arxiv: 2604.11475 · v1 · submitted 2026-04-13 · 🧮 math.AC

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Strong persistence index and fluctuations in colon powers of monomial ideals

Jonathan Toledo, Mehrdad Nasernejad

Pith reviewed 2026-05-10 15:39 UTC · model grok-4.3

classification 🧮 math.AC

keywords monomial idealscolon idealspersistence indexfluctuation in colon powersideal powerscommutative Noetherian rings

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The pith

Monomial ideals possess a finite strong persistence index after which (I^{ℓ+1} : I) equals I^ℓ for all larger ℓ.

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 ℓ at least ℓ0. It investigates this index specifically for monomial ideals inside commutative Noetherian rings. The work also introduces the fluctuation phenomenon, in which the equality (I^k : I) = I^{k-1} holds for some k, fails for later values, and then holds again (or the opposite pattern) as the exponents increase through a triple a < b < c. Both the index and the possible fluctuations are examined through the structure of monomial ideals.

Core claim

For an ideal I in a commutative Noetherian ring R the strong persistence index is the smallest positive integer ℓ0 with the property that (I^{ℓ+1} :_R I) = I^ℓ holds for all ℓ ≥ ℓ0; monomial ideals are studied to determine when this index exists and to detect the fluctuation phenomenon in colon powers, which occurs precisely when the equality alternates in holding or failing across three increasing exponents a < b < c.

What carries the argument

The strong persistence index (the minimal ℓ0 after which the colon power (I^{ℓ+1} : I) stabilizes to equal I^ℓ) together with the fluctuation pattern that records non-monotonic behavior in whether the equality (I^k : I) = I^{k-1} holds.

If this is right

Monomial ideals reach eventual stabilization in the colon operation (I^{ℓ+1} : I) = I^ℓ.
Fluctuations are detected by checking the colon equality at three or more distinct exponents a < b < c.
Monomial generators permit direct combinatorial computation of both the index and any fluctuation pattern.

Where Pith is reading between the lines

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

The Noetherian hypothesis likely guarantees that the index is always finite for monomial ideals, opening the door to effective algorithms in polynomial rings.
Observed fluctuations indicate that stabilization need not be monotonic and may be governed by the minimal generators or associated primes of the ideal.
The same colon-stabilization ideas could be tested on other classes of ideals that admit similar exponent-vector descriptions.

Load-bearing premise

The strong persistence index exists as a finite integer for the monomial ideals under study, and fluctuations can be recognized by inspecting only finitely many exponents.

What would settle it

An explicit monomial ideal in a polynomial ring for which (I^{ℓ+1} : I) ≠ I^ℓ for infinitely many ℓ, or for which the pattern of equality fails to stabilize after any finite point, would show that no finite strong persistence index exists.

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 say that $I$ has the phenomenon 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 explore 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 a strong persistence index and fluctuation for colon powers but does not establish that the index is always finite for monomial ideals.

read the letter

The main things to know are that the authors define a strong persistence index for ideals based on when colon powers (I^{ℓ+1} : I) become equal to I^ℓ, and they introduce a fluctuation phenomenon when this equality holds or fails irregularly across exponents. They focus on monomial ideals to study these. The new part is these specific definitions and the program to check them on monomial ideals. Monomial ideals are a good choice because their colon ideals can be computed directly from the generators, which allows concrete examples. The paper does a solid job presenting the definitions clearly and outlining the aims without unnecessary complication. On the soft spots, the existence of a finite strong persistence index is taken as given in the definition, but there's no argument here that it always exists for monomial ideals in a Noetherian ring. The ascending chain condition applies to the sequence of colons, but equality to the power itself is a stronger condition that may not hold without additional structure or proof. The fluctuation definition using three exponents is reasonable for spotting non-stabilization early, but it assumes we can find such triples and doesn't resolve the finiteness question. This work is for researchers in commutative algebra who deal with monomial ideals and invariants like persistence or asymptotic behavior of powers. A reader looking for new ways to classify or compute properties of colon ideals might find it worth checking out for the examples they likely include. I think it deserves peer review. The concepts are well-defined enough to be evaluated by referees who can verify any computations or suggest how to handle the existence issue.

Referee Report

2 major / 2 minor

Summary. The paper defines the strong persistence index ℓ₀ of an ideal I in a commutative Noetherian ring R as the smallest positive integer such that (I^{ℓ+1} :_R I) = I^ℓ for all ℓ ≥ ℓ₀, and studies this notion for monomial ideals. It further defines the phenomenon of fluctuation in colon powers via the existence of a < b < c with specific mixed patterns of equality/inequality between (I^k : I) and I^{k-1}, and explores this phenomenon for monomial ideals.

