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On the strong persistence property and normally torsion-freeness of square-free monomial ideals

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Pith's one-line read This paper proves that every square-free monomial ideal in five variables has the strong persistence property, giving a boundary case for powers of monomial ideals, and derives criteria for Conforti-Cornuejols counterexamples and for…

desk verdict The n=5 strong persistence theorem is the real result and looks solid; Theorem 5.5's proof has a genuine gap that the stress-test doesn't fix. read the letter →

arxiv 2411.14227 v1 pith:67HRRIPH submitted 2024-11-21 math.AC

Alain Bretto , Mehrdad Nasernejad , Jonathan Toledo This is my paper · ORCID
classification math.AC MSC 13B2513F2005C2505E40
keywords strongpersistencepropertysquare-freemonomialidealsnormallytorsion-freeassociatedprimesofpowerscluttersConforti-Cornuejolsconjecturesymbolic
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 tries to prove that five variables are the exact boundary for the strong persistence property among square-free monomial ideals. Concretely, it claims that for every square-free monomial ideal $I$ in $K[x_1,x_2,x_3,x_4,x_5]$, the quotient $(I^{s+1}:I)$ equals $I^s$ for all $s\geq 1$, so the associated primes of powers cannot appear unexpectedly. The same argument is then turned toward normal torsion-freeness: the paper gives a criterion that any minimal counterexample to the Conforti-Cornuejols conjecture must fail, and a necessary and sufficient condition for a linear combination $x_iI+x_jJ$ of two normally torsion-free square-free monomial ideals to be normally torsion-free. A reader should care because the first claim would pin down the smallest number of variables in which strong persistence can fail, and the latter two connect prime filtrations to the max-flow min-cut property of hypergraphs.

What carries the argument

The machinery is the clutter of a square-free monomial ideal together with the colon identity $(I^{s+1}:I)=I^s$ that defines strong persistence. For the five-variable theorem, the proof splits according to whether edges have size 1, 2, 3, or 4 and then applies structural lemmas: cones and chains of edges, the fact that polymatroidal ideals are strongly persistent, and reductions to edge ideals of bipartite graphs. For normal torsion-freeness, the load-bearing equivalence is the standard one recorded in Theorem 2.12, connecting the containment $\operatorname{Ass}(R/I^s)\subseteq\operatorname{Ass}(R/I)$ with $I^k=I^{(k)}$ for all $k$ and with the Mengerian/max-flow-min-cut property of the clutter. The proof of Theorem 5.5 compares the symbolic power $L^{(k)}=(x_iI,x_j)^{(k)}\cap(x_iI,J)^{(k)}$, expanded through $(I\cap J)^{(k)}=I^{(k)}\cap J^{(k)}$, with the ordinary power $(x_iI+x_jJ)^k$ using a binomial expansion and a cancellation step in the quotient by $x_j^{k-\alpha}$.

What would settle it

Enumerate all clutters on five vertices with edges of size 2 or 3 (the cases left in Theorem 3.18) and test the colon equality $(I^{s+1}:I)=I^s$ for $s=1,2,3$; the theorem predicts the equality holds in every case, so a single exceptional square-free monomial ideal would falsify it.

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Extended reading notes

Core claim

The paper's central discovery is that the five-variable ring is the dividing line for the strong persistence property of square-free monomial ideals. Theorem 3.18 asserts that every square-free monomial ideal $I$ in $K[x_1,\ldots,x_5]$ satisfies $(I^{s+1}:I)=I^s$ for all $s\geq 1$, while a displayed ten-generator cubic ideal in six variables has $(I^3:I)\neq I^2$ and hence fails. The proof reduces the associated clutter to a finite list of edge-cardinality cases and handles each with lemmas about cones, chains, polymatroidal exchange, and graph edge ideals. Separately, Corollary 4.8 states that in a minimal counterexample to the Conforti-Cornuejols conjecture, every square-free monomial $v\in I^\ell$ must lie in $p^{\ell+1}$ for some minimal prime $p$ of $I$. Theorem 5.5 then gives the iff criterion that $L=x_iI+x_jJ$ is normally torsion-free exactly when both $x_iI+J$ and $I+x_jJ$ are, under the support-disjointness assumptions $\gcd(x_j,u)=1$ and $\gcd(x_i,v)=1$.

Load-bearing premise

The weakest load-bearing premise is the uncomputed colon step in the proof of Theorem 5.5: after multiplying a monomial by $x_j^{k-\alpha}$, the paper concludes it lies in $J^{k-\alpha}I^\alpha$ without computing the quotient ideal $(I+x_jJ)^k : x_j^{k-\alpha}$.

