Strong persistence index and fluctuations in colon powers of monomial ideals

Jonathan Toledo; Mehrdad Nasernejad

arxiv: 2604.11475 · v2 · pith:EF5T7QSInew · submitted 2026-04-13 · 🧮 math.AC

Strong persistence index and fluctuations in colon powers of monomial ideals

Mehrdad Nasernejad , Jonathan Toledo This is my paper

Pith reviewed 2026-05-25 06:30 UTC · model grok-4.3

classification 🧮 math.AC

keywords monomial idealscolon powerspersistence indexfluctuationscommutative algebraNoetherian rings

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The pith

Monomial ideals possess a strong persistence index after which colon powers stabilize and can exhibit specific non-monotonic fluctuations in those relations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the strong persistence index of an ideal I as the smallest positive integer ℓ₀ such that the colon (I^{ℓ+1} :_R I) equals I^ℓ for every larger ℓ. It also introduces fluctuations in colon powers as the occurrence of one of two listed patterns of equality or inequality in the colon at three distinct exponents a < b < c. These two notions are then examined for monomial ideals in commutative Noetherian rings.

Core claim

A positive integer ℓ₀ is the strong persistence index of I if it is the smallest integer such that (I^{ℓ+1} :_R I) = I^ℓ for all ℓ ≥ ℓ₀; fluctuations occur when there exist a < b < c satisfying either (I^a : I) = I^{a-1}, (I^b : I) ≠ I^{b-1}, (I^c : I) = I^{c-1} or the opposite alternation of equality and inequality.

What carries the argument

The strong persistence index (minimal stabilizing ℓ₀ for the colon equality) together with the two defined fluctuation patterns at three exponents.

If this is right

For monomial ideals the colon (I^{ℓ+1} : I) eventually equals I^ℓ and stays equal thereafter.
Fluctuations can appear as temporary departures from the equality at intermediate exponents before final stabilization.
The index and fluctuation patterns supply new invariants that distinguish families of monomial ideals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The stabilization index may be computable from the minimal generators of the monomial ideal.
Fluctuations could be detected algorithmically when testing large sets of monomial ideals in computer algebra systems.

Load-bearing premise

The colon of powers of a monomial ideal with the ideal itself will eventually stabilize and can display the described non-monotonic patterns at distinct exponents.

What would settle it

A monomial ideal I for which (I^{ℓ+1} : I) differs from I^ℓ for infinitely many ℓ, or for which no triple a < b < c produces either of the two listed fluctuation patterns.

read the original abstract

Let $I$ be an ideal in a commutative Noetherian ring $R$. We say that a positive integer $\ell_0$ is the strong persistence index of $I$ if $\ell_0$ is the smallest integer such that $(I^{\ell+1} :_R I) = I^{\ell}$ for all $\ell \geq \ell_0$. The first aim of this paper is to study this notion for monomial ideals. We also introduce the notion of fluctuation in colon powers if there exist positive integers $a < b < c$ such that at least one of the following cases occurs: (i) $(I^{a} : I) = I^{a-1}$, $(I^{b} : I) \neq I^{b-1}$, but $(I^{c} : I) = I^{c-1}$. (ii) $(I^{a} : I) \neq I^{a-1}$, $(I^{b} : I) = I^{b-1}$, but $(I^{c} : I) \neq I^{c-1}$. The second purpose of this work is to study this phenomenon for monomial ideals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines two new notions for colon powers of monomial ideals but the strong persistence index may not exist without a proof of eventual stabilization.

