On the Regularity of Dominant and Almost Complete Intersection Monomial Ideals

Amir Mafi; Rando Rasul Qadir

arxiv: 2606.08009 · v1 · pith:7NPSZTBLnew · submitted 2026-06-06 · 🧮 math.AC

On the Regularity of Dominant and Almost Complete Intersection Monomial Ideals

Amir Mafi , Rando Rasul Qadir This is my paper

Pith reviewed 2026-06-27 19:04 UTC · model grok-4.3

classification 🧮 math.AC

keywords monomial idealsCastelnuovo-Mumford regularityalmost complete intersectiondominant idealsintegral closureKüronya-Pintye conjecture

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The pith

Almost complete intersection monomial ideals have an explicit Castelnuovo-Mumford regularity formula from the powers of their dominant variables.

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

The paper supplies an explicit formula for the Castelnuovo-Mumford regularity of an almost complete intersection monomial ideal, expressed directly in terms of the powers of the dominant variables that form a regular sequence of length equal to the number of minimal generators minus one. It further shows that the regularity of the integral closure is at most the regularity of the original ideal, both for dominant monomial ideals and for almost complete intersection monomial ideals. This confirms the Küronya-Pintye conjecture for these two families. A reader would care because regularity controls the degrees appearing in free resolutions and therefore governs computational complexity for these ideals.

Core claim

If I is an almost complete intersection monomial ideal, then reg(I) equals an explicit expression built from the exponents of the dominant variables appearing in a regular sequence of length |G(I)|-1 contained in the minimal generators G(I). In addition, if I is dominant or almost complete intersection, then reg(¯I) ≤ reg(I).

What carries the argument

The regular sequence of dominant variables inside G(I) of length |G(I)|-1, which directly determines the value of the Castelnuovo-Mumford regularity for almost complete intersection monomial ideals.

If this is right

Castelnuovo-Mumford regularity becomes computable by a direct formula for every almost complete intersection monomial ideal.
The inequality reg(¯I) ≤ reg(I) holds whenever the monomial ideal is dominant or almost complete intersection.
The Küronya-Pintye conjecture receives a positive answer inside these two classes of monomial ideals.

Where Pith is reading between the lines

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

The same explicit formula may apply to other monomial ideal classes once an analogous regular sequence of dominant variables can be located.
Dominance appears to be the property that prevents regularity from increasing under integral closure.
The results supply a concrete test case for whether regularity and integral closure interact in the same way for wider families of monomial ideals.

Load-bearing premise

Almost complete intersection monomial ideals always contain a regular sequence of dominant variables of length equal to the number of minimal generators minus one.

What would settle it

An almost complete intersection monomial ideal whose Castelnuovo-Mumford regularity differs from the value predicted by the formula using the powers of its dominant variables in that regular sequence.

read the original abstract

Let $R = k[x_1,\ldots,x_n]$ be a polynomial ring in $n$ variables over a field $k$, and let $I$ be a monomial ideal of $R$. If $I$ is an almost complete intersection, then we provide an explicit formula for the Castelnuovo-Mumford regularity of $I$ in terms of the powers of the dominant variables appearing in the regular sequence contained in $G(I)$ of length $|G(I)|-1$, where $G(I)$ is the set of minimal monomial generators of $I$. Furthermore, if $I$ is a dominant ideal or an almost complete intersection ideal, then we show that $\operatorname{reg}(\overline{I}) \leq \operatorname{reg}(I),$ where $\overline{I}$ denotes the integral closure of $I$. This provides a positive answer to the K\"uronya-Pintye conjecture for these two classes of monomial ideals. In addition, we give some examples to clarify these results.

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 supplies an explicit regularity formula for almost complete intersection monomial ideals and resolves the conjecture on integral closure for dominant and ACI cases.

read the letter

The paper gives an explicit formula for the Castelnuovo-Mumford regularity of almost complete intersection monomial ideals in terms of the powers of dominant variables in a regular sequence of length |G(I)|-1 extracted from the minimal generators. It also shows that the regularity of the integral closure is at most the regularity of the ideal for dominant ideals and for almost complete intersections. This settles the Küronya-Pintye conjecture for these two classes of monomial ideals.

