REVIEW 7 minor 3 references
Monomial ideals with arbitrarily high tiny powers in any number of variables
T0 review · 0 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read For any number of variables, a monomial ideal can have more generators than its first d powers
desk verdict A clean, correct construction that generalizes tiny squares to any number of variables and any finite depth; the only real defects are a sloppy abstract and two elementary facts stated without proof. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central mechanism is the skeleton-plus-filler construction: the skeleton $J$ has power-generator counts $|G(J^i)|$ that depend only on $n$ and $i$, not on $t$, and the filler monomials are taken from $Q = \langle \mu^2\rangle$, where $\mu = x_1^t\cdots x_n^t$. Lemma 2.1 shows $JQ \subseteq J^2$ and $Q^2 \subseteq J^2$, and Corollary 2.2 then gives $(J+Q')^i = J^i$ for every $i \geq 2$ and every $Q' \subseteq Q$. This means the added monomials are invisible in all higher powers. The monomials of $Q \setminus J$ are exactly the lattice points in $[2t,3t-1]^n$, and choosing them from a central integer cross-section of that cube keeps them mutually non-dividing. In the two-variable part, the improved tiny-square theorem uses a self-dual set of five divisibility conditions, together with a monotonicity lemma, to show that $I^2$ has at most nine generators.
What would settle it
Independently recompute $|G(J^i)|$ for the skeleton ideal at two different values of $t$ for some fixed $n$ and $d$; if the counts depend on $t$, the construction breaks. Alternatively, recompute Example 2.4: the paper predicts $|G(I)|=26$, $|G(I^2)|=9$, $|G(I^3)|=13$, $|G(I^4)|=17$, $|G(I^5)|=21$, and $|G(I^6)|=25$, so a single mismatch in a computer algebra system would refute the claim.
Extended reading notes
Core claim
For any integers $n,d \geq 2$ there is an $\mathfrak m$-primary monomial ideal $I \subset \mathbb K[x_1,\ldots,x_n]$ such that $|G(I)| > |G(I^i)|$ for every $i = 2,\ldots,d$. The proof constructs $I = J + Q'$, where $J = \langle x_1^{4t},\ldots,x_n^{4t}, x_1^{2t}\mu,\ldots,x_n^{2t}\mu\rangle$ with $\mu = x_1^t\cdots x_n^t$, and $Q'$ consists of monomials on a central integer cross-section of the cube $[2t,3t-1]^n$. The key identity is $(J+Q')^i = J^i$ for all $i \geq 2$, which follows from $JQ \subseteq J^2$ and $Q^2 \subseteq J^2$ where $Q = \langle \mu^2\rangle$. Because the number of integer points on that central cross-section grows without bound as $t$ grows, $t$ can be chosen large enough that $|G(I)|$ exceeds $A(n,d) = \max_{1 \le i \le d} |G(J^i)|$, while every tested power $I^i$ has the same generator count as $J^i$. Thus the generator count can drop immediately and stay low for any prescribed finite number of powers.
Load-bearing premise
The load-bearing premise is that the central slice of the cube $[2t,3t-1]^n$ contains arbitrarily many lattice points as $t$ grows, while the skeleton's power-generator counts stay fixed; if either of those elementary facts failed, the extra generators could not be added and the inequality would not follow.
Editorial extensions
If this is right
- For monomial ideals in any fixed number of variables $n \geq 2$, there is no general inequality $|G(I^i)| > |G(I)|$ for any fixed small $i$.
- The generator-count sequence $|G(I^i)|$ can begin with a strict drop of arbitrary finite length before the known asymptotic polynomial growth takes over.
- The construction gives explicit ideals with $|G(I)|$ arbitrarily large compared to the generator counts of all powers up to a prescribed $d$.
- In two variables, the paper provides a three-parameter family of ideals satisfying the improved five-condition criterion and therefore having exactly nine generators in their square.
- The five-condition theorem strengthens the earlier nine-condition result, showing that several of the old conditions were redundant.
Reading between the lines
- Editorial inference: because the identity $(J+Q')^i = J^i$ holds for every $i \geq 2$, the same ideal works for all $d$ simultaneously; the upper bound $d$ in the statement is only there because the proof records the maximum up to $d$.
- Editorial inference: the skeleton-filler method may transfer to other classes of ideals where a controlled subset of generators can be made 'invisible' in powers, potentially forcing generator counts to oscillate rather than grow monotonically.
- Editorial inference: the improved two-variable criterion suggests that self-dual divisibility conditions are the natural minimal hypothesis for pinning down the size of $I^2$, and the same dual-condition pattern might characterize other small-power generator sets.
- Editorial inference: algorithmic work on monomial ideals that assumes generator counts grow under powering should be adjusted, since even very large finite drops are possible before the asymptotic regime.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the minimal number |G(I^i)| of generators of powers of a monomial ideal I. Its main theorem (Section 2) constructs, for every n,d ≥ 2, an m-primary monomial ideal I in K[x_1,...,x_n] such that |G(I)| > |G(I^i)| for 2 ≤ i ≤ d. The construction starts with a skeleton ideal J generated by x_i^{4t} and x_i^{2t}μ, where μ = x_1^t...x_n^t, and then adds s monomials all lying on a central cross-section of [2t,3t-1]^n. Lemma 2.1 and Corollary 2.2 show that adding such monomials does not change any power J^i for i ≥ 2; Lemma 2.3 identifies exactly which monomials can be added. By taking t large enough, the number s of added monomials exceeds A(n,d)-2n, where A(n,d) = max_{i≤d} |G(J^i)|, so |G(I)| = 2n+s > A(n,d) ≥ |G(I^i)| for all 2 ≤ i ≤ d. Section 3 revisits and improves a theorem of Eliahou, Herzog, and Saem on planar monomial ideals with |G(I^2)| = 9, giving a shorter set of sufficient conditions. Several explicit examples in n = 2 and n = 3 are worked out with concrete generator lists and counts.
