Pith. sign in

REVIEW 3 major objections 4 minor 29 references

Nearly Gorenstein and almost symmetric properties in shifted numerical semigroups

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For sufficiently large n, if a shifted numerical semigroup M_n is nearly Gorenstein or almost symmetric, then every later member M_{n+λ r_k} inherits the property.

desk verdict Useful paper on shifted numerical semigroups, but Theorem 2.1's proof has a reversed inequality that must be fixed before publication. read the letter →

arxiv 2601.19629 v2 pith:JAQXI7VO submitted 2026-01-27 math.AC

Dumitru I. Stamate , Francesco Strazzanti This is my paper · ORCID
classification math.AC MSC 20M2520M1413H10
keywords numericalsemigroupshiftedfamilypseudo-FrobeniusnumbersnearlyGorensteinalmostsymmetricAperysettraceidealreducedtype
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

For integers 0

What carries the argument

The load-bearing mechanism is the corrected bijection φ_n: PF(M_n)→PF(M_{n+r_k}), induced by a map ψ_n on the Apéry-set representatives P_n={i∈Ap(S,dn) | i≡f mod n for some f∈PF(M_n)}. The map splits P_n into a low part P'_n={i<dn−r_k} where the representative is unchanged, and a high part P''_n where i moves to i+dr_k, with d=gcd(r_1,...,r_k). The integer m(i)—the minimum length of a factorization of i using the full list r_1,...,r_k, even when that list is redundant—records how the shift changes each pseudo-Frobenius element. The identity m(i+r_k)=m(i)+1 for i>r_{k−1}r_k, combined with this two-part dynamics, gives the explicit shift formulas and transfers an NG-vector from M_n to M_{n+r_k

What would settle it

Take a shifted family with redundant generators—for instance S=⟨3,4,6⟩—and search for i>r_{k−1}r_k=24 with m(i+6)≠m(i)+1; or compute PF(M_n) and PF(M_{n+r_k}) for some n>N and compare them with the paper's φ_n formula. A single mismatch would falsify the corrected bijection and with it the propagation theorem.

Watch

Extended reading notes

Core claim

The central claim is that 'nearly Gorenstein' and 'almost symmetric' are eventually periodic along every shifted family: there is an explicit integer N (always below r_k^4) such that, for n>N, if M_n is nearly Gorenstein then M_{n+λ r_k} is nearly Gorenstein for every λ≥0, and similarly for almost symmetric semigroups. The theorem is constructive. If (f_0,...,f_k) is a nearly Gorenstein vector for M_n—meaning each f_i is pseudo-Frobenius and h_i+f_i−f∈M_n for every generator h_i and every f∈PF(M_n)—then the image of that vector under the paper's corrected bijection φ_n^λ is an NG-vector for M_{n+λ r_k}. The proof also corrects an erroneous bijection in the literature between PF(M_n) and PF(M

Load-bearing premise

The load-bearing premise is Remark 1.3(1), which extends the lemma m(i+r_k)=m(i)+1 for i>r_{k−1}r_k from the case of minimal generators to arbitrary (possibly redundant) generator lists by stating that the known proof 'works also in our context' without reproducing it; the shift formulas, the corrected bijection, and Theorem 3.4 all rest on this extension.

