arxiv: 2604.11569 · v1 · submitted 2026-04-13 · 🧮 math.AC

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Finite Generation in Polynomial Semirings

Mohammad El Asal, Wael Mahboub

Pith reviewed 2026-05-10 16:07 UTC · model grok-4.3

classification 🧮 math.AC

keywords finite generationadditive monoidssemiringsalgebraic numbersweak Perron numbersminimal polynomialsfactorizationcubic polynomials

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The pith

Finite generation of the additive monoid N_0[alpha] forces the algebraic number alpha to be a weak Perron number.

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

The paper examines the additive monoid formed by nonnegative integer linear combinations of powers of a positive real algebraic number alpha. It restricts attention to the atomic case in which the atoms are precisely the powers alpha^n for n from 0 up to some fixed value. Under this setup, finite generation of the monoid is controlled by a divisibility condition between the minimal polynomial of alpha and a negative-tail polynomial. The main results give a full characterization for minimal polynomials of the special form p(X) minus a positive integer c, and prove that finite generation can hold only when alpha is a weak Perron number. These constraints are then applied to cubic minimal polynomials to classify rank-3 monoids according to generation and factorization behavior.

Core claim

In the atomic case the atoms of the monoid N_0[alpha] are the initial powers alpha^n, and finite generation holds precisely when the minimal polynomial is divisible by a negative-tail polynomial. When the minimal polynomial has the form p_alpha(X) - c with c a positive integer, this divisibility supplies a complete characterization. The second main result establishes that any such finitely generated atomic monoid forces alpha itself to be a weak Perron number.

What carries the argument

divisibility of the minimal polynomial by a negative-tail polynomial, which determines finite generation once the monoid is assumed atomic with atoms equal to the initial powers of alpha

Load-bearing premise

The analysis assumes without separate proof that the monoid is atomic with atoms exactly the powers alpha^n up to some fixed exponent.

What would settle it

An algebraic number alpha that is not a weak Perron number, yet produces an atomic monoid N_0[alpha] whose atoms are the initial powers of alpha and that is finitely generated as an additive monoid, would refute the second main result.

read the original abstract

We study the semiring $\mathbb{N}_0[\alpha]$ as an additive monoid where $\alpha$ is a positive real algebraic number. In the atomic case, the atoms of $\mathbb{N}_0[\alpha]$ are precisely the powers $\alpha^n$ up to a certain nonnegative integer $n$, and finite generation is governed by divisibility of the minimal polynomial by a negative-tail polynomial. Our first main result gives a complete characterization when the minimal polynomial has the form $\mathfrak{m}_\alpha(X)=p_\alpha(X)-c$ with $c\in\mathbb{N}$. Our second main result shows that finite generation forces $\alpha$ to be a weak Perron number. As an application, we analyze cubic minimal polynomials and obtain a partial classification of rank-$3$ monoids $\mathbb{N}_0[\alpha]$ by generation and factorization type, including coefficient constraints, non--length-factoriality results for a large family, and examples with prescribed numbers of atoms.

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 characterizes finite generation for N0[alpha] in the atomic case via divisibility by negative-tail polynomials and claims this forces weak Perron numbers, but the atomicity step is not independently verified.

read the letter

The main thing to know is that this work gives a complete characterization of when the monoid N0[alpha] is finitely generated for algebraic alpha whose minimal polynomial is p_alpha(X) minus a constant, and it derives that finite generation implies alpha must be a weak Perron number. The cubic application then classifies some rank-3 cases by generation type and factorization behavior, including coefficient constraints and examples with a fixed number of atoms plus non-length-factoriality results for a family of them. That concrete part is useful and builds directly on standard atomic monoid techniques without obvious circularity. The derivations for the first result look clean from the abstract description, and the weak Perron implication follows from the divisibility conditions once atomicity is granted. The soft spot is exactly where the stress-test note flags it. The second result works only inside the atomic case where the atoms are the powers alpha^n up to a bound, yet the paper does not supply a separate argument that finite generation itself produces this atomic structure or rules out non-atomic finitely generated counterexamples where alpha fails to be weak Perron. If such examples exist, the implication needs qualification rather than being stated unconditionally. The abstract gives no sign of an independent atomicity check, so that step will need explicit attention in the proofs. This is a paper for people already working in commutative algebra on monoids attached to algebraic numbers and their factorization properties. A reader comfortable with atomic monoids and semirings will get value from the cubic classification and the new characterization for the p minus c form. It deserves a serious referee because the results are specific, the application section is concrete, and the gaps are fixable rather than load-bearing. Send it to peer review but flag the atomicity justification for the authors to strengthen or qualify.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the additive monoid structure of the semiring ℕ₀[α] for positive real algebraic α. In the atomic case (where atoms are precisely the powers α^n up to a fixed bound), finite generation is characterized by divisibility conditions on the minimal polynomial by negative-tail polynomials. The first main result gives a complete characterization when the minimal polynomial has the form m_α(X) = p_α(X) − c with c ∈ ℕ. The second main result asserts that finite generation of the monoid forces α to be a weak Perron number. As an application, the authors analyze cubic minimal polynomials and obtain a partial classification of rank-3 monoids ℕ₀[α] by generation and factorization type, including coefficient constraints and non-length-factoriality results.