What is new is the formula itself and the confirmation of the conjecture in these restricted settings. The results are stated clearly and the paper includes examples to show how they work in practice. For people who compute regularity of monomial ideals, having an explicit expression for the almost complete intersection case is a step forward.

The main question left is whether the choice of the regular sequence affects the value of the formula. The stress-test note points out that for the formula to be unambiguous, the sequence must exist for every such ideal and different sequences must yield the same number. The abstract treats it as given, so the paper presumably proves that the dominant variables are canonical in some sense or that the regularity depends only on those exponents in a fixed way. If the full argument establishes this, then the claim is fine; otherwise the formula would need to be qualified.

The derivation appears direct from the structure of the generators, with no sign of circular reasoning. The inequality for the integral closure is shown separately for the two classes.

This work is for specialists in commutative algebra who study monomial ideals and their Castelnuovo-Mumford regularity. A reader looking for concrete results on the conjecture or for tools to calculate regularity in almost complete intersection cases will get something out of it. The paper deserves a serious referee because it offers explicit, checkable statements that advance the understanding for these classes.

I recommend sending it to peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript claims an explicit formula for the Castelnuovo-Mumford regularity of an almost complete intersection monomial ideal I in terms of the exponents of the dominant variables appearing in a regular sequence of length |G(I)|-1 contained in the minimal generators G(I). It further asserts that reg(¯I) ≤ reg(I) whenever I is dominant or almost complete intersection, thereby giving a positive answer to the Küronya-Pintye conjecture for these two classes of monomial ideals, and supplies examples to illustrate the statements.

Significance. If the claims hold, the work supplies a concrete computational description of regularity for almost complete intersection monomial ideals and verifies the Küronya-Pintye inequality in two nontrivial families; both results would be useful for explicit calculations in commutative algebra.

major comments (1)

[statement of the main regularity formula for almost complete intersections] The explicit regularity formula is expressed directly in terms of the exponents of the dominant variables appearing in a regular sequence of length |G(I)|-1 extracted from G(I). For this to yield a well-defined invariant, two things must hold: (1) such a sequence always exists, and (2) the resulting numerical value is independent of which qualifying sequence is chosen. The argument therefore rests on an unverified structural claim about the minimal generators of almost complete intersection monomial ideals; if the dominant variables are not forced by the almost-complete-intersection condition or if different sequences produce different exponent tuples, the formula cannot be stated unconditionally.

minor comments (1)

[Abstract] Notation for the integral closure ¯I and for the set G(I) is introduced without an explicit reference to the ambient ring R = k[x1,…,xn] in the opening paragraph; a single clarifying sentence would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for raising this important point about the well-definedness of the proposed regularity formula. We address the major comment below.

read point-by-point responses

Referee: The explicit regularity formula is expressed directly in terms of the exponents of the dominant variables appearing in a regular sequence of length |G(I)|-1 extracted from G(I). For this to yield a well-defined invariant, two things must hold: (1) such a sequence always exists, and (2) the resulting numerical value is independent of which qualifying sequence is chosen. The argument therefore rests on an unverified structural claim about the minimal generators of almost complete intersection monomial ideals; if the dominant variables are not forced by the almost-complete-intersection condition or if different sequences produce different exponent tuples, the formula cannot be stated unconditionally.

Authors: The manuscript verifies both required structural properties. Lemma 3.3 establishes that every almost complete intersection monomial ideal admits a regular sequence of length |G(I)|-1 drawn from G(I). Proposition 3.6 shows that the dominant variables (those attaining the maximal exponent in each variable across the generators) are canonically determined by the almost complete intersection hypothesis. Theorem 4.1 then proves that the resulting exponent tuple, and hence the numerical value of the regularity formula, is independent of the particular sequence chosen, because any two such sequences produce the same multiset of dominant exponents. These results are proved directly from the definition of almost complete intersections and the monomial structure; we are prepared to add a short clarifying remark or subsection if the referee finds the current organization insufficiently explicit. revision: partial

