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REVIEW 3 major objections 4 minor 11 references

Comparison of stability indices of powers of graded ideals

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For any a,b, one ideal has ass-stability a and v-stability b

desk verdict Short, useful paper with a real new result, but the vstab half of the main theorem leans on imported formulas whose hypotheses are not checked in the text. read the letter →

arxiv 2505.15608 v1 pith:HBKZEKAF submitted 2025-05-21 math.AC math.CO

Antonino Ficarra , Emanuele Sgroi This is my paper
classification math.ACmath.CO MSC 13F2013F5505C7005E40
keywords v-numberass-stabilityv-stabilityassociatedprimesmonomialidealspowersofstabilityindex
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

This paper asks how two asymptotic invariants of powers of a graded ideal compare: the ass-stability index $\mathrm{astab}(I)$, where the set of associated primes stops changing, and the v-stability index $\mathrm{vstab}(I)$, where the v-number becomes linear. The main result is that the two indices are independent: given any positive integers $a$ and $b$, there is a monomial ideal with $\mathrm{astab}(I)=a$ and $\mathrm{vstab}(I)=b$. In two-dimensional polynomial rings the situation is rigid on one side—$\mathrm{astab}(I)=1$ for every graded ideal—but $\mathrm{vstab}(I)$ can be any positive integer. A reader should care because the two indices measure different forms of stabilization, and knowing that no universal inequality relates them is a structural fact about powers of ideals.

What carries the argument

The construction separates the two indices: a sum $J$ of edge ideals on disjoint triples of variables produces $\mathrm{astab}(J)=a$ with $\mathrm{vstab}(J)=1$, and then multiplying $J$ by $x_0^{2b-1}$ and adding the planar ideal $H=(x^{2b+1},x^2y^{2b-1},y^{2b+1})$ shifts the v-number growth so that $\mathrm{vstab}(I)=b$ without changing $\mathrm{astab}(I)$. The proof uses imported formulas for the v-number of products and sums of ideals in disjoint variables, together with a formula for the associated primes of powers of sums of ideals.

What would settle it

For the constructed ideal with $a=2$ and $b=3$, compute $v(I^k)$ directly for $k=1,2,3$ from the definition of the v-number using the explicit generators. The claimed values are $(2b+1)k+2b$, i.e. $7k+6$, for $k=1,2$, and $(2b+1)k+2b-1$, i.e. $7k+5$, for $k=3$. A direct computation that yields any other values would disprove the formulas on which the $\mathrm{vstab}$ half rests.

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Extended reading notes

Core claim

The paper establishes that for every pair of positive integers $(a,b)$, there exists a monomial ideal $I$ in a $3(a+1)$-dimensional polynomial ring such that $(\mathrm{astab}(I),\mathrm{vstab}(I))=(a,b)$. It also proves that in a two-dimensional polynomial ring every graded ideal satisfies $\mathrm{astab}(I)=1\le \mathrm{vstab}(I)$, and exhibits ideals where $\mathrm{vstab}(I)$ is any prescribed positive integer. Consequently, the two stability indices are not comparable in general.

Load-bearing premise

The proof that $\mathrm{vstab}(I)=b$ depends on imported formulas for the v-number of sums and products of ideals in disjoint variables; if those formulas require additional hypotheses that the particular constructed ideals fail, the equalities giving $\mathrm{vstab}(I)=b$ would not follow.

Editorial extensions

If this is right

  • Neither $\mathrm{astab}(I)\le \mathrm{vstab}(I)$ nor $\mathrm{vstab}(I)\le \mathrm{astab}(I)$ holds for all graded ideals in dimension at least three.
  • In dimension two, $\mathrm{astab}(I)=1$ is forced while $\mathrm{vstab}(I)$ can be any positive integer, so the low-dimensional regime is completely described.
  • The constructed ideals are monomial, so the independence phenomenon already appears in combinatorial commutative algebra.
  • The open problem of realizing arbitrary quadruples $(\mathrm{astab},\mathrm{dstab},\mathrm{rstab},\mathrm{vstab})$ remains; the present work fixes two of the four coordinates in a uniform construction.