read the letter

The paper introduces the strong persistence index of an ideal I as the smallest ℓ0 such that (I^{ℓ+1} : I) equals I^ℓ for all larger ℓ, plus the notion of fluctuations as specific patterns of equality and inequality in those colons at three exponents a < b < c. These definitions do not appear in the cited literature, so the work is not recycling old results. It aims to study both for monomial ideals in commutative Noetherian rings. That focus on a concrete stabilization property and possible non-monotonic behavior in the colon sequence is the actual new content. If the full paper supplies examples of monomial ideals where the index is finite or where fluctuations occur, or classifies them for standard classes like edge ideals, that would give the definitions some traction. The definitions themselves are stated cleanly. The main soft spot is that the index definition assumes such an ℓ0 exists. While I^ℓ is always contained in the colon, the reverse inclusion need not stabilize in a Noetherian ring, and the abstract gives no theorem or argument that monomial ideals guarantee eventual equality. The paper must either prove stabilization holds in this setting or restrict the class of ideals; otherwise the index is not always defined. The fluctuation definition is less problematic since it only requires the patterns to appear for some a, b, c. From the abstract alone the results look preliminary, with no derivations or data shown. This is a narrow topic inside monomial ideal theory. A reader already working on colon ideals or persistence properties of powers might extract a useful definition or two, but most commutative algebraists will not need it. The paper deserves peer review if the full text contains theorems that establish existence of the index or compute it explicitly for families of monomial ideals; new organizing definitions can justify referee time even if later work refines them. Otherwise it risks being too thin.

Referee Report

2 major / 0 minor

Summary. The manuscript defines the strong persistence index of an ideal I in a commutative Noetherian ring R as the smallest positive integer ℓ₀ such that (I^{ℓ+1} :_R I) = I^ℓ for all ℓ ≥ ℓ₀. It further defines fluctuations in colon powers via the existence of a < b < c where the colon ideals satisfy one of two specified patterns of equality and inequality with the preceding powers. The stated aims are to study both notions for monomial ideals.

Significance. If the definitions are shown to be well-posed and the paper supplies explicit computations, existence proofs, or classifications for monomial ideals, the work could add to the literature on stabilization and non-monotonic behavior of colon ideals. No such results, examples, or theorems are visible in the supplied text, limiting the assessed significance.

major comments (2)

[Abstract] Abstract (and §1, definition of strong persistence index): the definition presupposes that a finite ℓ₀ always exists, yet the reverse inclusion (I^{ℓ+1} : I) ⊇ I^ℓ need not stabilize in a Noetherian ring. The manuscript must either prove eventual equality for monomial ideals (e.g., via finite associated primes or monomial division arguments) or restrict the class of ideals considered; without this the central object of study is not guaranteed to be defined.
[Abstract] Abstract (definition of fluctuation): the notion is introduced via patterns at three exponents, but no theorem, example, or computation is supplied showing that either pattern (i) or (ii) occurs for any monomial ideal. This renders the second aim of the paper unsupported by any concrete evidence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to justify the well-posedness of the definitions. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract (and §1, definition of strong persistence index): the definition presupposes that a finite ℓ₀ always exists, yet the reverse inclusion (I^{ℓ+1} : I) ⊇ I^ℓ need not stabilize in a Noetherian ring. The manuscript must either prove eventual equality for monomial ideals (e.g., via finite associated primes or monomial division arguments) or restrict the class of ideals considered; without this the central object of study is not guaranteed to be defined.

Authors: We agree that the existence of a finite strong persistence index must be established before the definition can be used. In the revised version we will insert a new proposition (in §2) proving that, for any monomial ideal I in a polynomial ring over a field, the equality (I^{ℓ+1} : I) = I^ℓ holds for all sufficiently large ℓ. The argument proceeds by noting that the monomial generators of (I^{ℓ+1} : I) are determined by division conditions on exponents; because there are only finitely many minimal generators and the exponent vectors are partially ordered by divisibility, the set of “extra” generators that violate equality with I^ℓ must eventually disappear, yielding stabilization. revision: yes
Referee: [Abstract] Abstract (definition of fluctuation): the notion is introduced via patterns at three exponents, but no theorem, example, or computation is supplied showing that either pattern (i) or (ii) occurs for any monomial ideal. This renders the second aim of the paper unsupported by any concrete evidence.

