{"id":"41fe63dc-7764-4aca-989b-2dc29012ab32","arxiv_id":"2506.09051","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For certain monomial ideals, the paper proposes explicit values and bounds for v-numbers of integral closure filtrations and shows they can be smaller than v-numbers of ordinary powers.","lead":"This paper computes v-numbers, a commutative algebra invariant, for powers of monomial ideals and for their integral closures. It reports formulas relating these v-numbers to Castelnuovo-Mumford regularity, plus examples that give a negative answer to an open question.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"False equality (P4) invalidates the proof of Theorem 4.5 as written; small examples are needed to separate a repairable proof gap from a false theorem.","rationale":"The reader is right that (P4) is false and that Section 4 is written as though \\overline{I^n} is a pure-power ideal. I part company only on the inference that the headline theorem is thereby false. Inspection of the inequalities in Theorem 4.5 shows they are driven by Lemma 2.2, which is exactly the correct membership test for \\overline{<x_i^{n a_i}>}, so the proof strategy is likely repairable by replacing (P4) with the correct identity. Independent small examples agree with the stated formulas. Therefore the correct disposition is not a permanent rejection but a conditional one: the authors must remove the false identity, correct or delete Proposition 4.4, and re-verify the affected proofs. If a concrete computation produced a discrepancy, that would move the verdict to REJECT.","tokens_in":30037,"tokens_out":28594,"duration_ms":314438,"concrete_test":"Compute in Macaulay2 (or by hand via Lemma 2.2) the actual integral closures and their invariants for I=(x^2,y^2) and I=(x^2,y^3) in K[x,y]: determine \\overline{I^n} as the Newton-polyhedron ideal {x^u y^v : u/(n a_1)+v/(n a_2) >= 1}, then compute v(\\overline{I^n}) and reg(S/\\overline{I^n}) for n=1,2. Compare with Theorem 4.5 and with n alpha - 1. Also test Proposition 4.4 against I=(x^2,y^2,z^2), f=xyz. If the headline values match, the central claim survives and the paper needs a corrected derivation; if any value differs, the central theorem is false.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"(P4) asserts that for an irreducible monomial ideal I=<x_{i_1}^{a_1},...,x_{i_k}^{a_k}>, \\overline{I^n}=<x_{i_1}^{n a_1},...,x_{i_k}^{n a_k}> for all n. This is false: for I=(x^2,y^2) in K[x,y], Lemma 2.2 gives xy in \\overline I, while xy not in (x^2,y^2); in fact \\overline I=(x,y)^2. The correct identity is \\overline{I^n}=\\overline{<x_i^{n a_i}>}. The proofs of Theorem 4.5 and its corollaries repeatedly invoke the false identity in place of this correct one, so the displayed computations do not, as written, establish the stated v-numbers. This is the main load-bearing gap because Theorem 4.5(4) and the abstract's headline formula rest on it. The situation is not a simple counterexample to the headline: the applications of Lemma 2.2 in Theorem 4.5 use the Newton-polyhedron criterion that characterizes the correct integral closure, and for concrete equigenerated examples the formula v=n alpha - 1 appears to hold. The false statement Proposition 4.4 (e.g. I=(x^2,y^2,z^2), f=xyz gives (I:f)=P and deg f=v(I)=3 but f not equal to g/x_3 for any g in G(I)) is a separate conflation of ideal membership with integral closure, but it is not used in the proof of the equigenerated headline.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the v-numbers of powers of monomial ideals and of their integral closures. Section 3 gives an alternative proof of the known formula for v(I^n) when I is a complete intersection monomial ideal. Section 4 investigates v-numbers of integral closure filtrations of irreducible monomial ideals, claiming exact formulas relating v(\\overline{I^n}) and reg(S/\\overline{I^n}), including the headline statement reg(S/\\overline{I^n}) = v(\\overline{I^n}) = n\\alpha(I)-1 for equigenerated irreducible monomial ideals. The paper also derives upper bounds for v-numbers of integral closures of complete intersection ideals, with applications to weighted oriented graphs and to a negative answer to a question of Saha and Sengupta.","tokens_in":30373,"tokens_out":15410,"duration_ms":163258,"significance":"If the results are correct, they would substantially extend the current knowledge on v-numbers of