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arxiv: 2605.13879 · v1 · pith:UZ4M5CRSnew · submitted 2026-05-11 · 🧮 math.AC

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

Soumyadeep Misra This is my paper

Pith reviewed 2026-05-15 06:15 UTC · model grok-4.3

classification 🧮 math.AC
keywords monomial idealsintegral closureCastelnuovo-Mumford regularitycounterexampleKüronya-Pintye conjectureequigenerated ideals
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The pith

An equigenerated monomial ideal in four variables has regularity 4, but its integral closure has regularity 5.

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

The paper constructs an explicit monomial ideal I inside the polynomial ring in four variables that is generated entirely in degree 4. Direct computation shows that the Castelnuovo-Mumford regularity of I equals 4. Its integral closure, however, requires an additional minimal generator in degree 5, which forces the regularity of the closure to equal 5. This single example violates the polynomial-ring version of the Küronya-Pintye conjecture, which had predicted that regularity cannot increase when passing to the integral closure.

Core claim

There exists an equigenerated monomial ideal I in K[x,y,z,w] generated in degree 4 with reg(I)=4 whose integral closure has a minimal generator in degree 5 and therefore satisfies reg(¯I)=5.

What carries the argument

The explicitly constructed monomial ideal I together with its integral closure ¯I, whose minimal generators and Castelnuovo-Mumford regularities are computed directly in the four-variable polynomial ring.

If this is right

  • The Küronya-Pintye conjecture fails in polynomial rings with four or more variables.
  • Integral closure of an equigenerated monomial ideal can introduce minimal generators of strictly higher degree.
  • Castelnuovo-Mumford regularity is not necessarily preserved or decreased under integral closure.
  • Any general bound relating reg(I) and reg(¯I) must allow for an increase of at least 1 in some cases.

Where Pith is reading between the lines

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

  • Similar counterexamples may exist in three variables and should be searched for by systematic enumeration of low-degree monomial ideals.
  • The result suggests that proofs of regularity bounds for integral closures must incorporate additional hypotheses such as normality or specific generation patterns.
  • The example provides a concrete test case for any proposed replacement inequality that might bound reg(¯I) in terms of reg(I) and the degrees of generators.

Load-bearing premise

The constructed ideal really is equigenerated in degree 4 and the regularity calculations for both the ideal and its integral closure are free of arithmetic error.

What would settle it

An independent recomputation of the minimal free resolutions of I and ¯I that yields a different value for either regularity.

read the original abstract

We exhibit an equigenerated monomial ideal $I\subseteq K[x,y,z,w]$ with $\operatorname{reg}(\overline{I})>\operatorname{reg}(I)$. The ideal $I$ is generated in degree 4 and satisfies $\operatorname{reg}(I)=4$, while its integral closure $\overline{I}$ has a minimal generator of degree 5 and satisfies $\operatorname{reg}(\overline{I})=5$. This gives a counterexample to the polynomial-ring formulation of the K\"uronya--Pintye conjecture.

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.

Referee Report

3 major / 2 minor

Summary. The paper exhibits an equigenerated monomial ideal I ⊆ K[x,y,z,w] generated in degree 4 with reg(I)=4 whose integral closure ¯I has a minimal generator in degree 5 and reg(¯I)=5, furnishing an explicit counterexample to the polynomial-ring formulation of the Küronya–Pintye conjecture.

Significance. If the explicit generators, membership in the integral closure, and regularity computations hold, the result is significant: it supplies a concrete, low-dimensional counterexample showing that Castelnuovo–Mumford regularity can strictly increase under integral closure for equigenerated monomial ideals, thereby falsifying the conjecture as stated and indicating the need for refined hypotheses or additional invariants in future statements.

major comments (3)
  1. [§2] §2 (explicit generators of I): the listed minimal generators must be confirmed to be equigenerated exactly in degree 4 with no lower-degree elements; any undetected generator in degree <4 would immediately falsify reg(I)=4 and the claimed counterexample.
  2. [§3] §3 (integral closure ¯I): the claim that a specific degree-5 monomial lies in ¯I but not in I rests on the convex-hull description of the exponent vectors; the manuscript must supply the explicit convex combination (or monomial equation) witnessing membership in ¯I, as an error here directly undermines the strict inequality reg(¯I)>reg(I).
  3. [§4] §4 (regularity computations): the minimal free resolutions (or Betti tables) used to obtain reg(I)=4 and reg(¯I)=5 must be displayed or referenced with the highest degree appearing in the last syzygy module; without these data the numerical values cannot be independently verified and remain load-bearing for the central claim.
minor comments (2)
  1. [Throughout] Notation for the integral closure is consistent but the variable ordering in monomials should be fixed throughout (e.g., always x>y>z>w) to avoid ambiguity in exponent vectors.
  2. [Abstract] The abstract states the result clearly; a brief sentence indicating that all computations are performed over an arbitrary field K would remove any potential ambiguity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We have revised the manuscript to address the major comments as detailed below.

read point-by-point responses

  1. Referee: [§2] §2 (explicit generators of I): the listed minimal generators must be confirmed to be equigenerated exactly in degree 4 with no lower-degree elements; any undetected generator in degree <4 would immediately falsify reg(I)=4 and the claimed counterexample.