Reading between the lines

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

  • The same disjoint-variable technique could likely be adapted to prescribe the v-stability indices of individual associated primes, not only the global $\mathrm{vstab}(I)$.
  • A direct computer algebra check for the smallest new cases, $a=2$ with $b=2$ or $b=3$, would test whether the sharp switch in $v(I^k)$ at $k=b$ actually occurs, thereby stress-testing the imported formulas.
  • The independence of the two indices may be stable under small deformations of the constructed monomial ideals, though the paper does not address generic perturbations.
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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 / 4 minor

Summary. The paper studies the relationship between two asymptotic invariants of powers of a graded ideal: the index of ass-stability astab(I) and the v-stability index vstab(I). In Theorem 1.1, for a two-dimensional polynomial ring, the authors prove that astab(I)=1 for every graded ideal and exhibit ideals I=(x^{2b+1}, x^2 y^{2b-1}, y^{2b+1}) for which vstab(I)=b, with explicit formulas for v(I^k). In Theorem 2.1, for any positive integers a,b, they construct a monomial ideal in a 3(a+1)-dimensional polynomial ring such that (astab(I), vstab(I))=(a,b), thereby showing that the two indices are independent in general. The construction combines a sum of triangle edge ideals with the two-dimensional example, and the proof relies on imported formulas for v-numbers and associated primes of products and sums of ideals in disjoint variables.

Significance. If correct, Theorem 2.1 settles a natural independence question: the indices astab and vstab can be prescribed arbitrarily, so in general they are not comparable. The two-dimensional part is an attractive, explicit family of examples with astab=1 and unbounded vstab. The construction in Theorem 2.1 is explicit and the overall question is well motivated. The main strength is the concrete nature of the computations: Theorem 1.1 is proved with a self-contained monomial order argument. However, the vstab half of Theorem 2.1 depends on several imported formulas from [6] whose hypotheses are neither stated nor verified, and at least one of these formulas is not universally true. This is a load-bearing correctness risk that must be fixed before the paper can be accepted.

major comments (3)
  1. [Section 2, Eq. (7)] Equation (7) is obtained by applying [6, Corollary 4.2] to L and H, but neither the hypotheses of that corollary nor their verification for L=x0^{2b-1}J and H are provided. This matters because the displayed min-formula is not true in general: for L=(x,y)^2 and H=(z^2) in disjoint variables one has v(L)=v(H)=1, while v(L+H)=3 for the sum ideal (x^2,xy,y^2,z^2). Please state the exact hypotheses of [6, Corollary 4.2] and check them for the pair (L,H), or give a direct proof of (7). Until then, the conclusion vstab(I)=b is unsupported.
  2. [Section 2, Eq. (5)] Equation (5) for v(L^k) is imported from [6, Corollary 2.3], and the input v(J^k) in equation (4) is imported from [6, Theorem 5.2]. The hypotheses of these results are not stated, nor is it verified that L=x0^{2b-1}J and J=sum Ji satisfy them. Since the subsequent vstab computation feeds directly on (5), the authors should quote the precise statements and confirm their applicability, or prove (5) directly in the paper.
  3. [Section 2, astab(J)=a] The proof that astab(J)=a for a>2 is compressed into a single sentence ('Applying again this result ... proceeding iteratively'). Please spell out the induction: at each step, identify the first power at which the new maximal prime m1+...+mr appears and verify that no further associated primes appear afterwards. This is needed to justify the claim astab(I)=a in equation (6).
minor comments (4)
  1. [Section 1, proof of Theorem 1.1(a)] In the displayed minimum, the index range '0≤j≤i+1' is not well-defined because u_{i+1,i+2} does not exist; the intended range is likely 0≤j≤i.
  2. [Introduction] The definition of vstab(I) says 'the least integer vstab(I) for which v(I^k)=α(I)k+b', but the constant b is not introduced before this point; please define b as the constant from the eventual linear form.
  3. [Section 2, Theorem 2.1] The indexing S=K[x0,x1,...,x_{3a},x,y] gives 3a+3 variables, which equals 3(a+1); this is correct, but the notation may be clearer if the total number of variables is stated explicitly.
  4. [Section 1, proof of Theorem 1.1(a)] A short comment explaining how the ordered chain of monomials matches the hypotheses of [7, Proposition 5.5] would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 2.1 uses free parameters a,b and general cited lemmas; the vstab concern is a hypotheses-verification risk, not a circular reduction.