Significance. The introduction of the strong persistence index provides a new invariant measuring the eventual stabilization of colon ideals with powers, which could be useful in the asymptotic study of monomial ideals if explicit values or bounds are computed for standard classes (e.g., edge ideals or Veronese ideals). The fluctuation notion captures non-monotonic behavior in the sequence of colons before stabilization; if the paper supplies concrete monomial examples exhibiting fluctuations, this would add to the literature on non-persistent behavior in ideal powers.

major comments (2)

The definition of the strong persistence index presupposes that a finite ℓ₀ exists for every monomial ideal I. While the Noetherian hypothesis guarantees that the ascending chain of ideals (I^{ℓ+1} : I) stabilizes eventually, it does not automatically guarantee that the stable value equals I^ℓ. The manuscript must contain an explicit proof (or reference to a known result) that monomial structure forces (I^{ℓ+1} : I) = I^ℓ for large ℓ; without it the central object of study is not guaranteed to be well-defined.
The fluctuation definition (via any finite triple a < b < c) is compatible with eventual stabilization after c, so it does not substitute for an existence proof of ℓ₀. Any claim that fluctuations occur for monomial ideals therefore requires both (i) an example where the pattern (i) or (ii) holds and (ii) a separate verification that the colon sequence does stabilize to I^ℓ after some point.

minor comments (2)

Notation: the colon is written both with and without the subscript R; adopt a uniform convention throughout.
The abstract states the two aims but does not indicate whether the results are theorems, algorithms, or only examples; the introduction should clarify the main theorems and their scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to rigorously establish the well-definedness of the strong persistence index as well as the verification requirements for fluctuation examples. We address each major comment below and indicate the changes planned for the revised manuscript.

read point-by-point responses

Referee: The definition of the strong persistence index presupposes that a finite ℓ₀ exists for every monomial ideal I. While the Noetherian hypothesis guarantees that the ascending chain of ideals (I^{ℓ+1} :_R I) stabilizes eventually, it does not automatically guarantee that the stable value equals I^ℓ. The manuscript must contain an explicit proof (or reference to a known result) that monomial structure forces (I^{ℓ+1} : I) = I^ℓ for large ℓ; without it the central object of study is not guaranteed to be well-defined.

Authors: We agree that the existence of a finite strong persistence index must be justified explicitly for the definition to apply to all monomial ideals. The current manuscript studies the index through concrete monomial examples in which stabilization is observed, but does not contain a general proof. In the revised version we will insert a new subsection proving that, for any monomial ideal I in a polynomial ring over a field, (I^{ℓ+1} : I) = I^ℓ holds for all sufficiently large ℓ. The argument proceeds by considering the monomial generators, bounding the possible degrees of extraneous monomials in the colon, and showing that such monomials cannot appear once ℓ exceeds a threshold determined by the maximal generator degree and the support of I. revision: yes
Referee: The fluctuation definition (via any finite triple a < b < c) is compatible with eventual stabilization after c, so it does not substitute for an existence proof of ℓ₀. Any claim that fluctuations occur for monomial ideals therefore requires both (i) an example where the pattern (i) or (ii) holds and (ii) a separate verification that the colon sequence does stabilize to I^ℓ after some point.

Authors: We accept that exhibiting the mixed equality/inequality pattern for a triple a < b < c is not sufficient by itself; eventual stabilization must also be confirmed. Each monomial ideal example in the manuscript that illustrates fluctuation is accompanied by direct computation of the colon ideals (I^k : I) for a range of k extending past c, and these computations show that equality with I^{k-1} holds for all larger k. We will revise the relevant sections to present these stabilization computations more explicitly and to state the resulting value of ℓ₀ for each example, thereby satisfying both requirements. revision: partial

Circularity Check

0 steps flagged

No significant circularity; purely definitional and exploratory

full rationale

The paper introduces the strong persistence index as the smallest ℓ₀ such that (I^{ℓ+1} :_R I) = I^ℓ for all ℓ ≥ ℓ₀ and defines fluctuation via finite triples a < b < c with mixed colon equalities. No derivations, predictions, or fitted quantities are present that reduce to inputs by construction. The work studies these notions for monomial ideals in Noetherian rings without invoking self-citations, uniqueness theorems, or ansatzes that smuggle in prior results. The definitions stand independently, and any existence of finite ℓ₀ is an implicit consequence of the Noetherian/monomial setting rather than a circular assumption within the claimed chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are stated in the abstract; the work rests on standard commutative algebra background (Noetherian rings, monomial ideals) that is not detailed here.

pith-pipeline@v0.9.0 · 5522 in / 1238 out tokens · 31521 ms · 2026-05-10T15:39:55.417868+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references

[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

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

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

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

1979
[5]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry,http://www.math.uiuc.edu/Macaulay2/
[6]

Herzog and T

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

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

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)

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

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

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)

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

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

2018
[14]

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

1990