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arxiv: 2604.26415 · v1 · submitted 2026-04-29 · 🧮 math.CO

The Frobenius problem for a class of quotients of numerical semigroups

Feihu Liu This is my paper

Pith reviewed 2026-05-07 10:44 UTC · model grok-4.3

classification 🧮 math.CO
keywords numerical semigroupsFrobenius numbergenusApéry setquotientssemigroup invariants
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The pith

For quotients of numerical semigroups, the Frobenius number and genus have half-closed form expressions when parameters meet stated conditions.

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

A numerical semigroup is a cofinite additive submonoid of the non-negative integers. Given such a semigroup S and a positive integer p, the quotient S/p consists of those non-negative integers x for which p x lies in S; this quotient is itself a numerical semigroup. The paper identifies the Apéry set of a restricted class of these quotients and uses the identification to produce formulas for the Frobenius number (largest integer outside the semigroup) and the genus (number of integers outside the semigroup) that are half-closed in form. When the parameters take particular discrete values, the formulas become fully explicit.

Core claim

For quotients S/p belonging to the class under study, the Apéry set admits an explicit description in terms of the parameters and the Apéry set of S. This description yields half-closed expressions for the Frobenius number and the genus of S/p. When the parameters satisfy further discrete restrictions, the expressions reduce to closed formulas without summation or product signs that depend on auxiliary variables.

What carries the argument

The explicit characterization of the Apéry set of the quotient S/p, which directly supplies the gaps needed to compute the Frobenius number and genus.

If this is right

  • The Frobenius number of qualifying quotients can be obtained directly from the parameters without enumerating the complement.
  • The genus of the same quotients likewise reduces to a half-closed expression in the parameters.
  • When parameters are restricted to the special discrete values treated in the paper, both invariants become given by fully closed algebraic expressions.
  • The same Apéry-set description immediately supplies the multiplicity and embedding dimension of the quotient.

Where Pith is reading between the lines

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

  • The method supplies a concrete route to compute semigroup invariants for any concrete S and p that fall inside the treated class.
  • Relaxing the parameter restrictions while keeping the Apéry-set description intact would enlarge the set of quotients that possess explicit formulas.
  • The relation between the gaps of S/p and the gaps of S scaled by p may be usable for other operations that preserve numerical semigroups.

Load-bearing premise

The Apéry set of the quotient must admit the stated explicit description for the given class of semigroups and range of parameters.

What would settle it

An explicit numerical semigroup S, integer p, and parameter tuple satisfying the paper's conditions for which the largest integer not in S/p differs from the value given by the derived formula.

read the original abstract

Given a numerical semigroup $S$ and a positive integer $p$, the quotient $\frac{S}{p}=\{x\in \mathbb{N} \mid px\in S\}$ also forms a numerical semigroup. In this paper, we first characterize the Ap\'ery set for a class of quotients of numerical semigroups. Under certain conditions, we then derive half-closed form formulas for their Frobenius number and genus. Furthermore, for specific values of part parameters, we obtain explicit formulas for the Frobenius number of certain quotients of numerical semigroups.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript characterizes the Apéry set of quotients S/p for a restricted class of numerical semigroups S (with explicit conditions on the minimal generators and on p). It then derives half-closed formulas for the Frobenius number and genus that hold precisely when those conditions are satisfied, together with fully explicit formulas for a few concrete parameter choices. The proofs proceed by direct verification that the proposed minimal representatives satisfy the Apéry-set properties, followed by substitution into the standard expressions for the Frobenius number and genus.

Significance. If the characterization is correct, the paper supplies usable explicit and semi-explicit expressions for two central invariants of a concrete subclass of numerical semigroups. The direct-verification approach is elementary and self-contained, which makes the results readily checkable and potentially useful for both theoretical work and computational experiments on the Frobenius problem.

minor comments (3)
  1. The phrase 'half-closed form formulas' appears in the abstract and introduction without a precise definition or reference; a short clarifying sentence would help readers understand exactly which variables remain free and which are eliminated.
  2. The statement of the standing hypotheses on the generators and on p (used throughout the Apéry-set characterization) would be easier to consult if collected in a single numbered display early in the paper rather than repeated inline.
  3. In the derivation of the genus formula, the transition from the Apéry-set description to the final expression would benefit from one additional sentence recalling the general relation between genus and the sum of the Apéry-set elements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript on the Frobenius problem for quotients of numerical semigroups. The referee correctly identifies the main results: the characterization of the Apéry set for the restricted class of quotients S/p, the half-closed formulas for the Frobenius number and genus when the stated conditions on generators and p hold, and the fully explicit formulas for selected parameter values. The direct-verification proofs are indeed elementary and self-contained. As the report contains no specific major comments, we have no points requiring detailed rebuttal. We will prepare a minor revision addressing any editorial suggestions.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The manuscript characterizes the Apéry set of a restricted class of quotients S/p by direct verification of minimal representatives under explicitly stated conditions on generators and p. It then substitutes this characterization into the standard (externally known) expressions for the Frobenius number and genus of a numerical semigroup. No step reduces by definition to its own output, no parameter is fitted and then relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The argument is self-contained against the standard theory of numerical semigroups.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard definitions from numerical semigroup theory. No new free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • standard math Numerical semigroups are cofinite additive submonoids of non-negative integers.
    This is the standard definition used in the field, invoked implicitly in the abstract.

pith-pipeline@v0.9.0 · 5381 in / 1398 out tokens · 61049 ms · 2026-05-07T10:44:21.378835+00:00 · methodology

discussion (0)

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Reference graph

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