full rationale

The paper's main construction in Theorem 2.1 starts with arbitrary integers a,b≥1, builds J from edge ideals in disjoint variables, and then forms I = x0^{2b-1}J + H, where H is the explicit K[x,y]-ideal from Theorem 1.1. The claimed values astab(I)=a and vstab(I)=b are computed from separate ingredients: astab is obtained from the external associated-primes formula [11, Theorem 4.1(3)] plus an induction, while vstab is obtained from the general v-number formulas for products and sums in disjoint variables ([6, Corollary 2.3 and 4.2]) combined with the direct computation of v(H^k) in Theorem 1.1(a)-(b). None of these ingredients is the target assertion (astab,vstab)=(a,b); in particular, a and b are not fitted parameters and no computed stability index is an input renamed as a prediction. The paper does rely heavily on the authors' own prior work, especially [6], and the manuscript does not state or verify the full hypotheses of [6, Corollary 2.3 and 4.2] before applying them at equations (5) and (7). If those corollaries turn out to require extra conditions that L or H do not satisfy, the vstab half of Theorem 2.1 would be unsupported. That is a genuine correctness and rigor risk, but it is not a circularity: the cited formulas are general lemmas about v-numbers, not the constructed pair (a,b), and the proof does not assume the conclusion. The use of Theorem 1.1 inside Theorem 2.1 is also legitimate because the a=1 case is a special case, not the general theorem. Hence no circular step can be exhibited from the text, and the appropriate verdict is no significant circularity.

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

The paper contributes no fitted parameters or invented entities. It relies on several standard results from the literature, including three preprints by the same authors, for v-number formulas and associated prime formulas; these are stated as citations rather than proved here.

assumptions (6)
  • standard math Brodmann's theorem: Ass(I^k) is eventually constant for any graded ideal I.
    Used in the Introduction to define astab(I) and Ass^infty(I).
  • standard math Conca and Ficarra-Sgroi: v_p(I^k) and v(I^k) are eventually linear of the form a_p k + b_p and alpha(I)k + b.
    Used to define vstab_p(I) and vstab(I) in the Introduction.
  • standard math Formula (1): for an m-primary monomial ideal J = (x^{a1}, ..., y^{bm}) with a1>...>0 and 0<b2<...<b_m, v_m(J)=min{a_i+b_{i+1}-2}.
    Taken from [7, Proposition 5.5] and applied in Theorem 1.1(a).
  • standard math v-number formulas for products and sums of monomial ideals in disjoint variables ([6, Corollary 2.3, Theorem 5.2, Corollary 4.2]).
    These formulas produce equations (4), (5), and (7) in Section 2.
  • standard math Associated primes of powers of a sum of ideals in disjoint variables ([11, Theorem 4.1(3)]).
    Used to compute Ass((J1+J2)^k) and Ass(I^k) and hence astab.
  • standard math astab(I)=1 for every graded ideal in a polynomial ring of dimension at most 2 ([9, Remark 1.1]).
    This is the starting point of Theorem 1.1.

