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Existence of a far-flung Gorenstein numerical semigroup attaining the Herzog--Kumashiro--Stamate bound

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Pith's one-line read For every type t≥2, a far-flung Gorenstein numerical semigroup attains the multiplicity upper bound n(t) coming from the Rohrbach problem.

desk verdict Clean, elementary existence proof that every Rohrbach number n(t) is attained by a far-flung Gorenstein numerical semigroup of type t, plus uniform counterexamples for the residue-versus-length inequality when t≥5. read the letter →

arxiv 2607.11093 v1 pith:72RUURCC submitted 2026-07-13 math.AC

Akihiro Sugawara This is my paper · ORCID
classification math.AC MSC 13H1020M1420M25
keywords numericalsemigroupfar-flungGorensteinreducedtypemaximalRohrbachproblemmultiplicitytraceidealcanonicalmodule
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Numerical semigroups that are far-flung Gorenstein are known to have multiplicity bounded above by the Rohrbach number $n(t)$ of their type $t$. The paper answers whether this bound is sharp for every $t$ by giving an explicit construction: take any extremal set $A$ of size $t$ and form the semigroup obtained by deleting from the half-line of multiplicity $n(A)$ the points that correspond to the elements of $A$. The resulting semigroup is far-flung Gorenstein of type exactly $t$ and multiplicity exactly $n(t)$. The same family also produces, for every $t \ge 5$, examples in which the residue of the trace ideal strictly exceeds the number of non-semigroup elements of the canonical ideal, answering a second open question in the negative.

What carries the argument

The family $S(m,A) = \Delta(m) \setminus \{(2m-1)-a : a \in A\}$, where $A$ is an extremal set of size $t$ containing $0$ and $m=n(A)$. Proposition 2 and Corollary 4 show that this semigroup is far-flung Gorenstein of maximal reduced type precisely when $m=n(A)$.

What would settle it

Exhibit a single extremal set $A$ of size $t \ge 2$ for which $\max A + 2 > n(A)$; then $S(n(A),A)$ fails to have maximal reduced type and the multiplicity-attaining claim for that $t$ collapses.

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Extended reading notes

Core claim

For every integer $t \ge 2$ and every extremal finite set $A$ of cardinality $t$, the numerical semigroup $S=S(n(A),A)$ is far-flung Gorenstein, has type $t$, and attains the multiplicity bound $m(S)=n(t)$. In particular, the Herzog–Kumashiro–Stamate upper bound is sharp for every type.

Load-bearing premise

Every extremal set of size at least 2 has its largest element at least two less than its Rohrbach number; without that gap the constructed semigroup need not have maximal reduced type.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper answers Question 5.5 of Herzog–Kumashiro–Stamate by constructing, for every t≥2, a far-flung Gorenstein numerical semigroup S of type t that attains the multiplicity bound m(S)=n(t) coming from the Rohrbach problem. The construction is the family S(m,A)=Δ(m)\{(2m-1)-a:a∈A} with m=n(A) and A any extremal set of cardinality t. After establishing a characterization of the far-flung Gorenstein property via rPF(S) (Lemma 1) and the basic arithmetic of S(m,A) (Proposition 2), the authors prove that the two numerical conditions max A+1≤n(A) and max A+n(A)<2m force rPF(S)=PF(S) (Theorem 1 and Corollary 4); these conditions hold automatically for extremal A by Lemma 3, yielding Theorem 2. The same family is then used to produce, for every t≥5, a far-flung Gorenstein semigroup with res(S)>l(S), thereby answering the residual case of the Herzog–Hibi–Stamate question and supplying an explicit counter-example for type 4.

Significance. The result settles a clean existence question that links the algebraic notion of far-flung Gorenstein rings to a classical problem in additive number theory. The construction is elementary, completely explicit, and works uniformly for all t≥2; it simultaneously yields an infinite family of counter-examples to the inequality res≤l. Because the proofs rely only on the definitions of n(A), rPF and the conductor, the argument is self-contained and immediately usable by other researchers working on numerical semigroups or one-dimensional Cohen–Macaulay rings.

