Some asymptotic results on $p$-lengths of factorizations for numerical semigroups and arithmetical congruence monoids

Christopher O'Neill; Eli B. Dugan; Emi Lycan; Sarah Mendoza De La Cruz; Shadi Gaskari; Spencer Chapman; Vadim Ponomarenko

REVIEW 4 major objections 4 minor 17 references

Some asymptotic results on $p$-lengths of factorizations for numerical semigroups and arithmetical congruence monoids

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Extremal p-lengths of factorizations in numerical semigroups are eventually quasipolynomial for p = 0, 1, 2, and ∞, with explicit periods and leading coefficients.

desk verdict Genuinely new p-length invariant with mostly clean asymptotics; the ℓ^m_2 proof has a real gap and the examples need repair, but this deserves refereeing. read the letter →

arxiv 2411.17010 v1 pith:VF7IPPSL submitted 2024-11-26 math.AC math.CO

Spencer Chapman , Eli B. Dugan , Shadi Gaskari , Emi Lycan , Sarah Mendoza De La Cruz , Christopher O'Neill , Vadim Ponomarenko This is my paper · ORCID

classification math.ACmath.CO MSC 20M1420M13

keywords p-lengthfactorizationlengthnumericalsemigroupquasipolynomialarithmeticalcongruencemonoidApérysetℓ_p-normasymptoticgrowth

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

This paper introduces p-length, a generalization of the classical factorization length that replaces the sum of exponents (the 1-norm) with the ℓ_p-norm of the exponent vector. It establishes that for every numerical semigroup S = ⟨g1,...,gk⟩, the minimum and maximum p-lengths of factorizations of n are eventually quasipolynomial functions of n for p ∈ {0,1,2,∞}, and it computes the degree, period, and leading coefficient of each. For example, the maximum ∞-length satisfies ℓ^M_∞(n) = (n - a_i)/g1 for all sufficiently large n, where a_i is the Apéry element of S congruent to n modulo g1. For arithmetical congruence monoids, the paper shows that in the singular monoid M_{4,6}, the minimal 1- and ∞-lengths of x^n grow linearly for every x = 2^a 5^b 7^c, whereas the maximum 0-length grows like Θ($n^{{1/2}}$) for x = 28 and 40 and Θ($n^{{2/3}}$) for x = 70. The result matters because it gives a complete asymptotic picture for a new invariant of factorization in the two most-studied monoid families, and it exposes a boundary: for p ≥ 3 the minimum p-length of a numerical semigroup need not be eventually quasipolynomial.

What carries the argument

The machinery is the pair of extremal functions ℓ^m_p(n) and ℓ^M_p(n) on the factorization set Z(n) = {z ∈ Z≥0^k : z1g1 + ··· + zkgk = n}, together with three structural identities that turn the discrete optimization into a quasipolynomial: the Apéry-set formulas ℓ^M_∞(n) = (n - a_i)/g1 and ℓ^m_∞(n) = (n + a_i)/g, where a_i is the element of Ap(S;g1) or Ap(S;g) in the appropriate residue class; the translation property for the Euclidean norm, that adding (g1,...,gk) to a minimizer preserves minimality, which yields ℓ^m_2(n+N) - ℓ^m_2(n) = 2n + N; and the one-variable reduction for p ≥ 2 that the maximum p-length is achieved by the factorization with maximal first coordinate. For ACMs the load-bearing tool is a classification of the atoms of M_{4,6} (an atom divides 2^a 5^b 7^c only in restricted forms), which permits counting distinct atoms in factorizations of x^n.

What would settle it

Take the numerical semigroup S = <2,3> and compute, for each n up to a large bound (say $10^{5}$), the unique integer solution (x,y) to 2x + 3y = n minimizing $x^{2}$ + $y^{2}$. If there are infinitely many n with the minimizing solution having x < 0 or y < 0, or if the second difference of the true minimum ℓ^m_2(n) ever fails to equal 2($2^{2}$ + $3^{2}$) = 26 for large n, then the claimed quasipolynomiality with period 13 is false. For a semigroup with three generators, the analogous check is whether the integer minimizer of $x1^{2}$ + $x2^{2}$ + $x3^{2}$ subject to g1x1 + g2x2 + g3x3 = n eventually has all coordinates nonnegative.