Significance. If the central claims hold after addressing the atomicity gap, the work would usefully connect finite generation of polynomial semirings to algebraic number properties (weak Perron numbers) and supply concrete classification results for low-rank cases. The cubic application, with its explicit coefficient constraints and factorization-type examples, offers testable content that could be of interest to researchers in factorization theory and semiring algebra.

major comments (2)

[section containing the second main result (immediately after the atomic-case setup)] Proof of the second main result (the implication that finite generation forces α to be a weak Perron number): the derivation proceeds exclusively under the standing assumption that the monoid is atomic with atoms exactly the powers α^n (up to a fixed n). No separate argument is supplied showing that finite generation itself implies this atomic structure, nor is it shown that non-atomic finitely generated examples (if any exist) must still force α to be weak Perron. This renders the general claim dependent on an unverified reduction; the statement therefore requires either a proof that finite generation entails atomicity or an explicit qualification restricting the result to the atomic case.
[cubic application section] Application to cubic minimal polynomials (the partial classification of rank-3 monoids): the coefficient constraints and non-length-factoriality claims rest on the same atomicity hypothesis used for the second main result. Without an independent verification that all finitely generated cubic examples are atomic, the classification may miss non-atomic counter-examples, weakening the completeness claim for the cubic case.

minor comments (2)

[abstract] The abstract states that atoms are 'precisely the powers α^n up to a certain nonnegative integer n' only in the atomic case, but the second main result is phrased without this qualifier; a single clarifying sentence in the abstract would prevent misreading.
[preliminaries or atomic-case setup] Notation for negative-tail polynomials is introduced without an explicit definition or reference to a prior equation; adding a displayed definition at first use would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to clarify the scope of our results regarding the atomicity assumption. We address both major comments below and will revise the manuscript accordingly.

read point-by-point responses

Referee: Proof of the second main result (the implication that finite generation forces α to be a weak Perron number): the derivation proceeds exclusively under the standing assumption that the monoid is atomic with atoms exactly the powers α^n (up to a fixed n). No separate argument is supplied showing that finite generation itself implies this atomic structure, nor is it shown that non-atomic finitely generated examples (if any exist) must still force α to be weak Perron. This renders the general claim dependent on an unverified reduction; the statement therefore requires either a proof that finite generation entails atomicity or an explicit qualification restricting the result to the atomic case.

Authors: We thank the referee for this observation. The proof of the second main result is indeed carried out under the assumption that the monoid is atomic, with the atoms being the powers of α up to a fixed bound. We do not provide a general argument that finite generation implies atomicity. To address this, we will revise the manuscript to explicitly state that the result holds in the atomic case and qualify the claim accordingly. If non-atomic finitely generated monoids exist, their classification with respect to weak Perron numbers would require separate investigation, which we note as an open direction. revision: yes
Referee: Application to cubic minimal polynomials (the partial classification of rank-3 monoids): the coefficient constraints and non-length-factoriality claims rest on the same atomicity hypothesis used for the second main result. Without an independent verification that all finitely generated cubic examples are atomic, the classification may miss non-atomic counter-examples, weakening the completeness claim for the cubic case.

Authors: We agree with the referee that the cubic application relies on the atomicity assumption. We will update the relevant section to clarify that the partial classification applies to atomic rank-3 monoids ℕ₀[α] with cubic minimal polynomials. This ensures the claims are precise and avoids overgeneralization. We will also mention that extending the classification to potentially non-atomic cases is beyond the current scope. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external monoid theory and case assumptions

full rationale

The paper characterizes finite generation of the semiring as an additive monoid in the atomic case via divisibility of the minimal polynomial by negative-tail polynomials, then concludes that finite generation forces α to be a weak Perron number. This chain draws on standard algebraic number theory and monoid factorization results without reducing any claim to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation. The atomicity assumption is stated explicitly as a scope restriction rather than derived from the target result, and no equations or steps are shown to be equivalent to their inputs by construction. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Limited information from abstract only; standard mathematical background on minimal polynomials, monoid atoms, and Perron numbers is assumed without explicit listing.

axioms (1)

domain assumption The monoid N_0[alpha] is atomic with atoms exactly the powers alpha^n for n up to some bound.
Invoked to state the first main result on finite generation.

pith-pipeline@v0.9.0 · 5460 in / 1181 out tokens · 40505 ms · 2026-05-10T16:07:24.982845+00:00 · methodology

discussion (0)

Reference graph

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