minor comments (4)
  1. In the statement of Lemma 3(2) the inequality is written max A+2≤n(A); the proof actually shows the slightly stronger max A+1≤n(A). A one-line remark clarifying the relation would avoid a momentary pause for the reader.
  2. The four explicit sets A5–A8 in the proof of Theorem 3 are asserted to satisfy 3(t-1)<n(At) and max At+2≤n(At). While these inequalities are elementary to check by hand, a short parenthetical verification (or a reference to the known values of n(t) for t≤8) would make the argument fully self-contained without external lookup.
  3. Example 5 claims that the listed generators give res(S)=8>7=l(S) and s(S)=t(S)=4. Since the verification is described as “we can check,” it would be helpful to record the sets K(S)\S and S\tr(K(S)) explicitly (they are already written in the text) and to note that the computation was performed with NumericalSgps, as is done for other examples.
  4. A few typographical inconsistencies appear: “F ar-flung” in the running head, “SUGA W ARA” with an extra space, and the occasional missing space after a period. These are purely cosmetic.

Circularity Check

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No circularity: existence constructions are direct verifications from classical Rohrbach extremal sets and elementary numerical-semigroup definitions.

full rationale

The paper answers Question 1 of Herzog–Kumashiro–Stamate by an explicit construction: for any extremal set A of cardinality t (taken as given by the classical Rohrbach problem), the numerical semigroup S = S(n(A), A) is shown to be far-flung Gorenstein of type t with multiplicity exactly n(t). All steps are self-contained set-arithmetic. Proposition 2 records the elementary identities m(S) = n(A), F(S) = 2n(A)−1 and rPF(S) = {(2n(A)−1)−a : a ∈ A}. Lemma 3 proves the two inequalities max A + 2 ≤ n(A) and 0,1 ∈ A by a short contradiction that uses only the definition of extremality. Corollary 4(3) then invokes these inequalities inside Theorem 1 to conclude rPF(S) = PF(S), and Theorem 2 assembles the pieces. The same construction supplies the counter-examples of Section 4 (Proposition 3, Theorem 3, Example 5) without feeding the target inequalities res(S) > l(S) or m(S) = n(t) back into the choice of A. Self-citations appear only for background classification results that are not used as load-bearing premises. Consequently the derivation never reduces a claimed prediction to its own inputs by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The paper works entirely inside the standard theory of numerical semigroups and the classical Rohrbach problem. No free parameters are fitted; the only external inputs are the definition of n(t) and the known existence of extremal sets of every finite cardinality. All other statements are proved from those ingredients plus elementary arithmetic.

assumptions (3)
  • domain assumption The Rohrbach number n(t) is well-defined and finite for every positive integer t, and extremal sets of every cardinality exist.
    Invoked throughout Section 2 and as the starting point of Theorem 2; taken from the classical additive-number-theory literature (Rohrbach 1937, OEIS A123509).
  • domain assumption A numerical semigroup S is far-flung Gorenstein if and only if {0,…,m(S)-1} lies in the sumset of {F(S)-f | f∈PF(S)} (Fact 3).
    Cited from Herzog–Kumashiro–Stamate and used as the base characterization that Lemma 1 refines to rPF.
  • standard math Standard facts: e(S)≤m(S), 1≤s(S)≤t(S), g(S)=n(S)+l(S), and the definitions of conductor, canonical ideal and trace ideal.
    Recalled in Section 1 and used without further proof.
invented entities (1)
  • The family of numerical semigroups S(m,A)=Δ(m)\{(2m-1)-a | a∈A} independent evidence
    purpose: Provides an explicit, uniformly defined source of far-flung Gorenstein semigroups of prescribed reduced type that attain the multiplicity bound when A is extremal.
    Introduced in Section 3; all subsequent existence and counterexample statements are instances of this single construction.

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Pith. "Pith review of Existence of a far-flung Gorenstein numerical semigroup attaining the Herzog--Kumashiro--Stamate bound." pith.science (2026). https://pith.science/paper/72RUURCC

@misc{pith2026260711093,
  author       = {Pith},
  title        = {Pith review of: Existence of a far-flung Gorenstein numerical semigroup attaining the Herzog--Kumashiro--Stamate bound},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/72RUURCC}},
  note         = {Machine review of arXiv:2607.11093}
}
abstract

We address a question posed by Herzog, Kumashiro, and Stamate regarding the upper bound for the multiplicity of any far-flung Gorenstein numerical semigroup. We answer this question in the affirmative by presenting a method for constructing certain far-flung Gorenstein numerical semigroups with maximal reduced type. Furthermore, using this method, we consider a question raised by Herzog, Hibi, and Stamate. More precisely, for any integer $t\ge5$, we construct a certain far-flung Gorenstein numerical semigroup with type $t$, which is a counterexample to the question.

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