Watch

Extended reading notes

Core claim

The central discovery is that extremal p-lengths are asymptotically rigid for numerical semigroups: for each p ∈ {0, 1, 2, ∞}, both the minimal and the maximal p-length functions ℓ^m_p(n) and ℓ^M_p(n) are eventually quasipolynomial, meaning that for all n beyond a finite threshold they equal a polynomial in n whose coefficients depend periodically on n. The paper determines the full set of attributes: ℓ^M_p has degree p, period g1, and leading coefficient 1/g1^p for p ≥ 1 (degree 0 and period 1 for p = 0), while ℓ^m_1 has degree 1, period gk, and leading coefficient 1/gk, ℓ^m_2 has degree 2, period N = $g1^{2}$ + ··· + $gk^{2}$, and leading coefficient 1/N, and ℓ^m_∞ has degree 1, period g = g1 + ··· + gk, and leading coefficient 1/g. The proof for p = ∞ uses Apéry sets to determine the constant term, and the proof for p = 2 uses a translation identity that reduces the problem to second differences. For arithmetical congruence monoids, the paper does not attempt quasipolynomial formulas but instead establishes explicit growth rates for the singular monoid M_{4,6}.

Load-bearing premise

The proof that the minimum 2-length is eventually quasipolynomial assumes, without proof, that for all sufficiently large n there is a factorization of n that minimizes the Euclidean norm over all integer solutions to the same linear equation; if that largeness condition ever fails, the quasipolynomiality of ℓ^m_2 rests on nothing.

Editorial extensions

If this is right

For any numerical semigroup, the value of the extremal p-length for every sufficiently large n is given by an explicit quasipolynomial, so the invariant is effectively computable rather than merely asymptotic.
The period and leading coefficient of each p-length function constitute new arithmetic invariants of the semigroup, such as N = g1^2 + ··· + gk^2 for the minimum 2-length and the Apéry elements for the maximum ∞-length.
The p = ∞ formulas tie extremal factorization lengths directly to the semigroup's Apéry structure, giving a clean geometric meaning to the periodic constant terms.
The ACM results show that the maximum number of distinct atoms in a factorization of x^n can grow like n^{1/2} or n^{2/3} depending on the base element, so the growth rate is a discriminating invariant of the base element even within a single monoid.
The Θ(n) lower bound for minimal 1- and ∞-lengths in M_{4,6} confirms that singular ACMs can mimic the linear growth of regular ACMs despite their unusual factorization behavior.

Reading between the lines

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

The unproved largeness step in Theorem 2.10 could likely be supplied by a lattice-geometry argument: the Euclidean minimizer over all integer solutions converges to the ray in direction (g1,...,gk) as n grows, so for all n beyond a bound depending only on the generator set, the minimizer lies in the nonnegative orthant; if that holds, the second-difference proof of quasipolynomiality is complete.
The special role of p = 2 may be explained by the fact that the ℓ_2-norm is the only ℓ_p-norm whose sublevel sets are rational ellipsoids, making the translation identity (z + (g1,...,gk)) preserve the norm difference in a linear way; p ≥ 3 norms lack such an identity, which is consistent with the paper's example that ℓ^m_3 for S = <2,3> involves a floor of an irrational multiple.
The growth exponents 1/2 and 2/3 for ℓ^M_0(x^n) in M_{4,6} are reminiscent of the divisor function's behavior in short intervals, suggesting a possible connection between factorization counting in singular ACMs and analytic number theory; examining other singular ACMs might yield other rational exponents.
The good/evil atom dichotomy used for M_{4,6} could be adapted to other finite-atomic ACMs to obtain similar Θ(n^α) bounds, potentially answering Question 3.8 for a larger class.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The p-length invariant is a good idea, and the numerical semigroup part mostly delivers what it promises. The paper introduces ℓ_p-extremal factorization lengths, proves eventual quasipolynomiality with explicit degree, period, and leading coefficient for p = 0, 1, 2, ∞, and shows ℓ^m_3 is not eventually quasipolynomial via an example. That last result is the most interesting thing here: the minimal cube-sum problem on ⟨2,3⟩ has irrational slopes, and the paper’s claim that this kills quasipolynomiality is exactly the kind of subtlety the field needs. I also like the ACM growth-rate results; the contrast between regular and singular ACMs is real.