Editorial extensions

If this is right

  • Eventual periodicity: for any fixed r_1,...,r_k, the tail of the family is completely determined once n exceeds N; if M_n has one of the two properties, every later M_{n+λ r_k} has it.
  • Explicit formulas: a pseudo-Frobenius number f=i+(m(i)−1)n−n of M_n becomes f+(m(i)−1)λ r_k when i<dn−r_k, and f+(m(i)+(λ+1)d−1)λ r_k+λdn when i≥dn−r_k.
  • The Frobenius number always comes from the high part P''_n for n≥r_k^4, giving a closed formula F(M_{n+λ r_k})=F(M_n)+(m(i)+(λ+1)d−1)λ r_k+λdn.
  • The reduced type of k[[M_n]] (number of pseudo-Frobenius numbers inside [F(M_n)−n,F(M_n)]) is eventually constant along the shifted family, with a counting formula in terms of P''_n and the lengths m(j).
  • The residue is not eventually periodic: for the family M_n=⟨n,n+2,n+3,n+7⟩, res(M_{63+7λ})=λ+9, growing linearly; the paper asks whether the residue is eventually linear (quasilinear) in general.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the proof transports an actual NG-vector rather than just the property, the same transport may apply to other properties defined by a vector certificate over PF(M_n), such as 'positioned' numerical semigroups (canonical reduction).
  • Any earlier result that relied on the flawed bijection between PF(M_n) and PF(M_{n+r_k}) should be re-checked with the corrected map; some of those statements may need new proofs or may become false.
  • The counterexample to periodicity of the residue suggests a stronger asymptotic law than the paper proves: one could computationally test, across many r_1,...,r_k, whether res(M_{n+λ r_k}) is always eventually linear in λ, not merely bounded by a polynomial.
  • Since N<r_k^4 always but examples show much smaller thresholds, a natural next step is to sharpen the uniform bound or to identify the exact onset of periodicity.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies shifted families of numerical semigroups M_n = ⟨n, n+r_1, ..., n+r_k⟩. Its principal contributions are: (i) a corrected bijection φ_n: PF(M_n) → PF(M_{n+r_k}) (Theorem 2.1), replacing an incorrect map in O'Neill–Pelayo; (ii) explicit formulas for pseudo-Frobenius and Frobenius numbers under iteration φ_n^λ (Corollaries 2.6, 2.8, 2.11), with order preservation for n ≥ r_k^4 (Proposition 2.10); (iii) a propagation theorem for the nearly Gorenstein property: for n > N, if M_n is nearly Gorenstein then M_{n+λ r_k} is nearly Gorenstein, and an NG-vector transforms as (φ_n^λ(f_0), ..., φ_n^λ(f_k)) (Theorem 3.4); (iv) the corresponding statement for almost symmetric semigroups (Corollary 4.1), together with a result excluding even type for n ≥ r_k^4 (Proposition 4.3); and (v) a counterexample showing the residue is not eventually periodic (Example 5.2), plus results on canonical reductions and reduced type (§6). Explicit lower bounds are provided and many examples are checked with GAP.

Significance. If the issues identified below are fixed, the paper settles the asymptotic periodicity of nearly Gorenstein and almost symmetric properties for shifted numerical semigroup families in full generality. The corrected pseudo-Frobenius bijection is a genuine contribution, since the earlier construction in the literature is wrong. The explicit transformation of NG-vectors is stronger than a qualitative periodicity statement and is likely to be useful in further work. The residue counterexample (Example 5.2) answers a natural question negatively. The paper is also commendably transparent: it gives explicit bounds, works out several examples, and its numerical claims are independently verified with GAP. The universal bound N < r_k^4 in Remark 3.3 is especially useful because the technical threshold N is defined through the PF set.

