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The Bi-UF Positive Conjecture for quadratic monogenic semirings and related progress

T0 review · 2 major / 3 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read For every positive quadratic algebraic number alpha, the monogenic semiring N0[alpha] cannot be both additively and multiplicatively factorial.

desk verdict Solid, genuinely new results on the Bi-UF Positive Conjecture, but Theorem 3.6 has a local proof gap in one subcase and the abstract overclaims 'quadratic' for 'positive quadratic.' read the letter →

arxiv 2607.22669 v1 pith:I5SMB4IY submitted 2026-07-06 math.GM

Felix Gotti , Omar Graia , Darren Han , Hengrui Liang This is my paper · ORCID
classification math.GM MSC 16Y6013F1513A0511R0913G05
keywords Bi-UFPositiveConjecturemonogenicsemiringuniquefactorizationmonoidhalf-factorialsemidomainquadraticalgebraicnumberatomiccomplex
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

The paper proves that the Bi-UF Positive Conjecture — which says that the nonnegative integers are the only positive semiring whose additive and multiplicative monoids both have unique factorization — holds for every quadratic monogenic semiring. It does so by assuming additive factoriality and deriving explicit non-unique multiplicative factorizations from the coefficient identities forced by alpha^2 = m alpha + n. The paper also shows that if the constant term of the primitive minimal polynomial of a positive non-rational algebraic generator is composite, the semiring is not factorial, giving another class satisfying the conjecture. It extends the conjecture from positive to complex semirings and narrows the search to finite N0-spans of algebraic numbers. In the relaxed half-factorial setting, it proves that N0 is the only positive rational bi-HFS and constructs two families of bi-HFS examples distinct from N0.

What carries the argument

The load-bearing tool is a characterization (Theorem 1.1) stating that for a positive algebraic number alpha, the additive monoid of S_alpha is a UFM exactly when its atoms are the powers 1, alpha, ..., alpha^{d-1}. In the quadratic case this reduces every element to a unique expression c + d alpha with c,d in N0, and the relation alpha^2 = m alpha + n converts any multiplicative product into two coefficient equations. These equations are used to certify atoms and to manufacture explicit non-unique factorizations. For the non-quadratic Theorem 3.8, the primitive minimal polynomial and Gauss's lemma control divisibility, showing that alpha is an atom that fails to be prime.

What would settle it

Find one positive quadratic algebraic number alpha for which S_alpha is both additively and multiplicatively factorial; the paper's Theorem 3.6 says none exists. Concretely, for alpha satisfying alpha^2 = m alpha + n, test the explicit candidates in the proof — for odd m the element (alpha + k)^2 with k = (r-m)/2, r = m^2 + 4n, and for even positive m the element (alpha + k)^2 with k = m1^2 - m1 + n, m1 = m/2 — and check whether the displayed factorizations actually give two distinct atomic factorizations; any failure would expose a gap.

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Extended reading notes

Core claim

The central theorem (Theorem 3.6) states: if alpha is a positive quadratic algebraic number such that the additive monoid of S_alpha = N0[alpha] is a unique factorization monoid, then the multiplicative monoid of S_alpha is not factorial. Hence no positive quadratic monogenic semidomain is a bi-UFS (Corollary 3.7). The proof splits into cases by the parity of the linear coefficient m in the minimal polynomial x^2 - m x - n and produces, in each case, two distinct factorizations of one element into atoms using identities such as (alpha + k)^2 = r(alpha + ...) and (alpha+k)^2 = r(2alpha + ...). A second theorem (Theorem 3.8) shows that if w_alpha(0), the constant term of the primitive integer

Load-bearing premise

The whole quadratic argument rests on the imported characterization that additive factoriality of S_alpha forces the additive atoms to be exactly the powers 1, alpha, ..., alpha^{d-1}; if that characterization fails for some positive quadratic alpha, the coefficient identities and the non-unique factorizations built from them no longer apply.

Editorial extensions

If this is right

  • The Bi-UF Positive Conjecture is now known to hold for every quadratic monogenic semidomain; the open cases are higher-degree generators.
  • Any bi-UFS with reduced, finite-rank additive monoid must be isomorphic to a finite N0-span of algebraic numbers in a number field (Theorem 4.2), so proving the complex conjecture reduces to ruling out factoriality in those spans.
  • If the additive monoid has rank 1, the only bi-UFS is N0 (Corollary 4.3).
  • The half-factorial analogue is strictly weaker: bi-HFS examples exist beyond N0, including Laurent-polynomial semirings, while positive rational bi-HFS still force N0.
  • The composite-constant-term theorem gives a purely arithmetic obstruction: any positive algebraic generator whose primitive polynomial has composite constant term generates a non-factorial monogenic semiring.

