{"id":"8fd29a3f-f51b-4f9a-83f5-a7f86921b9b3","arxiv_id":"2501.04990","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Quasi-atomicity does not ascend to polynomial extensions in general, and neither almost nor quasi-atomicity ascend to monoid domains over finite fields.","lead":"This paper studies whether two weak forms of atomicity, almost atomicity and quasi-atomicity, survive when passing to polynomial rings and monoid algebras. It gives positive ascent under extra conditions, a counterexample showing quasi-atomicity can fail to ascend to polynomial extensions, and counterexamples for monoid domains over finite fields.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.1 relies on the false assertion that R[x,y] is a UFD: for R=Z+Zx+x^2 R[x], the element sqrt(2)x^2 is infinitely divisible by 2, so R[x,y] is not even atomic and the proof of non-ascent is unsupported.","rationale":"The reader's weakest assumption was the unproved ACCP-based boundedness of factorization lengths in Theorem 3.2. That concern is real and should be addressed; ACCP alone does not bound lengths in arbitrary commutative monoids, and the proof of Theorem 3.2 uses that boundedness to choose a maximal decomposition. However, the single most load-bearing issue I find is different: Theorem 4.1's proof asserts that R[x,y] is a UFD when R is the subring in (4.1). That assertion is demonstrably false, as shown by the infinite divisibility of sqrt(2)x^2 by 2. Since the central negative result on non-ascent of quasi-atomicity to polynomial extensions depends on this UFD step, the proof as written does not establish the theorem. The underlying claim may still be true, and the example may be repairable via Roitman-type arguments, but a substantial revision is required. I therefore keep the verdict conditional: accept only after the UFD assertion is corrected or replaced and the Theorem 3.2 boundedness gap is closed.","tokens_in":21110,"tokens_out":8111,"duration_ms":84149,"concrete_test":"Test the atomicity of h = sqrt(2)x^2 in R[x,y] with R as in (4.1). Exhibit the chain h = 2^n * (sqrt(2)/2^n x^2) for every n >= 1; since 2 and each cofactor are nonunits, h has no factorization into finitely many irreducibles. This directly falsifies the UFD premise used at equations (4.3) and (4.4). Then either identify the intended ambient UFD and repair the passage from irreducibles in R[y] to that UFD, or supply an alternative proof of Theorem 4.1 that does not invoke unique factorization in R[x,y].","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem 4.1 (Section 4), the authors state: \"Since R[x,y] is an atomic domain (indeed, R[x,y] is a UFD).\" Here R is the ring defined in (4.1), R = Z + Zx + x^2 R[x], a subring of the real polynomial ring. This assertion is false. The element h = sqrt(2)x^2 lies in R and in R[x,y], and it admits the factorization h = 2 * (sqrt(2)/2 x^2). Both factors are nonunits: 2 is not invertible because 1/2 is not in R, and sqrt(2)/2 x^2 has zero constant term and is not a unit. The same factorization can be iterated, giving h = 2^n * (sqrt(2)/2^n x^2) for every n >= 1. Thus h has arbitrarily long factorizations into nonunits and cannot be a finite product of irreducibles. Hence R[x,y] is not atomic, and certainly not a UFD. The UFD premise is load-bearing: it is used to pass from irreducibles in R[y] to the factorization ay + b = r * product a_{i,j}(x,y) appearing in equation (4.4), and then to isolate a single y-degree-one irreducible factor. Without a genuine UFD (or an alternative argument) this step collapses. If the authors intended R[x,y] to denote the real polynomial ring, then the notation is ambiguous and the conclusion about the polynomial extension of the subring R does not follow. Either way, Theorem 4.1 is not proved by the written argument.