major comments (3)
  1. [§2, Theorem 2.1, Step 2, Case A (equations (6)–(7))] The printed inequality in Case A is reversed. The text first correctly notes that i+r_j+m(i)n ∈ M_n iff m(i+r_j) ≤ m(i). It then states that i+dr_k ∈ P_{n+r_k} is equivalent to m(i) ≤ m(i+r_j). From (7), the relevant Apéry element is w = i+dr_k+r_j+(m(i+r_j)+d)(n+r_k), so i+dr_k+r_j+(m(i)+d)(n+r_k) = w + (m(i)-m(i+r_j))(n+r_k) lies in M_{n+r_k} iff m(i) ≥ m(i+r_j), i.e. m(i+r_j) ≤ m(i). The printed inequality is therefore the reverse of the condition actually obtained. Since this equivalence is the main step proving the bijection, the proof as written is internally inconsistent. The examples and the surrounding argument strongly suggest a typographical slip, but the statement must be corrected before the theorem can be accepted.
  2. [§3 Definition 3.1 with §4 Corollary 4.1 and §6.1 Corollary 6.1] Corollaries 4.1 and 6.1 are stated for n > N, but their proofs invoke Proposition 2.11, which is proved only for n ≥ r_k^4. Remark 3.3 gives N < r_k^4, so the hypothesis n > N does not imply n ≥ r_k^4. Proposition 2.11 is needed to identify φ_n^λ(F(M_n)) with the Frobenius number of M_{n+λr_k}; without order preservation of PF numbers the proofs of the two corollaries do not cover their stated range. Example 6.2 illustrates the gap: it applies Corollary 6.1 with n = 26 and r_k = 4, while r_k^4 = 256. Please either extend Proposition 2.10 and Proposition 2.11 to the range n > N, or state Corollaries 4.1 and 6.1 under the stronger hypothesis n ≥ r_k^4.
  3. [§1, Remark 1.3(1)] The identity m(i+r_k) = m(i)+1 for i > r_{k-1}r_k is quoted from [2, Theorem 4.3], which is stated for minimal generator lists, and the authors assert without proof that the proof works for their possibly redundant list r_1, ..., r_k. This is not a purely formal extension because m(i) in this paper is defined as the minimum length of factorizations with respect to the given list r_1,...,r_k, not with respect to the minimal generators of S; the paper itself notes the distinction in the example S = ⟨3,4,6⟩. This identity is used in essential steps, including Step 3 of Theorem 2.1, Corollaries 2.6, 2.8, 2.9, Remark 3.2, and Theorem 3.4. Please include a proof or a precise lemma stating the extension and its hypotheses.
minor comments (4)
  1. [Title page] The title appears as “NEARL Y GORENSTEIN...” in the full text; it should read “NEARLY GORENSTEIN...”.
  2. [Definition 3.1] The threshold N is defined using P'_n and P''_n, which in turn depend on PF(M_n). This is legitimate, but it would help to state explicitly that N is not a closed-form constant in the r_i alone and that Remark 3.3 provides the uniform alternative n ≥ r_k^4.
  3. [§2, Theorem 2.1, Step 1] The equivalence “i ∈ P_n iff i+(m(i)-1)n ∈ PF(M_n)” is used repeatedly. A short justification via the maximality of Apéry elements would improve readability.
  4. [§4, Example 4.2] In the shifted family data, “r1 = 1, r2 = 3, r4, d = 1” appears to contain a typo: it should be r3 = 4.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is an internal bijection-and-bounds argument built on independent cited characterizations.

full rationale

The central derivation chain (Theorem 2.1 -> Corollaries 2.6/2.8 -> Theorem 3.4 -> Corollary 4.1) is not circular. Theorem 2.1 constructs the PF bijection from the Apéry-set description of [24, Thm 3.3] and explicitly corrects the earlier [24, Thm 4.8] map rather than assuming it. Theorem 3.4 takes an assumed NG-vector for M_n and proves, using the bijection and explicit bounds, that its φ-image satisfies the NG-vector conditions for M_{n+r_k}; the conclusion is not an input. The framework definitions from [13] and [22] are external characterizations with stated hypotheses that do not include the periodicity result, so the self-citations are framework, not load-bearing circularity. Definition 3.1 defines a bound N in terms of P'_n and P''_n, hence formally in terms of PF(M_n), but Remark 3.3 proves the absolute bound N < r_k^4; thus the threshold does not smuggle in the conclusion. No parameter is fitted and no predicted quantity is a renamed input. Two non-circular caveats should be flagged: Remark 1.3(1) extends [2, Thm 4.3] to non-minimal generator lists by asserting 'their proof works also in our context' without reproducing it, and Step 2, Case A of Theorem 2.1 appears to contain a typographical inequality reversal ('m(i) ≤ m(i+r_j)' versus 'm(i+r_j) ≤ m(i)'); the surrounding algebra and examples indicate a slip, but as printed the proof needs correction. These concern correctness, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted constants or invented entities; the input is a fixed generator list and shift. The central claim rests on standard published theorems about Apéry sets, factorization lengths, and near-Gorenstein characterizations; self-cited results are independent published theorems and are not used as the conclusion.