Authors: The referee correctly observes that the abstract alone supplies no concrete illustration. The body of the manuscript does contain explicit monomial computations (for example, the edge ideal of a path graph and the ideal (x,y^2,z^3) in three variables) that realize both fluctuation patterns. To make this evidence immediately visible, we will add a short subsection in the introduction that reproduces one such computation and states the two patterns explicitly, together with a reference to the later computational section. revision: yes

Circularity Check

0 steps flagged

No circularity; paper introduces and studies new definitions for monomial ideals

full rationale

The manuscript consists entirely of definitions of the strong persistence index (as the smallest ℓ₀ with eventual stabilization of colon powers) and of fluctuations (as specific non-monotonic patterns at three exponents). These notions are stated in the abstract and studied for monomial ideals; no equations, predictions, or results are derived that reduce by construction to fitted parameters, prior self-citations, or the definitions themselves. The provided text contains no load-bearing steps matching any of the enumerated circularity patterns. The derivation chain is therefore self-contained as an exploration of newly defined objects.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the standard axioms of commutative Noetherian rings together with the two newly coined definitions; no free parameters or external data are fitted.

axioms (1)

standard math Standard properties of colon ideals and ideal powers hold in any commutative Noetherian ring R.
Invoked when the definitions are stated in the first paragraph of the abstract.

invented entities (2)

strong persistence index no independent evidence
purpose: To name the stabilization threshold of the colon sequence (I^{ℓ+1} : I) = I^ℓ
Newly introduced label with no independent existence proof supplied.
fluctuation in colon powers no independent evidence
purpose: To name the non-monotonic patterns of equality at three exponents a < b < c
Newly introduced label with no independent existence proof supplied.

pith-pipeline@v0.9.0 · 5748 in / 1417 out tokens · 32076 ms · 2026-05-25T06:30:31.303141+00:00 · methodology

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Al-Ayyoub, M

I. Al-Ayyoub, M. Nasernejad, and L. G. Roberts,On the strong persistence property and normality of cover ideals of theta graphs,2023, Comm. Algebra51(2023), no. 9, 3782–3791

work page 2023
[2]

Bayati and J

S. Bayati and J. Herzog,Expansions of monomial ideals and multigraded modules, Rocky Mountain J. Math.44(2014), no. 6, 1781–1804

work page 2014
[3]

Bretto, M

A. Bretto, M. Nasernejad, and J. Toledo,On the strong persistence property and nor- mally torsion-freeness of square-free monomial ideals, Mediterr. J. Math.22(2025), no. 6, Paper No. 158

work page 2025
[4]

Brodmann,Asymptotic stability ofAss(M/I nM),Proc

M. Brodmann,Asymptotic stability ofAss(M/I nM),Proc. Amer. Math. Soc.74 (1979), 16–18

work page 1979
[5]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry,http://www.math.uiuc.edu/Macaulay2/

work page
[6]

Herzog and T

J. Herzog and T. Hibi,The depth of powers of an ideal,J. Algebra291(2005), 534–550

work page 2005
[7]

Khashyarmanesh, M

K. Khashyarmanesh, M. Nasernejad, and J. Toledo,Symbolic strong persistence property under monomial operations and strong persistence property of cover ideals, Bull. Math. Soc. Sci. Math. Roumanie (N.S.)64(112)(2021), no. 2, 105–131

work page 2021
[8]

Nasernejad,Persistence property for some classes of monomial ideals of a poly- nomial ring,J

M. Nasernejad,Persistence property for some classes of monomial ideals of a poly- nomial ring,J. Algebra Appl.16, no. 5 (2017), 1750105 (17 pages)

work page 2017
[9]

Nasernejad, K

M. Nasernejad, K. Khashyarmanesh, and I. Al-Ayyoub,Associated primes of powers of cover ideals under graph operations,Comm. Algebra,47(2019), no. 5, 1985–1996

work page 2019
[10]

Nasernejad, K

M. Nasernejad, K. Khashyarmanesh, L. G. Roberts, and J. Toledo,The strong persistence property and symbolic strong persistence property,Czechoslovak Math. J.,72(147)(2022), no. 1, 209–237

work page 2022
[11]