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the ascent of almost atomicity and quasi-atomicity to polynomial and monoid semidomains. Theorem 3.2 states that if S is almost atomic (resp. quasi-atomic) and M is a reduced monoid satisfying the ACCP, with an MCD condition on coefficient sets, then S[M] is almost atomic (resp. quasi-atomic). Section 4 claims a counterexample to the ascent of quasi-atomicity to polynomial extensions, using R = Z + Zx + x^2 R[x] (with real coefficients) and proving that R[y] is not quasi-atomic. Section 5 constructs, for each prime p, a rank-one torsion-free atomic Puiseux monoid M_p such that F_p[M_p] is not quasi-atomic, yielding non-ascent of almost and quasi-atomicity to monoid domains over finite fields. The paper also provides examples of almost atomic non-atomic domains and quasi-atomic non-almost-atomic domains.","tokens_in":21481,"tokens_out":15760,"duration_ms":160320,"significance":"If the main results were established, they would be a meaningful contribution: Theorem 5.6 would strengthen the known non-ascent of atomicity to monoid domains, and Theorem 4.1 would settle negatively the ascent of quasi-atomicity to polynomial extensions. The explicit constructions and the use of external irreducibility criteria (Lidl-Niederreiter, Blake et al.) are appropriate and nontrivial. However, several load-bearing steps in the proofs are not justified as written, so the paper does not currently establish its central claims.","major_comments":[{"comment":"The sentence 'The fact that M satisfies the ACCP guarantees that the set L is bounded' is not a consequence of the ACCP. The ACCP stabilizes ascending chains of principal ideals; it does not bound the number of factors in a factorization of a fixed element in an arbitrary commutative monoid. Since the proof chooses ℓ as the largest element of L, this assertion is load-bearing and the argument collapses without it. The theorem may be salvageable with an additional bounded-factorization hypothesis, but as stated the proof is incomplete.","section":"Section 3, proof of Theorem 3.2"},{"comment":"The assertion 'R[x,y] is an atomic domain (indeed, R[x,y] is a UFD)' is false for the ring R defined in (4.1). The element h = sqrt(2)x^2 belongs to R and satisfies h = 2 * (sqrt(2)/2 x^2), where both 2 and (sqrt(2)/2)x^2 are nonunits in R; iterating gives h = 2^n * (sqrt(2)/2^n)x^2 for every n, so R (and hence R[x,y]) fails the ACCP and is not atomic. This invalidates the factorization-rearrangement used to pass from (4.3) to (4.4) and to isolate a y-degree-one irreducible factor. The notation in (4.1) is also ambiguous because the right-hand side uses the same symbol for the real numbers and for the ring being defined.","section":"Section 4, proof of Theorem 4.1"},{"comment":"The claim that 'the equality B(x) = G(x^{(pd)^{-n}}) guarantees that B(x) is not a unit' is not justified. If t = 1 and g(x) is a nonzero constant, then G(x) is a unit and B(x) is a unit; the displayed equality alone does not exclude this case. The contradiction to the irreducibility of a_j requires B(x) to be a nonunit, so this is a load-bearing gap. The authors need to prove that for the chosen index j either t ≥ 2 or g(x) is nonconstant.","section":"Section 5, final paragraph of the proof of Theorem 5.6"}],"minor_comments":[{"comment":"Please disambiguate the use of R for the ring under construction and blackboard R for the real numbers in the defining expression; the current notation is circular if read literally.","section":"Section 4, equation (4.1)"},{"comment":"There are numerous typographical and extraction artifacts ('A TOMICITY', 'elem ent', 'semi domain', 'must exits'); a careful proofreading pass is needed.","section":"Throughout"},{"comment":"The statement that the set of lengths of decompositions into nonconstant polynomials is bounded should be justified explicitly by a degree argument; the bound is clear in the ordinary polynomial ring but should be stated.","section":"Section 3, proof of Lemma 3.4(1)"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses natural and timely questions, and the constructions in Section 5 are ambitious. My recommendation reflects that the central proofs are not currently valid. I would encourage the authors to repair or restructure the affected arguments; if Theorem 4.1 cannot be repaired, that claim should be withdrawn or made conditional."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"For my money the paper is worth engaging with, but not as written. The two headline negative results—quasi-atomicity not ascending to polynomial extensions (Theorem 4.1) and neither almost nor quasi-atomicity ascending to monoid domains over finite fields (Theorem 5.6)—are both plausible and interesting, but each has a proof gap that the authors need to close. The ascent theorem for monoid semidomains (Theorem 3.2) is also under-supported.