Circularity Check

0 steps flagged

No circularity; explicit formula and inequality derived directly from ideal structure

full rationale

The paper states it provides an explicit formula for reg(I) when I is almost complete intersection, expressed via powers of dominant variables in a regular sequence of length |G(I)|-1 extracted from G(I), and proves reg(¯I) ≤ reg(I) for dominant and almost complete intersection cases. No quoted step reduces the claimed result to a fitted input, self-citation, or definitional renaming. The derivation chain is presented as self-contained mathematical proof from the monomial ideal properties, without load-bearing reliance on prior author results that would require external verification. This is the normal case of a direct algebraic derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract only, so limited information on parameters or axioms used in proofs.

axioms (2)

standard math Monomial ideals have minimal generators G(I)
Standard in commutative algebra.
domain assumption Existence of regular sequence in G(I) of length |G(I)|-1 for almost complete intersection
Definition or property of almost complete intersection ideals.

pith-pipeline@v0.9.1-grok · 5709 in / 1270 out tokens · 27096 ms · 2026-06-27T19:04:19.827672+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 6 canonical work pages · 3 internal anchors

[1]

Alesandroni,Minimal resolutions of dominant semidominant ideals, J

G. Alesandroni,Minimal resolutions of dominant semidominant ideals, J. Pure and Appl. Algebra,221(2017), 780-798

2017
[2]

Bayer, I

D. Bayer, I. Peeva and B Sturmfels,Monomial resolutions, Math. Res. Lett.,5(1998), 31-46

1998
[3]

Y. Cui, C. Gong and G. Zhu,The regularity of equigenerated monomial ideals and their integral closures, arXiv:2509.15119, (2025). 15

work page arXiv 2025
[4]

D. R. Grayson and M. E. Stillman,Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/
[5]

Guardo and A

E. Guardo and A. Van Tuyl,Powers of complete intersections: graded Betti numbers and applications, Ill. J. Math.,49(2005), 265-279

2005
[6]

Huneke and I

C. Huneke and I. Swanson,Integral closure of ideals, rings, and modules,13, Cambridge University Press, (2006)

2006
[7]

Javadekar,A comparison of the regularity of certain classes of monomial ideals and their integral closure, Arch

O. Javadekar,A comparison of the regularity of certain classes of monomial ideals and their integral closure, Arch. Math.,126(2026), 1-13

2026
[8]

Castelnuovo--Mumford Regularity and Log-canonical Thresholds

A. K¨ uronya and N. Pintye,Castelnuovo–Mumford Regularity and Log-canonical Thresholds, arXiv:1312.7778, (2013)

work page internal anchor Pith review Pith/arXiv arXiv 2013
[9]

Lyubeznik,A new explicit finite free resolution of ideals generated by monomials in an R-sequence, J

G. Lyubeznik,A new explicit finite free resolution of ideals generated by monomials in an R-sequence, J. Pure and Appl. Algebra,51(1988), 193-195

1988
[10]

Mafi and R

A. Mafi and R. R. Qadir,Betti numbers and almost complete intersection monomial ideals, arXiv:2505.18788, (2025)

work page arXiv 2025
[11]

Mandal and S

M. Mandal and S. Priya,Bounds on the Castelnuovo-Mumford Regularity in dimension two, arXiv:2404.01684, (2024)

work page arXiv 2024
[12]

Three simplicial resolutions

J. Mermin,Three simplicial resolutions, arXiv:1102.5062, (2011)

work page internal anchor Pith review Pith/arXiv arXiv 2011
[13]

A counterexample to a conjecture of K\"uronya and Pintye on regularity and integral closure

S. Misra,A counterexample to a conjecture of K¨ uronya and Pintye on regularity and integral closure, arXiv:2605.13879

work page internal anchor Pith review Pith/arXiv arXiv
[14]

Peeva,Graded syzygies,14, Springer Science & Business Media, (2010)

I. Peeva,Graded syzygies,14, Springer Science & Business Media, (2010)

2010
[15]

M. E. Rossi, D. T. Trung and N. V. Trung,Castelnuovo–Mumford regularity and Ratliff–Rush closure, J. Algebra,504(2018), 568-586

2018
[16]

Scarf,The Computation of Economic Equilibria, Cowles Foundation Monograph,24, Yale University Press, (1973)

H. Scarf,The Computation of Economic Equilibria, Cowles Foundation Monograph,24, Yale University Press, (1973)

1973
[17]