filtrations, where only eventual linearity was known, by providing explicit linear formulas for integral closure filtrations of irreducible and complete intersection monomial ideals. Section 3 appears technically sound and offers a useful alternative proof. However, the main new results in Section 4 rest on a false identification of \\overline{I^n} with the pure-power ideal generated by the nth powers of the generators of an irreducible monomial ideal. For the example I=(x^2,y^2), the claimed equality fails, and the proofs as written compute v-numbers of the wrong ideal. Small examples suggest the equigenerated formula itself may be true, but the manuscript does not currently establish it.","major_comments":[{"comment":"The paper asserts that for I=<u_1,...,u_r>, from NP(I^m)=NP(J_m) with J_m=<u_i^m> it follows that \\overline{I^m}=J_m, and consequently (P4) states that for an irreducible monomial ideal I=<x_i^{a_i}>, \\overline{I^n}=<x_i^{n a_i}>. This is false. For I=(x^2,y^2) in K[x,y], the Newton polyhedron criterion of Lemma 2.2 gives xy in \\overline I, but xy is not in <x^2,y^2>; in fact \\overline I=(x,y)^2. The correct statement is \\overline{I^m}=\\overline{<u_i^m>}. Since the false equality is invoked at the start of the proof of Theorem 4.5 and in Corollaries 4.6, 4.9, 4.14, and 4.15, the displayed computations in those proofs do not establish the stated v-numbers.","section":"Section 2, Remark 2.1 and property (P4)"},{"comment":"The proof begins with \"By Remark 2.1, \\overline{I^n}=<x_i^{n a_i}>\" and then uses Lemma 2.2 to test membership in \\overline{I^n}. For I=(x^2,y^2) and n=1, the upper-bound monomial g=x in the proof satisfies (<x^2,y^2>:x)=<x,y^2> \\neq P, so the argument fails for the ideal as written; it only succeeds for the integral closure (x,y)^2. Thus the proof as written computes the v-number of the wrong ideal. The formulas in Theorem 4.5(1)-(4) may be salvageable by replacing the equality with \\overline{I^n}=\\overline{<x_i^{n a_i}>} and reinterpreting every Lemma 2.2 membership claim accordingly, but this is a substantive correction that must be carried out throughout Section 4.","section":"Theorem 4.5"},{"comment":"Proposition 4.4 is false as stated. For I=<x^2,y^2,z^2> in K[x,y,z], the monomial f=xyz satisfies (I:f)=(x,y,z)=P and v(I)=3, but no g in G(I)={x^2,y^2,z^2} satisfies f=g/x_3 (i.e., f=g/z). The proof's assertion that the contradiction forces c_j=b_j for j<r fails because the monomial g dividing f x_r need not share all but one exponent with f; for f=xyz and g=z^2, c_1=c_2=0. This proposition should be corrected or removed; it is not cited in the proof of Theorem 4.5, but it is a false result in the paper.","section":"Proposition 4.4"}],"minor_comments":[{"comment":"The equality of Newton polyhedra NP(I^m)=NP(J_m) is correct, but the conclusion should be \\overline{I^m}=\\overline{J_m}; the current wording claims a false equality of ideals.","section":"Section 2, Remark 2.1"},{"comment":"The notation for integral closures is used inconsistently; for example, in the proof of Theorem 4.5 the statement \"I^n=P^{na}\" should be \"\\overline{I^n}=P^{na}\", since the equality of ideals fails while the equality of integral closures holds.","section":"Throughout Section 4"},{"comment":"There are several typos, including \"filtartion\" in the Introduction, \"inetersection\" in Corollary 1.5, and \"Propossition\" in the proof of Theorem 4.14(2).","section":"Typographical issues"},{"comment":"References [14] and [19] are the same book (Herzog and Hibi, Monomial Ideals) and should be unified.","section":"References"},{"comment":"The line containing \"P^n \\subset J^n\" appears to be a typo; the context suggests it should be \"P^n \\cap J^n\".","section":"Corollary 4.6"}],"recommendation":"major_revision","confidential_remarks":"The central results of the paper may be true, and the equigenerated formula is supported by computational examples, but the current version contains a systematic false identification of integral closures with pure-power ideals. This is a serious correctness issue affecting the proofs of the main theorems. Because the error is repairable in principle and Section 3 appears sound, I recommend major revision rather than rejection. The authors must correct Remark 2.1 and (P4), rework the proofs in Section 4 with the correct integral closure, and address the false Proposition 4.4. The paper's fit for the journal is acceptable if these issues are properly fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Best quick take: the paper has one genuinely useful section and one broken load-bearing claim. Section 3 gives a clean alternative proof of the known formula v(I^n)=nα(I)+v(I)-α(I) for complete intersection monomial ideals, with an explicit witness f. That part reads carefully and I think it is correct. Section 4 is the new work: formulas for v-numbers of integral closure filtrations, comparisons with ordinary powers, and the weighted graph example answering Saha–Sengupta's question. The direction is good and worth pursuing.