The central theorems in Section 2 (2.4, 2.6, 2.8) look right. The arguments are short and mostly self-contained, and the reliance on Apéry sets is clean. The reader’s spot checks didn’t turn up issues there, and I agree.

The soft spots are real but localized. Theorem 2.10’s proof has a load-bearing gap: it asserts that for large enough n ∈ S, some z ∈ Z(n) minimizes ℓ2 over all integer solutions. That is not justified in the text. The stress-test note is right: Proposition 2.9 gives a shift that raises every coordinate, so an iteration argument is plausible, but the paper needs to state it and prove a uniform threshold across residue classes. The garbled sentence about smallest coordinates doesn’t help. This gap underlies the entire ℓ^m_2 row of Table 1, so it needs fixing.

Example 2.11’s displayed minimizer formula is also wrong—Lagrange multipliers on z1^3 + z2^3 with 2z1 + 3z2 = n give c1 = n(3√6 − 4)/19, not (−8 ± n√130)/19. The conclusion that ℓ^m_3 is not eventually quasipolynomial probably survives with the corrected formula, but as printed it’s an error. Proposition 3.6’s x = 40 construction has a 2-adic imbalance: 40^n = 2^{3n}5^n, and the two displayed factorizations don’t have matching 2-exponents. That one I didn’t repair in my head; it may need a different construction.

None of this looks like a broken core. The paper is honest about which results are imported and which are new, and the numerical semigroup proofs are not circular. It’s a solid contribution that needs a careful revision.

I’d send this to a serious referee. A referee can ask for a real largeness lemma for Theorem 2.10 and corrected examples. For anyone working on factorization invariants or quasipolynomial phenomena, this is worth reading, but wait for the revision.

Referee Report

4 major / 4 minor

Summary. The paper introduces p-length invariants for factorizations in commutative monoids, defined as the ℓ_p-norm of the exponent vector, and studies their extremal values for large elements of numerical semigroups and arithmetical congruence monoids (ACMs). For numerical semigroups, the paper claims that the minimal and maximal p-length functions are eventually quasipolynomial for p ∈ {0,1,2,∞}, with the degrees, periods, and leading coefficients listed in Table 1; it also claims by example that for p ≥ 3 the minimal p-length need not be eventually quasipolynomial. For ACMs, the paper establishes linear growth of some p-length functions along powers x^n and exhibits two singular ACMs where the maximum 0-length grows as n^{1/2} and n^{2/3}.

Significance. If the results are correct, the paper introduces a genuinely new family of factorization invariants and gives clean, explicit asymptotics for numerical semigroups and for several ACMs. The table of quasipolynomial attributes is a useful contribution, and the ACM examples displaying sublinear growth are striking and well-motivated. The paper also benefits from using explicit external benchmarks for the p=0 and p=1 cases rather than fitting parameters, and many of the Section 2 arguments are simple and checkable. However, several load-bearing steps currently contain gaps or explicit computational errors, so the results cannot be accepted in their present form.

major comments (4)