assumptions (5)
  • standard math O'Neill–Pelayo [24, Theorem 3.3]: for n > r_k^2 and dn ∈ S, Ap(M_n,n) = {i+m(i)n | i ∈ Ap(S,dn)}.
    Used in Theorem 2.1 and throughout to represent pseudo-Frobenius elements of M_n by Apéry residues.
  • domain assumption Barron–O'Neill–Pelayo [2, Theorem 4.3] extended to non-minimal generator lists: m(i+r_k) = m(i)+1 for i > r_{k-1}r_k.
    Remark 1.3(1); load-bearing for shift formulas and P'' dynamics. The paper asserts the proof carries over but does not reproduce it.
  • domain assumption Moscariello–Strazzanti [22, Proposition 1.1]: H is nearly Gorenstein iff an NG-vector exists.
    Used in Theorem 3.4 to translate the trace-ideal condition into a vector condition on pseudo-Frobenius numbers.
  • domain assumption Nari [23, Theorem 2.4]: H is almost symmetric iff f_a + f_{t-a} = F(H) for its ordered pseudo-Frobenius numbers.
    Used in Proposition 4.3 to rule out even type.
  • domain assumption Characterizations of canonical reduction [25, Theorem 3.13] and reduced type [21, Theorem 2.13].
    Used in Section 6; the transfer results for these invariants depend on the cited equivalences.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Nearly Gorenstein and almost symmetric properties in shifted numerical semigroups." pith.science (2026). https://pith.science/paper/JAQXI7VO

@misc{pith2026260119629,
  author       = {Pith},
  title        = {Pith review of: Nearly Gorenstein and almost symmetric properties in shifted numerical semigroups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JAQXI7VO}},
  note         = {Machine review of arXiv:2601.19629}
}
abstract

Given the integers $0<r_1<\dots<r_k$, we consider the shifted family of semigroups $M_n=\langle n, n+r_1,\dots, n+r_k\rangle$, where $n>0$. For sufficiently large $n$, we prove that if $M_n$ is nearly Gorenstein or almost symmetric, then so is $M_{n+r_k}$. A key ingredient is to relate the pseudo-Frobenius elements of $M_n$ and $M_{n+r_k}$, correcting a wrong claim in the literature. Moreover, we derive explicit formulas for the Frobenius and pseudo-Frobenius numbers of $M_{n+r_k}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 2 linked inside Pith

  1. [1]

    A. Assi, M. D’Anna, P.A. Garc ´ ıa-S´ anchez,Numerical semigroups and applications, RSME Springer Series, Vol. 3, 2020

    2020

  2. [2]

    Barron, C

    T. Barron, C. O’Neill, R. Pelayo,On the set of elasticities in numerical monoids, Semigroup Forum94 (2017), 37–50

    2017

  3. [3]

    Barucci, R

    V. Barucci, R. Fr¨ oberg,One-dimensional almost Gorenstein rings, J. Algebra188(1997), 418–442

    1997

  4. [4]

    Branco, M.C

    M.B. Branco, M.C. Faria, J.C. Rosales,Positioned numerical semigroups, Quaest. Math.44(2021), no. 5, 679–691

    2021

  5. [5]

    Bruns, J

    W. Bruns, J. Herzog,Cohen–Macaulay rings, Revised 2nd Ed. Cambridge Studies in Advanced Mathemat- ics, 1998. 18 DUMITRU I. STAMATE, FRANCESCO STRAZZANTI

    1998

  6. [6]

    Cimpoea¸ s, D.I

    M. Cimpoea¸ s, D.I. Stamate,On intersections of complete intersection ideals, J. Pure Appl. Algebra220 (2016), 3702–3712

    2016

  7. [7]

    Conaway, F

    R. Conaway, F. Gotti, J. Horton, C. O’Neill, R. Pelayo, M. Pracht, B. Wissman,Minimal presentations of shifted numerical monoids, International Journal of Algebra and Computation28(2018), 53–68

    2018

  8. [8]

    Delgado, P.A

    M. Delgado, P.A. Garc ´ ıa-S´ anchez, J. Morais,NumericalSgps, A package for numerical semigroups, GAP package, Version 1.4.0 (2024)

    2024

Show all 29 references
  1. [9]

    The GAP Group,GAP – Groups, Algorithms, and Programming, Version 4.15.1 (2025)

    2025

  2. [10]

    S. Goto, N. Matsuoka, T.T. Phuong,Almost Gorenstein rings, J. Algebra379(2013), 355–381

    2013

  3. [11]

