REVIEW 4 major objections 4 minor 32 references
On the atomicity of one-dimensional monoid algebras
T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read Odd-denominator monoid algebras are atomic, without ACCP
desk verdict Real progress on Gotti's conjecture: the odd-denominator case is reduced to a handful of fillable gaps, not a broken strategy. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The paper's central object is the splitting sequence: starting from an irreducible polynomial $f$ in $\mathbb{Z}[x]$ and an exponent sequence $(e_n)$, each next term is an irreducible divisor of $f_{n-1}(x^{e_n})$; the binary string records whether each step is a 'lift' or a 'split.' The key estimate is Proposition 5.1, which bounds the number of irreducible divisors of $f$ in $\mathbb{Z}[\frac{1}{r^\lambda}\mathbb{N}_0]$ by a finite quantity $A(f,\pi)$ depending only on $f$, the prime set $\pi$ of $r$, and the leading coefficient of $f$. This bound is obtained by combining Capelli's Lemma, a degree-counting argument, and a new invariant $\Lambda^*(f,P)$ that measures how many initial splits a splitting sequence can have; Lemma 4.5 transfers the bound from $f$ to every polynomial in its composed divisor set. The uniform bound is what forces factorization to stabilize as the exponent lattice is refined, which is the mechanism that excludes an infinite ascending chain of principal ideals.
What would settle it
Find an irreducible polynomial $f$ in $\mathbb{Z}[x]$ outside $\{\pm x\} \cup \{\pm \Phi_y(x) : y \in \mathbb{N}\}$ for which the number of irreducible divisors of $f(x^{r^\lambda})$ in $\mathbb{Z}[x]$ grows faster than the bound $A(f,\pi)$ of Proposition 5.1 as $\lambda$ grows, or exhibit an ACCP-supported non-monomial in $\mathbb{Z}[M_q]$ with $q$ of odd denominator that has a strictly ascending chain of principal ideals that never stabilizes. Either observation would disprove the claim that $\mathbb{Q}[M_q]$ is atomic for this class.
Extended reading notes
Core claim
The central claim is Theorem 5.2: for every $q \in ((0,1) \cap \mathbb{Q}) \setminus \mathbb{N}^{-1}_{>1}$ with odd denominator, the monoid algebra $\mathbb{Q}[M_q]$ is atomic. Since $M_q$ is an atomic Puiseux monoid that fails the ACCP, and ACCP ascends from a monoid to its monoid algebra, $\mathbb{Q}[M_q]$ is a one-dimensional integral domain that is atomic but does not satisfy the ACCP. The paper establishes this by showing that every ACCP-supported polynomial in $\mathbb{Z}[M_q]$ satisfies the ACCP, hence is atomic, and by reducing atomicity of $\mathbb{Q}[M_q]$ to atomicity of $\mathbb{Z}[M_q]$ through a coefficient-clearing argument. The load-bearing technical step is a stabilization claim: for any irreducible $f$ in $\mathbb{Z}[\frac{1}{r^\ell}\mathbb{N}_0]$ avoiding $\pm x$ and $\pm$ cyclotomic polynomials, there is a threshold $\Gamma$ such that $f$ has the same factorization into irreducibles in $\mathbb{Z}[\frac{1}{r^{\ell+\gamma}}\mathbb{N}_0]$ for every $\gamma \geq \Gamma$. This claim, derived from a uniform bound on the number of irreducible divisors in these subalgebras, rules out infinite strictly ascending chains of principal ideals.
Load-bearing premise
The proof depends on the stabilization claim: for each irreducible polynomial $f$ in the smaller algebra, the number of ways it factors cannot increase without bound as the allowed exponent denominators are enlarged; if that claim fails, an infinite strictly ascending chain of principal ideals could survive and the main theorem would collapse.
Editorial extensions
If this is right
- For every odd-denominator $q$ in the parameter set, $\mathbb{Q}[M_q]$ is a one-dimensional atomic domain that fails the ACCP, so atomicity and the ACCP are genuinely independent even in Krull dimension one.
- Gotti's conjecture is now settled for an infinite family of parameters, with only the even-denominator cases left open.
- The splitting-sequence machinery, especially Proposition 5.1, gives uniform divisor bounds in $\mathbb{Z}[\frac{1}{r^\lambda}\mathbb{N}_0]$ that can be reused to study ascent of finite factorization and bounded factorization properties to monoid algebras.
- Since $\mathbb{Q}[M_q]$ is atomic, every nonunit element has at least one factorization into irreducibles, making the factorization theory of these one-dimensional domains explicitly accessible.
