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$\operatorname{v}$-numbers of integral closure filtrations of monomial ideals

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.09051 v2 pith:DSMMRXXV submitted 2025-06-10 math.AC

Vanmathi A , Parangama Sarkar This is my paper
classification math.AC MSC 13A1513A1813A02
keywords v-numberintegralclosuremonomialidealsCastelnuovo-MumfordregularitycompleteintersectionidealirreducibleNewtonpolyhedronpowersof
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 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.

What carries the argument

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 $(\overline{I^n}:f)=P$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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 $xy$.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 5 minor

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.

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 (3)
  1. [Section 2, Remark 2.1 and property (P4)] 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.
  2. [Theorem 4.5] 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.
  3. [Proposition 4.4] 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.
minor comments (5)
  1. [Section 2, Remark 2.1] 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.
  2. [Throughout Section 4] 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.
  3. [Typographical issues] There are several typos, including "filtartion" in the Introduction, "inetersection" in Corollary 1.5, and "Propossition" in the proof of Theorem 4.14(2).
  4. [References] References [14] and [19] are the same book (Herzog and Hibi, Monomial Ideals) and should be unified.
  5. [Corollary 4.6] The line containing "P^n \subset J^n" appears to be a typo; the context suggests it should be "P^n \cap J^n".

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the main v-number formulas are combinatorial consequences of stated premises, not of the conclusions; the sole self-citation is non-load-bearing.

full rationale

We find no circular reduction in the derivation chain. The central results (Theorems 3.3, 4.5, 4.9, and 4.14) are proved by direct combinatorial arguments using stated properties (P1)-(P5), Lemma 2.2, and external results; the target formulas for v-numbers and regularities are not assumed in the proofs. The one self-citation, [1, Theorem 3.2], appears only in Lemma 4.1 to identify the asymptotic slope of alpha(overline{I^n}) via v(overline{I^n}), and Lemma 4.1 is not used in the proofs of the headline theorems, so this citation is not load-bearing. There is, however, a serious correctness gap that is not a circularity: property (P4)/Remark 2.1 asserts that for an irreducible monomial ideal I=<x_i^{a_i}>, one has overline{I^n}=<x_i^{n a_i}> for all n, which is false in general (e.g., I=(x^2,y^2) has overline I=(x,y)^2, not (x^2,y^2)). This invalidates the proofs in Section 4 as written, because v-numbers of the true integral closure are computed through the pure-power ideal instead. That is an incorrect premise, not an instance of the conclusion being fed back as an input. Accordingly, the circularity score is low, reflecting only the minor non-load-bearing self-citation, while the correctness gap should be addressed separately by the authors.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new free parameters or invented entities. It depends on standard monomial ideal theory plus one false ad hoc assumption about integral closures of pure-power ideals, which is the main source of the proof failure.

assumptions (3)
  • ad hoc to paper For an irreducible monomial ideal I=<x_i^{a_i}>, overline{I^n}=<x_i^{n a_i}>.
    Asserted in property (P4), Section 2. False in general: I=(x^2,y^2) gives overline I=(x,y)^2, not (x^2,y^2). This assumption underlies the Section 4 computations.
  • standard math Newton polyhedron membership criterion: x^a is in overline{<x^{b_i}>} iff sum a_i/b_i >= 1.
    Used in Lemma 2.2 and throughout. This is a standard convex-geometric description of integral closure for monomial ideals.
  • domain assumption For complete intersection monomial ideals, ordinary symbolic powers equal ordinary powers and associated primes are stable.
    Used in Section 3 to justify I^{(n)}=I^n and the primary component arguments. This is a classical property of regular sequences.

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Cite this review

Pith. "Pith review of $\operatorname{v}$-numbers of integral closure filtrations of monomial ideals." pith.science (2026). https://pith.science/paper/DSMMRXXV

@misc{pith2026250609051,
  author       = {Pith},
  title        = {Pith review of: $\operatornamev$-numbers of integral closure filtrations of monomial ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DSMMRXXV}},
  note         = {Machine review of arXiv:2506.09051}
}
abstract

In this article, we investigate the $\operatorname{v}$-numbers of powers of monomial ideals and their integral closures in a polynomial ring $S$. We provide an alternative proof for determining the $\operatorname{v}$-numbers of powers of complete intersection monomial ideals. Furthermore, we analyze the $\operatorname{v}$-numbers associated to integral closure filtrations of irreducible monomial ideals and explore their relationship with the Castelnuovo-Mumford regularity of these ideals. Consequently, we obtain that for all $n\geq 1$, $\operatorname{reg}(S/\overline{I^n})=\operatorname{v}(\overline{I^n})=n\alpha(I)-1$ where $I$ is an equigenerated irreducible monomial ideal. Finally, we give an upper bound for $\operatorname{v}$-numbers associated to the integral closure filtrations of complete intersection monomial ideals and explicitly compute these $\operatorname{v}$-numbers in certain cases. As a consequence, we show that for any integer $a\geq 1$, there exists a height two equigenerated complete intersection monomial ideal $I$ such that $\operatorname{reg}(S/\overline{I^n})-\operatorname{v}(\overline{I^n})=a-1$ for all $n\geq 1$. Moreover, we establish that for complete intersection monomial ideals, the $\operatorname{v}$-numbers of powers of ideals can be arbitrarily larger than the $\operatorname{v}$-numbers of integral closures of their powers.

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Works this paper leans on

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