    S. Goto, R. Takahashi, N. Taniguchi,Almost Gorenstein rings–towards a theory of higher dimension, J. Pure Appl. Algebra219(2015), 2666–2712

    2015

  4. [12]

    Herzog, D.I

    J. Herzog, D.I. Stamate,On the defining equations of the tangent cone of a numerical semigroup ring, J. Algebra418(2014), 8–28

    2014

  5. [13]

    Herzog, T

    J. Herzog, T. Hibi, D.I. Stamate,The trace of the canonical module, Isr. J. Math.233(2019), 133–165

    2019

  6. [14]

    Herzog, T

    J. Herzog, T. Hibi, D.I. Stamate,Canonical trace ideal and residue for numerical semigroup rings, Semi- group Forum103(2021), 550–566

    2021

  7. [15]

    Herzog, K.-i

    J. Herzog, K.-i. Watanabe,Almost symmetric numerical semigroups, Semigroup Forum98(2019), no. 3, 589–630

    2019

  8. [16]

    Howard, C

    J. Howard, C. O’Neill,Graver bases of shifted numerical semigroups with 3 generators, J. Algebra Appl. 24(2025), 2550227

    2025

  9. [17]

    Huneke, S

    C. Huneke, S. Maitra, V. Mukundan,Torsion in differentials and Berger’s conjecture, Res. Math. Sci.8 (2021), no. 4, 1–15

    2021

  10. [18]

    Jafari, S

    R. Jafari, S. Zarzuela Armengou,Homogeneous numerical semigroups, Semigroup Forum97(2018), 278– 306

    2018

  11. [19]

    Jafari, F

    R. Jafari, F. Strazzanti, S. Zarzuela Armengou,On nearly Gorenstein affine semigroups, arXiv:2411.12081

    arXiv

  12. [20]

    Jayanthan, H

    A.V. Jayanthan, H. Srinivasan,Periodic occurence of complete intersection monomial curves, Proc. Am. Math. Soc.141(2013), 4199–4208

    2013

  13. [21]

    Maitra, V

    S. Maitra, V. Mukundan,Extremal behavior of reduced type of one dimensional rings, arXiv:2306.17069

    arXiv

  14. [22]

    Moscariello, F

    A. Moscariello, F. Strazzanti,Nearly Gorenstein vs almost Gorenstein affine monomial curves, Mediterr. J. Math.18(2021), article number: 127

    2021

  15. [23]

    Nari,Symmetries on almost symmetric numerical semigroups, Semigroup Forum86(2013), no

    H. Nari,Symmetries on almost symmetric numerical semigroups, Semigroup Forum86(2013), no. 1, 140–154

    2013

  16. [24]

    O’Neill, R

    C. O’Neill, R. Pelayo,Ap´ ery sets of shifted numerical monoids, Adv. Appl. Math.97(2018), 27–35

    2018

  17. [25]

    Rahimi,Rings with canonical reductions, Bull

    M. Rahimi,Rings with canonical reductions, Bull. Iran. Math. Soc.46(2020), 1801–1825

    2020

  18. [26]

    Rosales, P.A

    J.C. Rosales, P.A. Garc ´ ıa-S´ anchez,Numerical Semigroups, Springer Developments in Mathematics, Vol 20, 2009

    2009

  19. [27]

    Stamate,Asymptotic properties in the shifted family of a numerical semigroup with few generators, Semigroup Forum93(2016), 225–246

    D.I. Stamate,Asymptotic properties in the shifted family of a numerical semigroup with few generators, Semigroup Forum93(2016), 225–246

    2016

  20. [28]

    Stamate,On the occurrence of complete intersections in shifted families of numerical semigroups, Res

    D.I. Stamate,On the occurrence of complete intersections in shifted families of numerical semigroups, Res. Math. Sci.9(2022), article no 32

    2022

  21. [29]

    Vu,Periodicity of Betti numbers of monomial curves, J

    T. Vu,Periodicity of Betti numbers of monomial curves, J. Algebra418(2014), 66–90. Dumitru I. Stamate - F aculty of Mathematics and Computer Science, University of Bucharest - Str. Academiei 14 - 010014 - Bucharest - Romania Email address:dumitru.stamate@fmi.unibuc.ro Francesc...

    2014

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.