Reading between the lines

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

  • The coefficient-comparison technique used for quadratics may extend to higher-degree monogenic semirings, where additive factoriality gives a basis {1, alpha, ..., alpha^{d-1}} and multiplication by alpha yields a linear recurrence; analogous identities could yield non-unique factorizations in degree d.
  • The paper's Theorem 3.8 suggests a testable heuristic: for algebraic integers, composite field norm should imply failure of multiplicative factoriality, so bi-UFS candidates must have prime norm; this could be checked against known classes.
  • The Laurent-polynomial construction indicates that the bi-HF condition is far less restrictive than bi-UF; a broader classification of bi-HFS may be possible along block-monoid lines.
  • The structural theorem (Theorem 4.2) effectively converts the complex conjecture into a question about subrings of number fields, which may be approachable by existing results on arithmetic of orders.
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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

2 major / 3 minor

Summary. The paper studies complex semirings, focusing on monogenic semidomains S_α = N_0[α]. Its central claims are: (1) Theorem 3.6 shows that if α is a positive quadratic algebraic number for which the additive monoid (S_α,+) is factorial, then S_α is not multiplicatively factorial; hence no positive quadratic monogenic semidomain is a bi-UFS (Corollary 3.7). (2) Theorem 3.8 gives a second class: if α is a positive non-rational algebraic number whose primitive minimal polynomial has composite constant term, then S_α is not factorial. (3) Section 4 proves a structure theorem for semidomains whose additive monoid is a finite-rank free commutative monoid and uses it to formulate a Bi-UF Complex Conjecture. (4) Section 5 proves that N_0 is the only positive rational bi-HFS and constructs two families of bi-HFS semidomains distinct from N_0, one via pullbacks and one via Laurent polynomials.

Significance. If the proofs are completed, the paper gives substantial progress on the Bi-UF Positive Conjecture: it settles the conjecture for the natural class of quadratic monogenic semidomains and identifies another infinite class of monogenic semidomains that satisfy the conjecture. The structural theorem in Section 4 usefully narrows the scope of the extended complex conjecture, and the constructions in Section 5 enrich the known examples of bi-HFS semidomains. The paper is clearly written and builds carefully on the published characterization in [14]; I do not see circularity in that dependence. However, a key step in the proof of Theorem 3.6 is not fully justified, and since that theorem is the basis of Corollary 3.7, the main claim is not yet established as written.

major comments (2)
  1. [§3.1, Theorem 3.6, Case 2 (m=0, even n>2)] The proof that B = 2√n + (n+4)/2 is irreducible or twice an irreducible is not derived. From a factorization B=(a+b√n)(c+d√n), coefficient comparison gives ad+bc=2 and ac+nbd=(n+4)/2. Ruling out bd>0 only shows that, after swapping, one factor is a positive integer dividing 2. If that integer is 1, the factorization is trivial; if it is 2, one must prove that the cofactor √n + (n+4)/4 is actually an element of S_α (which requires (n+4)/4 to be an integer, i.e., n≡0 mod 4), and that this cofactor is irreducible, e.g. via Proposition 3.4. For n≡2 mod 4 the factor 2 cannot occur, so B is irreducible, but this also needs an argument. Without this step, the right-hand side of (3.6) is not shown to be a factorization into irreducibles distinct from (√n+2)^2. Since n ranges over an infinite family, this gap is load-bearing.
  2. [§3.1, Theorem 3.6, Case 2 (m even positive)] The same type of gap occurs for C = 2α + r - 2m_1 + 1. From ad+bc+2m_1bd=2 and m_1>1, the paper correctly gets bd=0 and ad=2, so after swapping a=1 or a=2. It then concludes 'C is either irreducible or 2(α+ℓ)' without justification. If a=2, the cofactor has constant term (r-2m_1+1)/2, which must be an integer; the parity condition is not discussed. If a=1, the factorization is by a unit, but one still has to argue that reducibility of C would force a factorization with a=2 and positive b,d. These details are needed to ensure that splitting C on the right-hand side of (3.7) yields a genuine factorization into irreducibles distinct from (α+k)^2.
minor comments (3)
  1. [Theorem 3.8, final paragraph] The last sentence reads 'Since every atom in a factorial monoid is prime, so S_α is factorial.' The intended conclusion is that S_α is not factorial. Please correct this typo; the surrounding argument clearly supports the negation.
  2. [Abstract] The phrase 'we extend the statement of the Bi-UF Positive Conjecture by motivated by a structural theorem' is grammatically awkward; suggest 'motivated by a structural theorem'.
  3. [Theorem 3.8, proof] There is a typo 'becase' in the sentence 'which is not possible becase α is not rational.' Also, in the proof of the claim, the induction step would be easier to follow if the sign condition on S_k were stated explicitly as an invariant, since it is used later to conclude b_k ∈ N_0.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main results are supported by an external published characterization and explicit coefficient-based constructions.