[Theorem 2.10 (Section 2)] The proof of Theorem 2.10 is incomplete at the step asserting that one may choose n ∈ S large enough so that some z ∈ Z(n) minimizes ℓ2(·) over all integer solutions to g1 x1 + ... + gk xk = n. This is the load-bearing point: it must show that for all sufficiently large n, and in every residue class modulo N = g1^2 + ... + gk^2, a global integer minimizer can be taken with nonnegative coordinates. Proposition 2.9 only propagates an existing integer minimizer from n to n+N; it does not establish the existence of such a minimizer for all large n. The sentence comparing the smallest coordinate of the minimizer for (2.3) with the smallest coordinate of the minimizer for (2.3) is garbled and does not prove a uniform threshold. Without this step, the recurrence ℓ^m_2(n+N) − ℓ^m_2(n) = 2n + N, and hence the degree-2 quasipolynomiality asserted in Table 1, is unsupported. A repair is available, for example by showing that every integer minimizer lies within bounded distance of the real minimizer (n/N)(g1,...,gk), but the proof as written needs a substantial addition.
[Example 2.11 (Section 2)] The displayed closed form for the ℓ^m_3(n) minimizer in S = <2,3> is incorrect. Solving the Lagrange multiplier condition for minimizing z1^3 + z2^3 subject to 2z1 + 3z2 = n gives z1 = n(3√6 − 4)/19 (up to the appropriate integer rounding), not (−8 ± n√130)/19. The qualitative conclusion may be salvageable with the corrected formula, but as written the example is false. Moreover, the phrase 'similar computation to the proof of Proposition 2.9' is misleading: over all integer solutions the cubic is unbounded below along the kernel direction (−3,2), so the minimization must be restricted to the nonnegative solution set Z(n).
[Proposition 3.6 (Section 3)] For x = 40 = 2^3·5, the two displayed factorizations of (2^3·5)^n contain only n total factors of 2, not 3n; they factor 10^n rather than 40^n. The exponent on the atom 2^2 should be (3n−k−1)/2 in the first case and (3n−k−2)/2 in the second. In addition, the second construction has final 5-exponent n−k^2−1, which is negative at the endpoint n = k^2 of the claimed interval. The Θ(n^{1/2}) claim may be true, but the proof as written is invalid and needs correction.
[Proposition 3.7 (Section 3)] The lower-bound construction for x = 70 contains negative exponents: the factor (2·5^{2i+1}7^{2a−2i−1}) has exponent 2a−2i−1 = −1 when i = a. The subsequent assertion that 'we may choose c appropriately so that ... for some n ≤ k^3' is not justified. Since this construction is the entire lower bound for the Θ(n^{2/3}) claim, the proof must be rewritten with a valid choice of exponents and an explicit verification that c is a nonnegative integer and that the resulting n lies in the claimed range.

minor comments (4)

[Section 1 and Section 2] The definition of ℓ0(z) via z_i^0 is ambiguous because 0^0 is undefined; the authors should define ℓ0(z) as the number of nonzero coordinates of z.
[Lemma 2.7] The proof assumes that g2 is the smallest among g2,...,gk; this should be stated explicitly, for instance by assuming the generators are ordered g1 < g2 < ... < gk.
[Theorem 2.6 proof] The line 'ℓm∞(n+ai) = l∞(qg) = q' contains a typo; it should be 'ℓm∞(qg) = q'.
[Theorem 3.5 proof] The chain 'a(Q′+R′) ≤ a(Q+R+Q′+R′) = 1/n a(b+c) = (P+P′)(b+c)' appears to contain a typo; it should read '= a n (b+c) = (P+P′)(b+c)'. Also, the derived inequality 2P′ < P actually yields m < k, so the stated m ≤ 2k is weaker than what the argument gives.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the new asymptotic results are derived from first principles, and the only substantive weakness (the largeness claim in Theorem 2.10) is an unproved proof step, not a circular reduction.

full rationale

The paper's new results (Theorems 2.4, 2.6, 2.8, 2.10, and 3.5-3.7) are obtained by direct arguments using Apéry sets, integer-optimization comparisons, and explicit atom characterizations; they are not fitted to the quantities they predict. The p=0 and p=1 entries in Table 1 are explicitly quoted from prior published work ([2], [7]) rather than re-derived, and the cited theorems are parameter-free results with stated assumptions that do not include the target asymptotics. Although some cited authors overlap with the present authors, this is ordinary reliance on prior literature, not a self-citation chain that forces the conclusion. Lemma 3.4 is imported from [6] and [5, Example 4.10], but it is an external published characterization of atoms of M4,6, not a restatement of the paper's own asymptotic claims. The genuine weakness is in the proof of Theorem 2.10: the sentence 'the smallest coordinate of the integer solution to (2.3) minimizing ℓ2(·) is strictly larger than the smallest coordinate of the integer solution to (2.3) minimizing ℓ2(·)' is garbled, and the assertion that one may choose n large enough so that some z in Z(n) minimizes ℓ2 over all integer solutions is not fully justified. This is a correctness gap in proving quasipolynomiality of ℓ^m_2, not a circular step: Proposition 2.9 supplies a real shift mechanism, and the claimed period N and leading coefficient 1/N are not assumed in the hypotheses. No derivation in the paper reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters. The paper's new results are self-contained except for standard semigroup facts and cited atom classifications. The 'good' and 'evil' atom labels in Theorem 3.5 are classification devices, not free parameters or invented entities.