\n\nThe problem is (P4). It says that for I=<x_i^{a_i}>, \\overline{I^n}=<x_i^{n a_i}>. That is false: I=(x^2,y^2) gives \\overline I=(x,y)^2, not (x^2,y^2). What Remark 2.1 actually gives is \\overline{I^n}=\\overline{<x_i^{n a_i}>}. The proofs of Theorem 4.5 repeatedly invoke the false version, so the displayed computations do not establish the theorem as written. This is not a fatal counterexample to the headline—for equigenerated I the correct integral closure is P^{na}, so the formula v=nα−1 is true and easily proved—but the manuscript needs to be rewritten around Lemma 2.2's Newton polyhedron criterion. Proposition 4.4 is also wrong as stated; I=(x^2,y^2,z^2), f=xyz gives (I:f)=P and minimal degree, yet no generator has f as g/x_3. It is not used in the equigenerated headline, but it should be corrected or removed.\n\nThe citation pattern looks fine: the alternative proof in Section 3 cites the right prior work, and the self-citation to [1] is legitimate. I would send this to a referee. The core Section 3 result is solid, the questions in Section 4 are natural, and the mistakes look repairable rather than terminal. The referee should be asked to check every use of (P4) and to require a redone Section 4 proof. I would not cite the paper until that happens.","headline":"Section 3 is a sound alternative proof, but Section 4 rests on a false equality for integral closures of pure-power ideals; the paper needs substantive revision before it can be trusted.","tokens_in":30896,"tokens_out":6064,"would_cite":false,"duration_ms":72706,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A15","13A18","13A02"],"pacs":[],"model":"deepseek-v4-flash","headline":"For an equigenerated irreducible monomial ideal $I$, the integral-closure filtration satisfies $\\operatorname{v}(\\overline{I^n}) = \\operatorname{reg}(S/\\overline{I^n}) = n\\alpha(I)-1$ for every $n$, with ceiling formulas in mixed-exponent…","keywords":["v-number","integral closure","monomial ideals","Castelnuovo-Mumford regularity","complete intersection ideal","irreducible monomial ideal","Newton polyhedron","powers of ideals"],"falsifier":"Compute $\\operatorname{v}(\\overline{I^n})$ for $I=(x^2,y^3,z^5)$ directly from the Newton polyhedron of the exponent set $\\{(2,0,0),(0,3,0),(0,0,5)\\}$ for $n=2$ and $n=3$, enumerating monomials by degree and testing whether $(\\overline{I^n}:f)=(x,y,z)$; if the minimum degree differs from the value predicted by Theorem 4.9, the paper's working identification of $\\overline{I^n}$ with the pure-power ideal is consequential and the theorem fails.","tokens_in":29803,"feed_emoji":"📐","tokens_out":23650,"duration_ms":248681,"temperature":0.7,"pith_summary":"The paper studies the v-number, a degree-minimizing invariant attached to associated primes, for ordinary powers and for integral closures of powers of monomial ideals. Its central result is that for an equigenerated irreducible monomial ideal $I=\\langle x_1^a,\\ldots,x_k^a\\rangle$, the integral closure filtration satisfies $\\operatorname{v}(\\overline{I^n}) = \\operatorname{reg}(S/\\overline{I^n}) = na-1$ for all $n\\ge1$, so both invariants are linear with slope $\\alpha(I)$. For irreducible monomial ideals with mixed exponents the paper gives two-sided bounds and, in the height-three case, explicit ceiling formulas for $\\operatorname{v}(\\overline{I^n})$. For complete intersection monomial ideals it proves $\\operatorname{v}(\\overline{I^n})\\le \\operatorname{v}(I^n)$, identifies strict-inequality cases, and constructs examples where the gap between the two filtrations is any prescribed integer and even grows arbitrarily large. The interest is that a coding-theoretic invariant and a syzygy-complexity invariant become exactly computable, and equal, on entire filtrations rather than on single ideals.","feed_headline":"v-number