- The atomicity conclusion implies that the failure of the ACCP in this setting does not obstruct the existence of irreducible factorizations, clarifying the boundary between the two conditions.
Reading between the lines
- A natural testable extension is to remove the odd-denominator restriction: if the same stabilization claim can be proved with $r$ even, Gotti's conjecture would follow in full, since the apparent obstruction is the parity of the exponent in the cyclotomic factorization steps.
- The uniform divisor bound suggests that the refined algebras $\mathbb{Z}[\frac{1}{r^\lambda}\mathbb{N}_0]$ form a chain whose atomic structure stabilizes entrywise; one could investigate whether the stabilized factorization sets define an invariant of the limit algebra, or of the original $\mathbb{Q}[M_q]$.
- The methods may transfer to other rank-one monoid algebras built from monotone or multi-geometric Puiseux monoids, where analogous splitting-sequence bounds could tame infinite ascending chains of principal ideals.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies atomicity of monoid algebras of exponentially cyclic Puiseux monoids M_q over Q. The main theorem (Theorem 5.2) asserts that Q[M_q] is atomic for every q in ((0,1) ∩ Q) \ N^{-1}_{>1} with odd denominator, extending the recent result for q = 3/4. The strategy is to reduce atomicity of Q[M_q] to atomicity of Z[M_q] via Proposition 2.1, use the almost ACCP of M_q to reduce to ACCP-supported polynomials, and then prove that every ACCP-supported polynomial in Z[M_q] satisfies the ACCP. The proof introduces splitting sequences, bounds their lengths through algebraic number theory (Lemma 4.4 and Lemma 4.5), and then uses a stabilization claim inside Theorem 5.2 to rule out an infinite ascending chain of principal ideals. The paper concludes with a contradiction involving cyclotomic factorizations and p-adic-type valuations.
Significance. If the proof is completed, the result provides the first infinite family of one-dimensional integral domains that are atomic but fail the ACCP, making substantial progress on Gotti's conjecture. The splitting-sequence machinery and the reduction to Z[M_q] are natural and potentially reusable tools for further cases of the conjecture. The paper contains no fitted parameters or empirical constants, and the overall proof architecture is coherent: it reduces the problem to concrete polynomial-factorization bounds. However, several load-bearing technical steps are currently either omitted or too terse to verify, so the main theorem is not yet fully established as written.
major comments (4)
- [Section 4, Lemma 4.5] The proof of inequality (4.4) in Case 2 is too terse to be checkable. In particular, the construction of the sequence S, the assertion that the exponents a_1,...,a_m can be chosen 'maximally,' and the contradictions in Subcases 2.1 and 2.2 are not fully argued; Subcase 2.2 appears to rely on an ordering of the primes in P that is not introduced. This lemma is load-bearing because it supplies the uniform bound for monic irreducible polynomials that Proposition 5.1 uses in the case lc(f) ∈ {±1}.
- [Section 5, Proposition 5.1] The proof of Proposition 5.1 explicitly treats only the case lc(f) ∉ {±1} and dismisses the case lc(f) ∈ {±1} with the statement that 'a similar proof structure may be used.' Since the monic bound r^{Λ*(f,π)} is exactly what the stabilization Claim in Theorem 5.2 invokes for non-exceptional irreducible polynomials, this case must be written out in full rather than omitted.
- [Section 5, equations (5.3)-(5.4)] The symbol v_{n(q)} is used for the valuation of an lcm and later of the exponent c, but Section 2.1 only defines p-adic valuations v_p for primes p. The final contradiction depends on comparing these valuations, so either v_m must be defined for composite m or the argument must be rewritten with an explicit prime divisor of n(q). In addition, the displayed equality (5.3) needs a proof explaining how the cyclotomic factorization and the condition gcd(n(q), r) = 1 imply the claimed equality of valuations.
- [Theorem 5.2, stabilization Claim] The step from the uniform divisor bound in Proposition 5.1 to the statement that f has 'the same factorization into irreducibles' for all sufficiently large γ is not fully justified. One needs to prove both that the number of irreducible factors of f in the tower Z[1/r^{ℓ+γ} N0] is nondecreasing in γ and that stabilization of that number, together with uniqueness of factorizations in these UFDs, forces the factorization itself to stabilize. Neither observation is stated or proved.
minor comments (4)
- [Section 2.1] The notation /llbracketa, b/rrbracket is used for discrete intervals with unusual delimiters; please ensure it is rendered consistently as a discrete interval in the published version.