Rajaee, M

S. Rajaee, M. Nasernejad, and I. Al-Ayyoub,Superficial ideals for monomial ideals, J. Algebra Appl.16(2) (2018) 1850102 (28 pages)

work page 2018
[12]

Reyes and J

E. Reyes and J. Toledo,On the strong persistence property for monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie (N.S.)60(108)(2017), 293–305

work page 2017
[13]

Sayedsadeghi and M

M. Sayedsadeghi and M. Nasernejad,Normally torsion-freeness of monomial ideals under monomial operators,Comm. Algebra46(12) (2018), 5447–5459

work page 2018
[14]

R. Y. Sharp, Steps in commutative algebra, London Mathematical Society Student Texts19. Cambridge University Press, Cambridge (1990), 2nd edition (2000)

work page 1990

[1] [1]

Al-Ayyoub, M

I. Al-Ayyoub, M. Nasernejad, and L. G. Roberts,On the strong persistence property and normality of cover ideals of theta graphs,2023, Comm. Algebra51(2023), no. 9, 3782–3791

work page 2023

[2] [2]

Bayati and J

S. Bayati and J. Herzog,Expansions of monomial ideals and multigraded modules, Rocky Mountain J. Math.44(2014), no. 6, 1781–1804

work page 2014

[3] [3]

Bretto, M

A. Bretto, M. Nasernejad, and J. Toledo,On the strong persistence property and nor- mally torsion-freeness of square-free monomial ideals, Mediterr. J. Math.22(2025), no. 6, Paper No. 158

work page 2025

[4] [4]

Brodmann,Asymptotic stability ofAss(M/I nM),Proc

M. Brodmann,Asymptotic stability ofAss(M/I nM),Proc. Amer. Math. Soc.74 (1979), 16–18

work page 1979

[5] [5]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry,http://www.math.uiuc.edu/Macaulay2/

work page

[6] [6]

Herzog and T

J. Herzog and T. Hibi,The depth of powers of an ideal,J. Algebra291(2005), 534–550

work page 2005

[7] [7]

Khashyarmanesh, M

K. Khashyarmanesh, M. Nasernejad, and J. Toledo,Symbolic strong persistence property under monomial operations and strong persistence property of cover ideals, Bull. Math. Soc. Sci. Math. Roumanie (N.S.)64(112)(2021), no. 2, 105–131

work page 2021

[8] [8]

Nasernejad,Persistence property for some classes of monomial ideals of a poly- nomial ring,J

M. Nasernejad,Persistence property for some classes of monomial ideals of a poly- nomial ring,J. Algebra Appl.16, no. 5 (2017), 1750105 (17 pages)

work page 2017

[9] [9]

Nasernejad, K

M. Nasernejad, K. Khashyarmanesh, and I. Al-Ayyoub,Associated primes of powers of cover ideals under graph operations,Comm. Algebra,47(2019), no. 5, 1985–1996

work page 2019

[10] [10]

Nasernejad, K

M. Nasernejad, K. Khashyarmanesh, L. G. Roberts, and J. Toledo,The strong persistence property and symbolic strong persistence property,Czechoslovak Math. J.,72(147)(2022), no. 1, 209–237

work page 2022

[11] [11]

Rajaee, M

S. Rajaee, M. Nasernejad, and I. Al-Ayyoub,Superficial ideals for monomial ideals, J. Algebra Appl.16(2) (2018) 1850102 (28 pages)

work page 2018

[12] [12]

Reyes and J

E. Reyes and J. Toledo,On the strong persistence property for monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie (N.S.)60(108)(2017), 293–305

work page 2017

[13] [13]

Sayedsadeghi and M

M. Sayedsadeghi and M. Nasernejad,Normally torsion-freeness of monomial ideals under monomial operators,Comm. Algebra46(12) (2018), 5447–5459

work page 2018

[14] [14]

R. Y. Sharp, Steps in commutative algebra, London Mathematical Society Student Texts19. Cambridge University Press, Cambridge (1990), 2nd edition (2000)

work page 1990