assumptions (4)

standard math Standard properties of numerical semigroups and Apéry sets (e.g., Ap(S;g) has exactly g elements, one per residue class modulo g)
Invoked in Theorems 2.4 and 2.6; follows from [4].
domain assumption Generators are ordered g1 < g2 < ... < gk, with g1 the multiplicity
Lemma 2.7 and Lemma 2.3 use g2 > g1; Table 1's leading coefficients use g1.
domain assumption Convention 0^0 = 0 for ℓ_0(z), so ℓ_0 counts nonzero coordinates
Definition in Section 2; not stated explicitly but required for ℓ^M_0(n) = k and the p = 0 results.
domain assumption The atom characterization of M_{4,6} in Lemma 3.4 is correct (cited from [6] and [5])
All Section 3 results for M_{4,6} rely on it; if it failed, Theorem 3.5 and Propositions 3.6-3.7 would need rework.

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Pith. "Pith review of Some asymptotic results on $p$-lengths of factorizations for numerical semigroups and arithmetical congruence monoids." pith.science (2026). https://pith.science/paper/VF7IPPSL

@misc{pith2026241117010,
  author       = {Pith},
  title        = {Pith review of: Some asymptotic results on $p$-lengths of factorizations for numerical semigroups and arithmetical congruence monoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VF7IPPSL}},
  note         = {Machine review of arXiv:2411.17010}
}

abstract

A factorization of an element $x$ in a monoid $(M, \cdot)$ is an expression of the form $x = u_1^{z_1} \cdots u_k^{z_k}$ for irreducible elements $u_1, \ldots, u_k \in M$, and the length of such a factorization is $z_1 + \cdots + z_k$. We introduce the notion of $p$-length, a generalized notion of factorization length obtained from the $\ell_p$-norm of the sequence $(z_1, \ldots, z_k)$, and present asymptotic results on extremal $p$-lengths of factorizations for large elements of numerical semigroups (additive submonoids of $\mathbb Z_{\ge 0}$) and arithmetical congruence monoids (certain multiplicative submonoids of $\mathbb Z_{\ge 1}$). Our results, inspired by analogous results for classical factorization length, demonstrate the types of combinatorial statements one may hope to obtain for sufficiently nice monoids, as well as the subtlety such asymptotic questions can have for general monoids.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Adams, R

D. Adams, R. Ardila, D. Hannasch, A. Kosh, H. McCarthy, V. Pon omarenko, and R. Rosenbaum, Bifurcus semigroups and rings , Involve 2 (2009), no. 3, 351–356

work page 2009

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I. Aliev, J. De Loera, C. O’Neill, and T. Oertel, Sparse solutions of linear Diophantine equations, SIAM Journal on Applied Algebra and Geometry 1 (2017), no. 1, 239–253

work page 2017

[3] [3]

D. F. Anderson and P. Pruis, Length functions on integral domain , Proc. Amer. Math. Soc. 113 (1991), 933–937

work page 1991

[4] [4]

Assi and P

A. Assi and P. Garc ´ ıa-S´ anchez,Numerical semigroups and applications , RSME Springer Series, 1. Springer, [Cham], 2016

work page 2016

[5] [5]

Baginski and S

P. Baginski and S. Chapman, Arithmetic congruence monoids: a survey , Combinatorial and additive number theory - CANT 2011 and 2012, 15–38, Springer Proc. Math. Stat., 101, Springer, New York, 2014

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Banister, J

M. Banister, J. Chaika, S. Chapman, and W. Meyerson, On the arithmetic of arithmetical congruence monoids, Colloq. Math. 108 (2007), no. 1, 105–118

work page 2007

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Barron, C

T. Barron, C. O’Neill, and R. Pelayo, On the set of elasticities in numerical monoids , Semigroup Forum 94 (2017), no. 1, 37–50

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Beck and S

M. Beck and S. Robins, Computing the continuous discretely. Integer-point enume ration in polyhedra, Undergraduate Texts in Mathematics. Springer, New York, 2007

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