\n\nWhat is genuinely useful: the paper identifies the right questions and provides a coherent strategy. The construction in Section 3 of almost/quasi-atomic domains is nice and correct as far as I checked. The finite-field monoid construction in Section 5 is a natural extension of Coykendall–Gotti, and the reduction to irreducibility of binomials/trinomials via Lidl–Niederreiter is clever. If the gaps close, the significance is real: these would settle natural analogues of Roitman's theorem.\n\nNow the soft spots, in order of severity.\n\nFirst, Theorem 4.1. The proof asserts that R[x,y] is a UFD, where R = Z + Zx + x^2 R[x] (here R means the real polynomial ring). That is false. The element h = sqrt(2)x^2 lies in R and equals 2 * (sqrt(2)/2 x^2); both factors are nonunits, and the same move repeats, so h is not atomic and R fails ACCP. Hence R[x,y] is not atomic, let alone a UFD. The UFD claim is load-bearing: it is used to obtain equation (4.4) and isolate a degree-one factor. Without it the proof does not go through. Maybe the authors meant the real polynomial ring in x,y; then the factor r in (4.4) would be an arbitrary real, not necessarily an element of R, and subsequent steps still need repair.\n\nSecond, Theorem 5.6. The claim that B(x) is not a unit is not justified when g(x) is a constant and t = 1; in that case B(x) is a unit and the factorization is trivial. The argument needs to handle this case separately.\n\nThird, Theorem 3.2's proof assumes that ACCP of M bounds the length of decompositions of a polynomial into nonconstant factors. That requires a short argument and may fail in general commutative monoids; in the rank-one linearly ordered setting it might be true, but it is not written.\n\nThese are not manufactured quibbles; the false UFD assertion in particular is a serious flaw in the current write-up. The paper should not be accepted as is, but it deserves a serious referee and a revision. If the authors fix these gaps, I would expect the results to hold and the paper to be a solid contribution.\n\nWho is this for? People working in factorization theory and monoid domains. A reading group could have productive discussion on the constructions and on what would be needed to repair them. But I would not cite the theorem statements as established until the proofs are corrected.","headline":"Plausible and worth exploring, but as written the two headline negative results are not proven: Theorem 4.1 rests on a false UFD assertion, and Theorem 5.6 has an unhandled unit case.","tokens_in":21971,"tokens_out":11188,"would_cite":false,"duration_ms":106445,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F15","13A05","20M25","06F05","11Y05","13G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that quasi-atomicity does not ascend to polynomial extensions and that neither almost atomicity nor quasi-atomicity ascends to monoid domains over finite fields.","keywords":["almost atomicity","quasi-atomicity","atomic domain","monoid semidomain","monoid domain","Puiseux monoid","polynomial extension","finite field"],"falsifier":"Exhibit a reduced monoid $M$ satisfying the ACCP and a polynomial $f \\in S[M]$ whose decompositions into nonconstant polynomials over $S$ have unbounded length; such an example would break the bounded-length step in Theorem 3.2. For the finite-field construction, a direct check in the $p=2$ case—deciding whether $x^2+x+1$ in $\\mathbb{F}_2[M_{2,3}]$ has a nonzero multiple that factors into irreducibles—would settle the corresponding claim of Theorem 5.6.","tokens_in":20930,"feed_emoji":"🔢","tokens_out":11161,"duration_ms":100782,"temperature":0.7,"pith_summary":"Two weakenings of atomicity—almost atomicity and quasi-atomicity—say that every element becomes factorable after multiplication by an atomic element, or by any element at all. This paper asks whether these properties are preserved when one passes from a semidomain to its polynomial extension or to a monoid semidomain. It establishes a positive result under extra structure: if the exponent monoid satisfies the ascending chain condition on principal ideals and the coefficient sets of indecomposable polynomials have maximal common divisors, then both weak forms of atomicity ascend. The paper then shows that the extra conditions are necessary: the quasi-atomic ring $\\mathbb{Z}+\\mathbb{Z}x+x^2\\mathbb{R}[x]$ has a polynomial extension that is not quasi-atomic, and for every prime $p$ there is a rank-one torsion-free atomic monoid $M_p$ for which $\\mathbb{F}_p[M_p]$ is not