full rationale

The paper's derivation chain is not circular. The most load-bearing imported result, Theorem 1.1, is quoted from [14, Theorem 5.4], a published characterization theorem coauthored by the first author of this paper. Although this is a self-citation, it is independent support: it is a parameter-free theorem about positive algebraic monogenic semidomains, its stated assumptions do not include the target non-factoriality conclusions, and it is externally published and falsifiable. Lemma 3.1 applies Theorem 1.1 only to obtain the quadratic relation alpha^2 = m alpha + n and the resulting coefficient identities; Propositions 3.4, 3.5, and Theorem 3.6 then use those identities to construct explicit non-unique multiplicative factorizations. Nothing is fitted and then renamed as a prediction, and no equation is equivalent to its input by construction. Theorem 3.8 similarly gives a direct construction of an atom that is not prime, using only the primitive minimal polynomial and Gauss's lemma. The paper is self-contained relative to the cited external characterization, and I find no circular step.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new algebraic entities. It relies on standard background results and one prior characterization theorem by an overlapping author; no derivation reduces to an input or fitted value.

assumptions (6)
  • standard math Theorem 1.1 ([14, Thm 5.4]): For a positive algebraic number α, (Sα,+) is a UFM iff its additive atoms are exactly {1, α, ..., α^{deg mα − 1}}.
    Imported from prior published work by overlapping authors. It is the foundation of Lemma 3.1 and all quadratic coefficient identities.
  • standard math A reduced finite-rank factorial monoid is a finite-rank free commutative monoid.
    Used in Corollary 4.3 via [15, Theorem 1.2.2] to reduce the Bi-UF Complex Conjecture to finite N0-spans.
  • standard math Gauss's lemma and primitive polynomial divisibility in Z[x].
    Used in Theorem 3.8 to transfer divisibility conditions from evaluation at α to divisibility in Z[x].
  • standard math Localization of an integral domain by Z\{0} is an integral domain; a finite-dimensional Q-algebra that is a domain is a field.
    Used in Theorem 4.2 to embed the Grothendieck group into a number field.
  • standard math Evaluation at a transcendental number is injective on Laurent polynomials.
    Used in Section 5.2 to identify the Laurent polynomial ring with its evaluation at t.
  • domain assumption All monoids are cancellative and commutative; complex semidomains are subsemirings of C, so cancellation holds.
    The paper's convention in Section 2.1; this underpins coefficient comparison and divisibility arguments throughout.

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Pith. "Pith review of The Bi-UF Positive Conjecture for quadratic monogenic semirings and related progress." pith.science (2026). https://pith.science/paper/I5SMB4IY

@misc{pith2026260722669,
  author       = {Pith},
  title        = {Pith review of: The Bi-UF Positive Conjecture for quadratic monogenic semirings and related progress},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I5SMB4IY}},
  note         = {Machine review of arXiv:2607.22669}
}
abstract

A complex semiring is a subset of the complex plane that is closed under the standard addition and multiplication of complex numbers and contains both $0$ and $1$. A complex semiring $S$ is called a bi-UFS if both its additive monoid $(S,+)$ and its multiplicative monoid $(S\setminus \{1\}, \cdot)$ are unique factorization monoids (UFM). The Bi-UF Positive Conjecture states that $\mathbb{N}_0$ is the only subsemiring of the nonnegative cone of the real line that is a bi-UFS. In this paper, we prove that no simple semiring extension of $\mathbb{N}_0$ by a quadratic algebraic number is a bi-UFS, identifying a natural class of complex semirings satisfying the statement of the Bi-UF Positive Conjecture. We also identify another class of complex semirings satisfying the statement of the Bi-UF Positive Conjecture. Then we extend the statement of the Bi-UF Positive Conjecture by motivated by a structural theorem we established for semidomains whose additive monoid are finite-rank free commutative monoids. Finally, we consider the bi-HF property, which is a relaxed version of the bi-UF property. We prove that $\mathbb{N}_0$ is the only positive rational semidomain having the bi-HF property, and we provide two methods to construct bi-HFS complex semirings that are distinct from $\mathbb{N}_0$.

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