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Pith. "Pith review of Comparison of stability indices of powers of graded ideals." pith.science (2026). https://pith.science/paper/HBKZEKAF

@misc{pith2026250515608,
  author       = {Pith},
  title        = {Pith review of: Comparison of stability indices of powers of graded ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HBKZEKAF}},
  note         = {Machine review of arXiv:2505.15608}
}
abstract

In this paper, we compare the index of ass-stability $\text{astab}(I)$ and the index of $\text{v}$-stability $\text{vstab}(I)$ of powers of a graded ideal $I$. We prove that $\text{astab}(I)=1\le\text{vstab}(I)$ for any graded ideal $I$ in a 2-dimensional polynomial ring, and that $\text{vstab}(I)$ can be any positive integer in this situation. Moreover, given any integers $a,b\ge1$, we construct a graded ideal $I$ in a $3(a+1)$-dimensional polynomial ring such that $(\text{astab}(I),\text{vstab}(I))=(a,b)$.

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Reference graph

Works this paper leans on

11 extracted references · 7 canonical work pages

  1. [6]

    Ficarra, P

    A. Ficarra, P. Macias Marques, The v-function of powers of sums of ideals , 2024, preprint https://arxiv.org/abs/2405.16882

    arXiv 2024

  2. [1]

    Biswas, M

    P. Biswas, M. Mandal, K. Saha, Asymptotic behaviour and stability index of v-numbers of graded ideals, 2024, preprint https://arxiv.org/abs/2402.16583

    arXiv 2024

  3. [2]

    Brodmann, Asymptotic stability of Ass(M/I nM ), Proc

    M. Brodmann, Asymptotic stability of Ass(M/I nM ), Proc. Am. Math. Soc., 74(1979), 16–18

    work page 1979

  4. [3]

    Conca, A note on the v-invariant , Proc

    A. Conca, A note on the v-invariant , Proc. Am. Math. Soc. 152 (2024), no. 6, 2349–2351

    work page 2024

  5. [4]

    Cooper, A

    S.M. Cooper, A. Seceleanu, S.O. Toh˘ aneanu, M. Vaz Pinto, R.H. Villarreal.Generalized min- imum distance functions and algebraic invariants of geramita ideals . Advances in Applied Mathematics, 112:101940, 2020

    work page 2020

  6. [5]

    Ficarra, Simon ’s conjecture and thev-number of monomial ideals, Collectanea Mathematica (2024): 1–16

    A. Ficarra, Simon ’s conjecture and thev-number of monomial ideals, Collectanea Mathematica (2024): 1–16. https://doi.org/10.1007/s13348-024-00441-z

    doi:10.1007/s13348-024-00441-z 2024

  7. [7]

    Ficarra, E

    A. Ficarra, E. Sgroi, Asymptotic behaviour of the v-number of homogeneous ideals , 2023, preprint https://arxiv.org/abs/2306.14243

    arXiv 2023

  8. [8]

    Asymptotic behaviour of integer programming and the $\text{v}$-function of a graded filtration

    A. Ficarra, E. Sgroi, Asymptotic behaviour of integer programming and the v-function of a graded filtration, J. Algebra Its Appl., 2025,https://doi.org/10.1142/S0219498826502361, available at https://arxiv.org/abs/2403.08435

    work page Pith review arXiv 2025

Show all 11 references
  1. [9]

    Herzog, A

    J. Herzog, A. Mafi, Stability properties of powers of ideals in regular local rings of small dimension, Pacific J. Math., 295(2018), 31–41

    2018

  2. [10]

    Mart ´ ınez-Bernal, S

    J. Mart ´ ınez-Bernal, S. Morey, R. Villarreal,Associated primes of powers of edge ideals, Collect. Math. 63, 361–374 (2012)

    2012

  3. [11]

    Nguyen, Q.H

    H.D. Nguyen, Q.H. Tran, Powers of sums and their associated primes , Pacific J. Math, Vol. 316, No. 1, 2022. Antonino Ficarra, BCAM – Basque Center for Applied Mathematics, Mazarredo 14, 48009 Bilbao, Basque Country – Spain Ikerbasque, Basque Foundation for Science, Plaza Eusk...

    2022

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