quasi-atomic. Thus two natural positive questions about weaker atomicity are answered in the negative, extending the known failure of atomicity itself.","feed_headline":"Quasi-atomicity does not ascend to polynomial rings","feed_subtitle":"A concrete ring shows quasi-atomicity breaks when a variable is added; finite-field monoid algebras fail too.","key_machinery":"The load-bearing mechanism is a substitution trick for monoid domains. Given a Puiseux monoid $M$ containing $\\mathbb{N}_0[1/p]$ and a low-degree polynomial $f_d(x)$ over $\\mathbb{F}_p$ whose dilation $f_d(x^{d^n})$ is irreducible for every $n$, a purported factorization of $f_d(x)$ in $\\mathbb{F}_p[M]$ is evaluated at $x \\mapsto x^{1/(pd)^n}$ for large $n$. This sends all exponents into $\\mathbb{N}_0$, so the factorization lands in the UFD $\\mathbb{F}_p[x]$; irreducibility of $f_d(x^{d^n})$ forces one substituted atom to be divisible by it, and pulling the divisibility back through the substitution shows the original atom was reducible. A secondary mechanism is the MCD/indecomposable-length argument in Theorem 3.2: an ACCP exponent monoid is used to select a longest factorization into nonconstant polynomials, and extraction of coefficient maximal common divisors turns indecomposables into atoms.","core_discovery":"The central claim is that the two weakest commonly studied factorization properties do not transfer automatically. Under the hypotheses of Theorem 3.2—a reduced monoid $M$ satisfying the ACCP and a coefficient semidomain whose indecomposable-polynomial coefficient sets admit maximal common divisors—$S[M]$ is almost atomic when $S$ is, and quasi-atomic when $S$ is. The paper proves this by factoring an arbitrary polynomial into indecomposables of maximal length, pulling out coefficient MCDs, and using divisor-closedness of the monomial submonoid. It then shows sharpness: the ring $R = \\mathbb{Z}+\\mathbb{Z}x+x^2\\mathbb{R}[x]$ is quasi-atomic, yet $R[y]$ is not; the proof forces the linear polynomial $\\alpha x^2 y + \\beta x^2$ to have no quasi-atomic multiple when $\\alpha,\\beta$ are algebraically independent irrationals. For finite fields, Theorem 5.6 constructs, for each prime $p$, an atomic Puiseux monoid $M_p$ with $\\mathbb{F}_p[M_p]$ not quasi-atomic, via a substitution argument that transfers any purported factorization into the UFD $\\mathbb{F}_p[x]$ and contradicts irreducibility of a chosen binomial or trinomial.","pith_inferences":["An extension not pursued here is to test whether the substitution method works over coefficient fields other than $\\mathbb{F}_p$, for instance over $\\mathbb{Q}$ or number fields, provided one can find polynomials whose $d$-th power dilations are irreducible.","The boundedness step in Theorem 3.2 uses a claim about ACCP monoids that is not generally valid; if a counterexample exists, the positive ascent theorem would hold under a stronger hypothesis such as bounded factorization rather than ACCP.","The non-ascent example suggests a recipe for producing more quasi-atomic rings whose polynomial extensions fail: concentrate the non-atomicity in monomials of degree at least 2 and then form a linear polynomial whose coefficients are algebraically independent monomials.","A concrete computational test of the $p=2$ case would decide whether $x^2+x+1$ in $\\mathbb{F}_2[M_{2,3}]$ admits a quasi-atomic multiple; a positive answer would force a revision of the finite-field construction."],"forward_implications":["If the hypotheses of Theorem 3.2 hold, almost atomicity and quasi-atomicity survive the passage to monoid semidomains, so the positive ascent results cover polynomial extensions with an MCD condition.","Quasi-atomicity is not preserved by adjoining a variable: the quasi-atomic ring $\\mathbb{Z}+\\mathbb{Z}x+x^2\\mathbb{R}[x]$ has a non-quasi-atomic polynomial extension.","For every prime $p$, there are rank-one torsion-free atomic monoids whose monoid algebras over $\\mathbb{F}_p$ fail even quasi-atomicity, so neither weak ascent property holds for monoid domains over finite fields.","The examples separate the hierarchy: $\\mathbb{Z}[x]+\\mathbb{Q}[x]x^2$ is almost atomic but not atomic, while $\\mathbb{Z}[x]+\\mathbb{R}[x]x^2$ is quasi-atomic but not almost atomic.","Whether almost atomicity ascends to ordinary polynomial extensions remains open, as the paper explicitly asks."],"supporting_citations":[{"why":"Introduces the two