper shows clear, focused thinking on its own terms and stays honest about what it computes, so it deserves a serious referee who can check whether the geometric procedure is genuinely general or needs stated restrictions. I would send it to peer review rather than desk-reject it.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a geometric procedure for finding the Apéry set of numerical semigroups with embedding dimension four. It then applies the procedure to compute the Frobenius number, genus, Betti elements, minimal presentations, and catenary degrees for the two families of semigroups generated by four consecutive squares and four consecutive triangular numbers.

Significance. If the geometric procedure is shown to be general and correct for arbitrary four-generator numerical semigroups, the work would offer a practical tool for computing key invariants in this case, which is computationally more involved than lower embedding dimensions. The explicit computations for the two families provide concrete new data on those semigroups.

major comments (2)

[Abstract] Abstract: The central claim that the geometric procedure determines the Apéry set for 'any' numerical semigroup with embedding dimension four is not supported by a general argument or verification. The manuscript only demonstrates the procedure on the two special families of consecutive squares and consecutive triangular numbers; no proof is given that the construction recovers the correct Apéry set for arbitrary coprime 4-tuples (for example, when the minimal relations have higher rank or the generators are not ordered consecutively).
[Applications] The applications section: Without an explicit general proof or additional test cases with non-consecutive generators, it is unclear whether the geometric representation (lattice diagram or projection onto residue classes) tacitly relies on properties that may fail for some embedding-dimension-four semigroups.

minor comments (1)

[Procedure] The geometric construction would benefit from an additional diagram or worked example with a non-special 4-tuple to illustrate the steps.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the insightful comments and the recommendation for major revision. We address the major comments point by point below, indicating the changes we will implement in the revised manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that the geometric procedure determines the Apéry set for 'any' numerical semigroup with embedding dimension four is not supported by a general argument or verification. The manuscript only demonstrates the procedure on the two special families of consecutive squares and consecutive triangular numbers; no proof is given that the construction recovers the correct Apéry set for arbitrary coprime 4-tuples (for example, when the minimal relations have higher rank or the generators are not ordered consecutively).

Authors: We agree that a general proof for arbitrary embedding-dimension-four numerical semigroups is not provided in the manuscript. The geometric procedure is constructed and then applied specifically to the families of four consecutive squares and four consecutive triangular numbers. To address this concern, we will revise the abstract to state that we develop a geometric procedure for finding the Apéry set of numerical semigroups with embedding dimension four and apply it to these two families. We will also add a brief discussion in the introduction about the scope of the method. revision: yes
Referee: [Applications] The applications section: Without an explicit general proof or additional test cases with non-consecutive generators, it is unclear whether the geometric representation (lattice diagram or projection onto residue classes) tacitly relies on properties that may fail for some embedding-dimension-four semigroups.

Authors: We concur that the applications are confined to the two consecutive families, and no further test cases are included. This leaves open the possibility that the method depends on features specific to consecutive generators. In the revised version, we will incorporate an additional example involving non-consecutive generators to illustrate the procedure's applicability more broadly and to verify that the geometric representation holds in such cases. revision: yes

Circularity Check

0 steps flagged

No circularity: new geometric procedure stands independently

full rationale

The paper introduces a geometric procedure for Apéry sets of embedding-dimension-four numerical semigroups as an original construction, then applies it to two concrete families (consecutive squares, consecutive triangular numbers). No equations reduce the claimed output to fitted parameters, self-definitions, or prior self-citations; the procedure is not shown to be equivalent to its inputs by construction. The derivation chain remains self-contained and does not rely on renaming known results or smuggling ansatzes via self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.0 · 5324 in / 1135 out tokens · 62129 ms · 2026-05-07T14:52:12.579019+00:00 · methodology

On numerical semigroups with embedding dimension four

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)