Editorial extensions

If this is right

  • If Theorem 3.18 is correct, the first possible failure of strong persistence for square-free monomial ideals occurs in six variables, and the cubic ideal displayed in the introduction is a witness.
  • Since strong persistence implies persistence, every square-free monomial ideal in at most five variables also has the persistence property for associated primes.
  • Corollary 4.8 gives an explicit obstruction: any square-free monomial ideal with some $v\in I^\ell$ lying outside every $p^{\ell+1}$, $p\in\operatorname{Min}(I)$, cannot be a minimal counterexample to the Conforti-Cornuejols conjecture.
  • Theorem 5.5 reduces the normal torsion-freeness of $x_iI+x_jJ$ to two smaller checks, $x_iI+J$ and $I+x_jJ$, and Example 5.6 shows how repeated application verifies a Mengerian 4-uniform hypergraph ideal.
  • By contrapositive of Theorem 5.5, if either $x_iI+J$ or $I+x_jJ$ is not normally torsion-free, then $L$ itself cannot be; Example 5.4 demonstrates that both failures can occur even when $I$, $J$, and $I+J$ are normally torsion-free.

Reading between the lines

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

  • Beyond the paper, the five-variable theorem suggests that the family of strongly persistent square-free monomial ideals has a finite combinatorial description in each fixed variable count, so the same case-split method could yield an explicit certificate for each clutter on five vertices.
  • The iff criterion of Theorem 5.5 is iterative in spirit: if one records which half-combinations are normally torsion-free, the criterion can be applied repeatedly to decide sums of many scaled ideals, at the cost of an exponential number of sub-checks.
  • A testable extension is that the strong persistence failure in six variables is not an artifact of the coefficient field: monomial colon equalities are preserved under field extension, so the same ten-generator ideal should fail over every field.
  • The uncomputed colon step in Theorem 5.5 can be probed independently on the examples in the paper; if the containment $I^\alpha \cap \sum_{\beta<\alpha} J^{k-\beta}I^\beta \subseteq J^{k-\alpha}I^\alpha$ ever fails, the proof would need repair even though the theorem statement might survive.
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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

2 major / 4 minor

Summary. The paper proves three results about square-free monomial ideals: (1) every square-free monomial ideal in K[x1,x2,x3,x4,x5] has the strong persistence property (Theorem 3.18); (2) a necessary condition for a minimal counterexample to the Conforti-Cornuéjols conjecture, namely that every square-free monomial v in I^ℓ must lie in p^{ℓ+1} for some minimal prime p (Corollary 4.8); and (3) a necessary and sufficient condition for the linear combination L = xiI + xjJ to be normally torsion-free, namely that both xiI + J and I + xjJ are normally torsion-free (Theorem 5.5). The proofs are largely combinatorial case analyses, supplemented by known results on strong persistence, symbolic powers, and normally torsion-free ideals, and by explicit Macaulay2 computations in examples.

Significance. If correct, Theorem 3.18 is a substantial structural result: it identifies n = 5 as the exact boundary for the strong persistence property among square-free monomial ideals, complementing the known six-variable counterexample. Theorem 5.5 gives a clean, checkable criterion for normally torsion-free linear combinations and is used to reprove that the 4-uniform hypergraph H3(C8) is Mengerian. The paper is careful to give concrete examples and to use Macaulay2 to verify computational claims, and the proofs are self-contained in their use of elementary monomial ideal techniques. However, the case analysis in Theorem 3.18 is long enough that completeness of the case split must be checked explicitly, and one step in Theorem 5.5 is asserted without the needed monomial-degree justification.