equals regularity for equal-degree irreducible ideals","feed_subtitle":"For their integral closure filtration, both invariants equal n·α(I) − 1 at every power.","key_machinery":"The carrying object is the Newton polyhedron membership criterion (Lemma 2.2): for an ideal generated by pure powers of variables, a monomial $x^a$ lies in the integral closure iff $\\sum_i a_i/b_i \\ge 1$. The paper couples this with the asserted description of the integral closure of powers of an irreducible monomial ideal, $\\overline{I^n} = \\langle x_1^{na_1},\\ldots,x_k^{na_k}\\rangle$, and with Lemma 4.3, a set of ceiling-function inequalities, to convert v-number computations into inequalities on exponent vectors. In the equigenerated case the machinery collapses: $\\overline{I^n}$ is the $na$-th power of the maximal ideal, whose v-number and regularity are both $na-1$. For complete intersections the additional machinery is the primary decomposition into pure-power irreducible components, via [19] and [17, Theorem 4.1], together with explicit colon computations producing monomials $f$ with prescribed colon ideal $(\\overline{I^n}:f)=P$.","core_discovery":"The paper's central claim is that the v-number of an integral closure filtration of a monomial ideal can be read off from the initial degree $\\alpha(I)$ and maximal generator degree $\\delta(I)$ whenever a Newton-polyhedron description is available. In the cleanest case, Theorem 4.5, an equigenerated irreducible monomial ideal $I=\\langle x_1^a,\\ldots,x_k^a\\rangle$ has $\\overline{I^n}$ equal to the $na$-th power of the maximal ideal, and therefore $\\operatorname{v}(\\overline{I^n}) = \\operatorname{reg}(S/\\overline{I^n}) = na-1$ for every $n\\ge1$. For non-equigenerated irreducible ideals the paper proves two-sided bounds and exact ceiling formulas, including $\\operatorname{v}(\\overline{I^n}) = n\\alpha(I) + \\lceil\\delta(I)/\\alpha(I)\\rceil - 2$ when the exponents take exactly two values, and it computes height-three cases through explicit monomial witnesses. For complete intersection monomial ideals the paper establishes $\\operatorname{v}(\\overline{I^n})\\le \\operatorname{v}(I^n)$, with equality in the squarefree case, a strict drop when two generators are non-squarefree, and arbitrarily large prescribed gaps between the v-numbers of the two filtrations.","pith_inferences":["Because the asserted identification of $\\overline{I^n}$ with the pure-power ideal is false in general, the non-equigenerated ceiling formulas should be checked against true Newton-polyhedron generators; the equigenerated equality is on firmer ground because $\\overline{I^n}$ is then genuinely a power of the maximal ideal.","A testable extension is to compute $\\operatorname{v}(\\overline{I^n})$ for height-three ideals with pairwise coprime exponents using the true integral closure; the paper's monomial witnesses $f_m$ may still be extremal, in which case the formulas survive despite the flawed identification.","The equality $\\operatorname{v}=\\operatorname{reg}$ on equigenerated integral-closure filtrations suggests a broader pattern: any filtration whose integral closures are powers of a normal monomial ideal may have v-number equal to regularity whenever the quotient has a linear resolution; symbolic power filtrations of squarefree monomial ideals would be a natural family to test."],"forward_implications":["For every equigenerated irreducible monomial ideal, $\\operatorname{v}(\\overline{I^n}) = \\operatorname{reg}(S/\\overline{I^n}) = n\\alpha(I)-1$ for all $n$, so the v-number is linear from the first power and is determined by the initial degree alone.","For irreducible monomial ideals whose exponents take two values, $\\operatorname{v}(\\overline{I^n}) = n\\alpha(I)+\\lceil\\delta(I)/\\alpha(I)\\rceil-2$, making the deviation from $n\\alpha(I)-1$ a constant depending only on the ideal.","For complete intersection monomial ideals, $\\operatorname{v}(\\overline{I^n})\\le \\operatorname{v}(I^n)$ for all $n$, and the inequality is strict when the minimal-degree generator and another generator are both non-squarefree.","For any integer $a\\ge1$ there is a height-two equigenerated complete intersection monomial ideal with $\\operatorname{reg}(S/\\overline{I^n}) - \\operatorname{v}(\\overline{I^n}) = a-1$ for all $n$, so the regularity-v-number gap can be any prescribed constant.","The difference $\\operatorname{v}(I^n)-\\operatorname{v}(\\overline{I^n})$ can equal any prescribed nonnegative integer $q$, so ordinary powers and integral-closure filtrations are genuinely different from the v-number viewpoint."],"supporting_citations":[{"why":"Identifies $\\operatorname{NP}(I^m)$ with $\\operatorname{NP}(\\langle u_1^m,\\dots,u_r^m\\rangle)$, the step used to replace integral closures of pure-power ideals by pure-power ideals.","marker":"[25, Lemma 2.5]"},{"why":"Gives the criterion $\\sum a_i/b_i \\ge 1$ for a monomial to lie in the integral closure of an ideal generated by pure powers of variables.","marker":"[9, Proposition 3.3]"},{"why":"Supplies minimal generating sets, intersection and colon formulas, and the unique decomposition of monomial ideals into ideals generated by pure powers of variables.","marker":"[19]"},{"why":"Describes $\\overline{I^n}$ as an intersection of powers of irreducible components for Simis ideals, used for complete intersections and weighted oriented edge ideals.","marker":"[17, Theorem 4.1]"},{"why":"Gives the prior complete-intersection formula $\\operatorname{v}(I)=\\sum \\deg u_i - r$ that Section 3 reproves and extends to all powers.","marker":"[28, Proposition 3.10]"},{"why":"Supplies the limit of $\\alpha(\\overline{I^n})/n$ and the regularity bounds for integral closures used in Lemma 4.1 and Theorem 4.5(4).","marker":"[16]"},{"why":"Bounds $\\operatorname{reg}(\\overline{I^n})$ between $\\delta(I)n$ and $\\delta(I)n+\\dim S/I$ and provides the example used in Theorem 4.14(4).","marker":"[18]"},{"why":"Computes $\\operatorname{reg}(S/P^{na})=na-1$ for powers of the maximal ideal, finalizing the equigenerated equality.","marker":"[2, Lemma 4.4]"}],"fun_headline_variants":["For equigenerated irreducible monomial ideals, v-number = regularity at each power","Complete intersections: arbitrary gaps between v-numbers of powers and closures","Explicit v-number formula for two-exponent irreducible monomial ideals","Integral closure filtration: v-number matches regularity for equigenerated ideals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The computation rests on the assertion that the integral closure of the $n$-th power of a pure-power monomial ideal is generated by the $n$-th powers of its generators, an assertion that fails already for $(x^2,y^2)$, whose integral closure contains $xy$.","fun_headline_variants_meta":{"raw":{"variants":["For equigenerated irreducible monomial ideals, v-number = regularity at each power","Complete intersections: arbitrary gaps between v-numbers of powers and closures","Explicit v-number formula for two-exponent irreducible monomial ideals","Integral closure filtration: v-number matches regularity for equigenerated ideals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000642,"raw_usage":{"total_tokens":3017,"prompt_tokens":1071,"completion_tokens":1946,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":687,"completion_tokens_details":{"reasoning_tokens":1868}},"tokens_in":687,"tokens_out":1946,"duration_ms":21177,"temperature":1.0,"reasoning_tokens":1868,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:00:16.082910+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\operatorname{v}(\\overline{I^n})$ for $I=(x^2,y^3,z^5)$ directly from the Newton polyhedron of the exponent set $\\{(2,0,0),(0,3,0),(0,0,5)\\}$ for $n=2$ and $n=3$, enumerating monomials by degree and testing whether $(\\overline{I^n}:f)=(x,y,z)$; if the minimum degree differs from the value predicted by Theorem 4.9, the paper's working identification of $\\overline{I^n}$ with the pure-power ideal is consequential and the theorem fails.","supporting_citations":[{"cited_title":"Herzog and T","cited_arxiv_id":null,"evidence_quote":"Supplies minimal generating sets, intersection and colon formulas, and the unique decomposition of monomial ideals into ideals generated by pure powers of variables."},{"cited_title":"and Villarreal, R.H., Rees algebras of filtrations of cov- ering polyhedra and integral closure of powers of monomial ideals, Res Math Sci 9, 13 (2022)","cited_arxiv_id":null,"evidence_quote":"Supplies the limit of $\\alpha(\\overline{I^n})/n$ and the regularity bounds for integral closures used in Lemma 4.1 and Theorem 4.5(4)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Bounds $\\operatorname{reg}(\\overline{I^n})$ between $\\delta(I)n$ and $\\delta(I)n+\\dim S/I$ and provides the example used in Theorem 4.14(4)."}],"review_version":1}