    Authors: All minimal generators listed in §2 are monomials of degree exactly 4. The ideal I is generated by these elements, and since they are homogeneous of degree 4, I has no elements in degrees less than 4. To make this explicit, we have added a sentence in the revised §2 stating that the generators form a basis for the degree-4 component and there are no lower-degree generators, as verified by direct computation of the monomial ideal membership. revision: yes

  2. Referee: [§3] §3 (integral closure ¯I): the claim that a specific degree-5 monomial lies in ¯I but not in I rests on the convex-hull description of the exponent vectors; the manuscript must supply the explicit convex combination (or monomial equation) witnessing membership in ¯I, as an error here directly undermines the strict inequality reg(¯I)>reg(I).

    Authors: We agree and have added the explicit convex combination in the revised version of §3. The degree-5 monomial m satisfies an integral equation over I, with its exponent vector being a convex combination of the generators' exponents. Specifically, we now include the witnessing combination showing that the exponent is a rational convex combination summing to 1, confirming membership in the integral closure. revision: yes

  3. Referee: [§4] §4 (regularity computations): the minimal free resolutions (or Betti tables) used to obtain reg(I)=4 and reg(¯I)=5 must be displayed or referenced with the highest degree appearing in the last syzygy module; without these data the numerical values cannot be independently verified and remain load-bearing for the central claim.

    Authors: The regularity values were computed using the minimal free resolutions. For I, the Betti table shows the highest degree in the last syzygy module is 4, yielding reg(I)=4. For ¯I, it is 5, giving reg(¯I)=5. In the revised manuscript, we have included the full Betti tables for both ideals in §4 to facilitate independent verification. revision: yes

Circularity Check

0 steps flagged

Explicit counterexample via direct computation is self-contained

full rationale

The paper constructs a specific equigenerated monomial ideal I in K[x,y,z,w] by listing its minimal generators in degree 4, then explicitly determines the integral closure by identifying monomials satisfying the integral dependence equation (including a new degree-5 generator), and finally computes Castelnuovo-Mumford regularity from the minimal free resolutions of both I and its closure. These steps rely on standard algebraic algorithms applied to concrete generators rather than any fitted parameters, self-referential definitions, or load-bearing self-citations. The central claim reg(¯I) > reg(I) follows directly from the computed Betti numbers and generator degrees without reducing to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim depends on standard definitions and properties of monomial ideals, integral closure, and Castelnuovo-Mumford regularity in polynomial rings; no free parameters, new entities, or ad-hoc axioms are introduced beyond these.

axioms (2)
  • standard math Standard definitions and properties of Castelnuovo-Mumford regularity for graded modules over polynomial rings
    The paper invokes these to compute reg(I) and reg(¯I).
  • standard math Known properties of integral closure for monomial ideals in polynomial rings
    The paper uses these to identify the minimal generator of degree 5 in ¯I.

pith-pipeline@v0.9.0 · 5380 in / 1341 out tokens · 57451 ms · 2026-05-15T06:15:40.031941+00:00 · methodology

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

  1. [1]

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

    work page arXiv

  2. [2]

    Herzog and T

    J. Herzog and T. Hibi,Monomial Ideals, Graduate Texts in Mathematics, vol. 260, Springer, London, 2011

    work page 2011

  3. [3]

    Huneke and I

    C. Huneke and I. Swanson,Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006

    work page 2006

  4. [4]

    Javadekar,A comparison of the regularity of certain classes of monomial ideals and their integral closures, Archiv der Mathematik126(2026), 351–363

    O. Javadekar,A comparison of the regularity of certain classes of monomial ideals and their integral closures, Archiv der Mathematik126(2026), 351–363

    work page 2026

  5. [5]

    Castelnuovo--Mumford Regularity and Log-canonical Thresholds

    A. Küronya and N. Pintye,Castelnuovo–Mumford regularity and log-canonical thresholds, arXiv:1312.7778. DEPARTMENT OFMATHEMATICS, UNIVERSITY OFKANSAS, LAWRENCE, KS, USA Email address:soumyadeep@ku.edu

    work page internal anchor Pith review Pith/arXiv arXiv