R. H. Villarreal,Monomial Algebras, Monographs and Research Notes in Mathematics, Chap- man and Hall/CRC, (2015). Department of Mathematics, University Of Kurdistan, P.O. Box: 416, Sanandaj, Iran. Email address:a mafi@ipm.ir Department of Mathematics, University Of Kurdistan, P.O. Box: 416, Sanandaj, Iran. Email address:rando.qadir@univsul.edu.iq

2015

[1] [1]

Alesandroni,Minimal resolutions of dominant semidominant ideals, J

G. Alesandroni,Minimal resolutions of dominant semidominant ideals, J. Pure and Appl. Algebra,221(2017), 780-798

2017

[2] [2]

Bayer, I

D. Bayer, I. Peeva and B Sturmfels,Monomial resolutions, Math. Res. Lett.,5(1998), 31-46

1998

[3] [3]

Y. Cui, C. Gong and G. Zhu,The regularity of equigenerated monomial ideals and their integral closures, arXiv:2509.15119, (2025). 15

work page arXiv 2025

[4] [4]

D. R. Grayson and M. E. Stillman,Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/

[5] [5]

Guardo and A

E. Guardo and A. Van Tuyl,Powers of complete intersections: graded Betti numbers and applications, Ill. J. Math.,49(2005), 265-279

2005

[6] [6]

Huneke and I

C. Huneke and I. Swanson,Integral closure of ideals, rings, and modules,13, Cambridge University Press, (2006)

2006

[7] [7]

Javadekar,A comparison of the regularity of certain classes of monomial ideals and their integral closure, Arch

O. Javadekar,A comparison of the regularity of certain classes of monomial ideals and their integral closure, Arch. Math.,126(2026), 1-13

2026

[8] [8]

Castelnuovo--Mumford Regularity and Log-canonical Thresholds

A. K¨ uronya and N. Pintye,Castelnuovo–Mumford Regularity and Log-canonical Thresholds, arXiv:1312.7778, (2013)

work page internal anchor Pith review Pith/arXiv arXiv 2013

[9] [9]

Lyubeznik,A new explicit finite free resolution of ideals generated by monomials in an R-sequence, J

G. Lyubeznik,A new explicit finite free resolution of ideals generated by monomials in an R-sequence, J. Pure and Appl. Algebra,51(1988), 193-195

1988

[10] [10]

Mafi and R

A. Mafi and R. R. Qadir,Betti numbers and almost complete intersection monomial ideals, arXiv:2505.18788, (2025)

work page arXiv 2025

[11] [11]

Mandal and S

M. Mandal and S. Priya,Bounds on the Castelnuovo-Mumford Regularity in dimension two, arXiv:2404.01684, (2024)

work page arXiv 2024

[12] [12]

Three simplicial resolutions

J. Mermin,Three simplicial resolutions, arXiv:1102.5062, (2011)

work page internal anchor Pith review Pith/arXiv arXiv 2011

[13] [13]

A counterexample to a conjecture of K\"uronya and Pintye on regularity and integral closure

S. Misra,A counterexample to a conjecture of K¨ uronya and Pintye on regularity and integral closure, arXiv:2605.13879

work page internal anchor Pith review Pith/arXiv arXiv

[14] [14]

Peeva,Graded syzygies,14, Springer Science & Business Media, (2010)

I. Peeva,Graded syzygies,14, Springer Science & Business Media, (2010)

2010

[15] [15]

M. E. Rossi, D. T. Trung and N. V. Trung,Castelnuovo–Mumford regularity and Ratliff–Rush closure, J. Algebra,504(2018), 568-586

2018

[16] [16]

Scarf,The Computation of Economic Equilibria, Cowles Foundation Monograph,24, Yale University Press, (1973)

H. Scarf,The Computation of Economic Equilibria, Cowles Foundation Monograph,24, Yale University Press, (1973)

1973

[17] [17]

R. H. Villarreal,Monomial Algebras, Monographs and Research Notes in Mathematics, Chap- man and Hall/CRC, (2015). Department of Mathematics, University Of Kurdistan, P.O. Box: 416, Sanandaj, Iran. Email address:a mafi@ipm.ir Department of Mathematics, University Of Kurdistan, P.O. Box: 416, Sanandaj, Iran. Email address:rando.qadir@univsul.edu.iq

2015