- [Example 3.4(3)] The expression F := f(x) ∘ (g(x^{3^{n-1}}))_{n∈N} is ambiguous about the starting index and the intended sequence of polynomials; please clarify the notation.
- [Lemma 2.5] The paragraph beginning 'Let P′(x) := f(x^p)/P(x)' appears to be part of the proof but is not labeled as such; it should be moved into the proof environment.
- [Definition 3.3] In Definition 3.3, Nspl(F) is described as the 'number of splits at the beginning' of F but is then defined as a minimal index; the wording should be aligned with the formal definition.
Circularity Check
No circularity: the proof derives atomicity of Q[M_q] from independent algebraic lemmas and does not reuse the target result, fitted parameters, or self-citations as inputs.
full rationale
No circular step is present. The main theorem (Theorem 5.2) is proved for all odd-denominator q directly; the previously known case Q[M_{3/4}] from [6] is not assumed, and the citation to [6] is only for proof strategy ('We follow the proof of [6, Proposition 4.8]'), not as a black-box input. The derivation uses standard independent facts: cyclotomic polynomial identities (Lemmas 2.3 and 2.4), Guersenzvaig's irreducibility criteria (Lemma 2.5), bounds on the number of irreducible divisors (Lemma 2.6), Capelli's Lemma (Lemma 2.7), and the almost-ACCP property of exponentially cyclic monoids. None of these presuppose atomicity of Q[M_q]. Proposition 5.1 gives explicit finite bounds on divisor counts via Lemmas 2.5, 2.6, 4.4, and 4.5; there is no fitted parameter and no 'prediction' that is statistically forced. The stabilization Claim inside Theorem 5.2 is a derived statement: its proof appeals to Proposition 5.1 plus the tower of UFD extensions, and its conclusion (eventual identity of factorizations) is stronger than, and not an input to, Proposition 5.1. There are no self-citations by the author and no imported uniqueness theorem. The only passage that warrants scrutiny is Proposition 5.1, where the case lc(f) ∈ {±1} is dismissed with 'A similar proof structure may be used for the case when lc(f) ∈ {±1}, although the quantities H and θ1 are not needed for the proof.' Since the stabilization Claim depends on that monic case, this is a genuine omitted-verification and correctness risk, but it is not circular: the monic bound is not assumed from the theorem, and no equation identifies the target with an input. Accordingly, the circularity score is 0.
Assumptions & free parameters
assumptions (5)
- domain assumption M_q satisfies the almost ACCP for q in ((0,1) cap Q) minus N^{-1}_{>1}
- standard math Capelli's Lemma and its consequences for irreducibility of f(x^r)
- standard math Guersenzvaig's irreducibility criteria for f(x^p) over Z (Theorems 4.3 and 4.4 of [25])
- standard math Cyclotomic factorization identities for Phi_y(x^r) and x^n - 1 (Lemmas 2.3 and 2.4)
- domain assumption Atomicity criteria for monoid algebras, e.g., [17, Theorem 3.3] used in Proposition 2.1
Cite this review
Pith. "Pith review of On the atomicity of one-dimensional monoid algebras." pith.science (2026). https://pith.science/paper/W54MLGUQ
@misc{pith2026241116151,
author = {Pith},
title = {Pith review of: On the atomicity of one-dimensional monoid algebras},
year = {2026},
howpublished = {\url{https://pith.science/paper/W54MLGUQ}},
note = {Machine review of arXiv:2411.16151}
}
abstract
The ascending chain condition on principal ideals (ACCP) is almost always complementary to atomicity within integral domains: in fact, Cohn initially stated that these two conditions were equivalent. This assertion has been shown to be false, however most counterexamples require technical algebraic constructions. In 2017, Gotti conjectured that for every $q$ in the set $S := ((0, 1) \cap \mathbb{Q}) \setminus {\mathbb{N}}^{-1}_{> 1}$, atomicity ascends from the exponentially cyclic Puiseux monoid $M_q$ to its monoid algebra over the field of rationals. If this conjecture were true, it would provide an extremely wide class of atomic domains of Krull dimension one not satisfying the ACCP, and so would be perhaps the simplest possible such examples. Bu et al. recently proved that the monoid algebra $\mathbb{Q} \left[M_{3/4} \right]$ is atomic, marking the first progress towards settling this conjecture. We strengthen this result and prove that $\mathbb{Q}[M_q]$ is atomic for all $q \in S$ having an odd denominator.
Reference graph
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