major comments (2)
  1. [§3, Theorem 3.18, Case 4] The classification of three 2-edges into types I, II, and III is not exhaustive. On X = {x1,...,x5}, take the 2-edges e1={x1,x2}, e2={x3,x4}, e3={x1,x5} and the 3-edges {x2,x3,x5}, {x2,x4,x5}. This is a valid clutter satisfying the standing assumptions of Case 4, including e_i^c ∉ E(C) and |e'∩e_i| = 1 for every 3-edge e', but the three 2-edges are none of the listed types. The subsequent analysis treats only type III and type I, so this configuration is omitted from the proof of Theorem 3.18. The omission is repairable: e1 and e2 are disjoint and every remaining edge meets e1, so Proposition 3.11 applies directly; the proof should state this case explicitly rather than asserting the three-type classification.
  2. [§5, Theorem 5.5, inclusion (23)] The deduction of inclusion (23) is not justified in the text. From x_j^{k-α}W ∈ (I+x_jJ)^k with W ∈ I^α, the paper concludes W ∈ J^{k-α}I^α. This conclusion is valid: since x_j is coprime to every generator of I and J, the monomial W has no factor x_j, and expanding (I+x_jJ)^k = Σ_{r=0}^k x_j^{k-r} I^r J^{k-r}, only the r = α summand has x_j-exponent k-α and can divide x_j^{k-α}W, forcing W ∈ J^{k-α}I^α. However, this monomial-degree argument is not supplied, and the same missing justification is needed when factoring x_i^α in the display immediately after (23). Because this step is load-bearing for the iff statement, it should be written out.
minor comments (4)
  1. [§4, Lemma 4.2] In the proof of Lemma 4.2, the symbol m is first used as an integer (|G(I^s)|) and then is used as a prime ideal ('If p = m', 'p ⊊ m'); this makes the argument very hard to read and should be rewritten with distinct notation.
  2. [§3, Theorem 3.18, Case 4] In the sentence 'In particular, if e′ ∈ E(C) with |e′| = 3, then |e∩ei| = 1 for each i', the first edge should be e′; also the notation e_i^c should be introduced before first use.
  3. [§3, Theorem 3.18, Case 4] The phrases 'we consider {y1,y2} = {x1,x5} in Proposition 3.13' and 'in Proposition 3.14' are imprecise, since those propositions are stated in terms of sets X and Y; the identification of variables should be spelled out.
  4. [Throughout] There are several English and grammar slips, e.g. 'Due to p is arbitrary' in Proposition 4.1, 'the proof is over' in Lemma 4.2, and 'the only possible for the edges' in Theorem 3.18; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are derived from prior published theorems and the paper's own monomial computations, not from the conclusions being proved.

full rationale

I walked the derivation chains of the three main results. Theorem 3.18 is proved by reducing a 5-variable square-free ideal to the existing 4-variable result (Theorem 2.3 from [36]), cone and two-edge reductions (Propositions 3.5 and 3.6), graph edge-ideal strong persistence, and the new 3-uniform lemmas. None of these inputs defines strong persistence in terms of the theorem's conclusion; the lemmas are proved directly from membership in (I^{k+1}:I). The dependence on [36] and [34] is self-citation only in the sense that a current coauthor is also an author of those papers; those citations are published, parameter-free theorems with stated assumptions that do not contain Theorem 3.18, so under the review rules they are independent evidence rather than circular imports. Section 4's criterion (Corollary 4.8) follows contrapositively from Theorem 4.6, whose proof uses the colon equalities (p^t:v)=p^{t-ℓ} and Proposition 2.10; these are proved in the paper or in [31] and do not presuppose the target normally torsion-free conclusion. Section 5's Theorem 5.5 is an iff statement; the necessity direction uses contraction/deletion stability from [37], and sufficiency derives L^k = L^{(k)} by binomial expansion and inclusion (23). The flagged inclusion (23) is a monomial ideal-theoretic step, potentially a soundness gap, but not circular: it is not obtained by assuming L is normally torsion-free, and the claimed equality L^{(k)}=(xiI+xjJ)^k is derived, not assumed. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from same-author work to forbid alternatives, and no ansatz is smuggled via citation. I therefore find no circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No empirical free parameters or newly invented entities appear. The mathematical argument is a chain of established monomial ideal facts plus a new combinatorial case analysis; the only nonstandard input is the hidden colon-division step in Theorem 5.5.

assumptions (4)
  • standard math Standard facts on associated primes, symbolic powers, and Brodmann stabilization.
    Used throughout Sections 3 through 5; quoted from Brodmann, Villarreal, and standard references.
  • domain assumption Square-free monomial ideals are identified with clutters.
    Section 3 translates the ideal-theoretic strong persistence property into combinatorial statements about hypergraphs.
  • standard math Theorem 2.12 equivalence of normal torsion-freeness, the Mengerian property, the max-flow min-cut property, and ordinary/symbolic power equality.
    Quoted as Villarreal's Theorem 14.3.6 and used repeatedly in Sections 4 and 5.
  • ad hoc to paper The hidden colon implication in Theorem 5.5 leading to inclusion (23).
    The proof jumps from x_j^(k-alpha) W in (I + x_j J)^k to W in J^(k-alpha) I^alpha without proving the relevant colon computation. This is a load-bearing assumption not derived in the paper.

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Pith. "Pith review of On the strong persistence property and normally torsion-freeness of square-free monomial ideals." pith.science (2026). https://pith.science/paper/67HRRIPH

@misc{pith2026241114227,
  author       = {Pith},
  title        = {Pith review of: On the strong persistence property and normally torsion-freeness of square-free monomial ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/67HRRIPH}},
  note         = {Machine review of arXiv:2411.14227}
}
abstract

In this paper, we first show that any square-free monomial ideal in $K[x_1, x_2, x_3, x_4, x_5]$ has the strong persistence property. Next we will provide a criterion for a minimal counterexample to the Conforti-Cornuejols conjecture. Finally we give a necessary and sufficient condition to determine the normally torsion-freeness of a linear combination of two normally torsion-free square-free monomial ideals.

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