weakenings of atomicity whose ascent behavior is studied here.","marker":"[4]"},{"why":"Supplies the original monoid-domain non-ascent construction and the $\\mathbb{F}_2$ example that Section 5 generalizes.","marker":"[10]"},{"why":"Proves the polynomial-semidomain ascent under maximal-common-divisor conditions that Theorem 3.2 extends to monoid semidomains.","marker":"[21]"},{"why":"Provides the prototypical almost-atomic and quasi-atomic non-atomic domains that anchor Sections 3 and 4.","marker":"[27]"},{"why":"Gives the binomial irreducibility criterion used in the $p \\equiv 1$ mod 4 case of the finite-field construction.","marker":"[28]"},{"why":"Gives the trinomial irreducibility criterion used in the $p \\equiv 3$ mod 4 case.","marker":"[3]"},{"why":"Proves that atomicity does not ascend to polynomial extensions and introduces the indecomposable-polynomial technique reused here.","marker":"[32]"},{"why":"Provides a recent rank-one atomic monoid whose monoid domain is not atomic, used as a comparative baseline.","marker":"[22]"}],"fun_headline_variants":["Quasi-atomicity does not ascend to polynomials","No ascent for almost-atomicity in monoid domains","Weak atomicity fails under polynomial extension","Counterexample: quasi-atomicity breaks in polynomial rings","Ascent fails for both weak atomicity notions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The positive ascent theorem rests on the claim that in a reduced monoid satisfying the ACCP, any fixed polynomial in $S[M]$ has a bounded number of nonconstant factors; this claim is not proved and does not follow from ACCP for arbitrary commutative monoids.","fun_headline_variants_meta":{"raw":{"variants":["Quasi-atomicity does not ascend to polynomials","No ascent for almost-atomicity in monoid domains","Weak atomicity fails under polynomial extension","Counterexample: quasi-atomicity breaks in polynomial rings","Ascent fails for both weak atomicity notions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000317,"raw_usage":{"total_tokens":1847,"prompt_tokens":1056,"completion_tokens":791,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":672,"completion_tokens_details":{"reasoning_tokens":717}},"tokens_in":672,"tokens_out":791,"duration_ms":7614,"temperature":1.0,"reasoning_tokens":717,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:25:43.379501+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a reduced monoid $M$ satisfying the ACCP and a polynomial $f \\in S[M]$ whose decompositions into nonconstant polynomials over $S$ have unbounded length; such an example would break the bounded-length step in Theorem 3.2. For the finite-field construction, a direct check in the $p=2$ case—deciding whether $x^2+x+1$ in $\\mathbb{F}_2[M_{2,3}]$ has a nonzero multiple that factors into irreducibles—would settle the corresponding claim of Theorem 5.6.","supporting_citations":[{"cited_title":"Boynton and J","cited_arxiv_id":null,"evidence_quote":"Introduces the two weakenings of atomicity whose ascent behavior is studied here."},{"cited_title":"Coykendall and F","cited_arxiv_id":null,"evidence_quote":"Supplies the original monoid-domain non-ascent construction and the $\\mathbb{F}_2$ example that Section 5 generalizes."},{"cited_title":"Gotti and H","cited_arxiv_id":null,"evidence_quote":"Proves the polynomial-semidomain ascent under maximal-common-divisor conditions that Theorem 3.2 extends to monoid semidomains."},{"cited_title":"Lebowitz-Lockard, On domains with properties weaker than atomicity , Comm","cited_arxiv_id":null,"evidence_quote":"Provides the prototypical almost-atomic and quasi-atomic non-atomic domains that anchor Sections 3 and 4."},{"cited_title":"Lidl and H","cited_arxiv_id":null,"evidence_quote":"Gives the binomial irreducibility criterion used in the $p \\equiv 1$ mod 4 case of the finite-field construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the trinomial irreducibility criterion used in the $p \\equiv 3$ mod 4 case."},{"cited_title":"Roitman, Polynomial extensions of atomic domains , J","cited_arxiv_id":null,"evidence_quote":"Proves that atomicity does not ascend to polynomial extensions and introduces the indecomposable-polynomial technique reused here."},{"cited_title":"Gotti and H","cited_arxiv_id":null,"evidence_quote":"Provides a recent rank-one atomic monoid whose monoid domain is not atomic, used as a comparative baseline."}],"review_version":1}