Significance. If the result is correct, and the referee finds no error in the central derivation, the paper settles a natural question in the negative: the expected growth |G(I^2)| > |G(I)|, and more generally |G(I^i)| growing with i, fails in every number of variables for arbitrarily high i. This generalizes the two-variable counterexample of [1] to arbitrary n. The main construction is explicit and self-contained: the key containment (J+Q')^i ⊆ J^i is proved by elementary divisibility, and the antichain conditions on the added monomials are checked carefully. The worked examples confirm the claimed counts. The paper also gives a cleaner, more symmetric set of conditions for the two-variable 'tiny squares' theorem. The only unproved supporting facts are two elementary assertions about the skeleton J and the central cross-section; both are true and easily supplied. This is a useful, citable contribution to the study of powers of monomial ideals.
minor comments (7)
- [Abstract] The abstract says the construction gives |G(I)| > |G(I^i)| for all i ≤ d, but this cannot hold for i = 1. The body of the paper correctly states the conclusion for 2 ≤ i ≤ d; please adjust the abstract accordingly.
- [Section 2] After defining J, the paper asserts that the number of generators of J^i depends only on i and n, not on t. This fact is used to define A(n,d), so it is load-bearing. Please add the one-line justification: multiplying every exponent vector by t gives an order-preserving bijection on the exponent vectors of generators, so the minimal generators of J_t^i are in bijection with those of J_1^i.
- [Section 2] The paper asserts that the number of integer points on a central cross-section of [2t,3t-1]^n can be made arbitrarily large for fixed n ≥ 2. This is also used in the construction. Please add a sentence explaining that this number is the central coefficient of (1+x+...+x^{t-1})^n and grows polynomially in t of degree n-1, hence is unbounded.
- [Section 2, Corollary 2.2] The notation 'Q' ⊆ Q' is informal: Q is an ideal, while Q' is later taken to be a set of monomials. Please clarify that Q' denotes the ideal generated by the chosen monomials, or state explicitly that (J+Q')^i means the sum of ideals.
- [Section 2, Lemma 2.3] In the ⊇ direction of the proof, 'every minimal generator of J has an exponent greater than or equal to 3t' is ambiguous: the generator x_i^{2t}μ has exponent t in other coordinates. What is meant is that each minimal generator has at least one exponent ≥ 3t, which is the property needed to prevent divisibility of monomials in [2t,3t-1]^n.
- [Section 3, Theorem 3.4] The proof concludes that |G(I^2)| ≤ 9 and cites [1] for the lower bound |G(I^2)| ≥ 9. Since the hypotheses here are weaker than those of [1], please state explicitly that the lower-bound argument in [1] does not use the additional conditions, or give a direct argument that the nine displayed monomials are incomparable.
- [Throughout] There are several typographical errors, for example 'att raction' in the abstract, 'the m back' in Example 2.5, and irregular spacing in 'w ill' and other words. A careful proofreading pass would improve the presentation.
Circularity Check
No significant circularity: the construction and proofs are self-contained, with all cited results external prior work.
full rationale
The central construction in Section 2 is not circular. The ideal I = J + Q' is built from an explicit skeleton J and extra monomials on a central cross-section of [2t, 3t-1]^n. Corollary 2.2 derives (J + Q')^i = J^i for i ≥ 2 directly from JQ ⊆ J^2 and Q^2 ⊆ J^2, both proved in Lemma 2.1. The extra monomials are chosen from Q \ J, so they add new minimal generators without changing higher powers. The threshold A(n,d) is computed from the skeleton J and is used only to choose t large enough; it is not a fitted parameter disguised as a prediction. The two facts stated without proof—that |G(J^i)| is independent of t and that the central cross-section has arbitrarily many points—are elementary, true, and not equivalent to the theorem's conclusion. Section 3 explicitly attributes the original theorem and the lower bound |G(I^2)| ≥ 9 to [1], which is external prior work; the improved conditions are proved independently from divisibility relations and Lemma 3.3 from [1]. There is no self-citation chain and no input is renamed as an output. The only notable defect is a verbal inconsistency in the abstract, which says |G(I)| > |G(I^i)| for all i ≤ d, whereas the body correctly states and proves the strict inequality only for i ≥ 2; this is a correctness/typographical issue, not circularity.
Assumptions & free parameters
free parameters (1)
- t (scaling parameter) =
chosen sufficiently large (e.g., t=22 in Example 2.4)
assumptions (5)
- standard math The minimal monomial generating set G(I) of a monomial ideal is unique.
- standard math For fixed n, |G(J^i)| does not depend on the scaling parameter t.
- standard math The central coefficient of (1+x+...+x^{t-1})^n grows without bound as t increases.
- standard math Lemma 3.3 from [1]: if f(v) divides f(v1) and f(v2) with v1 ≤ v2, then f(v) divides f(v') for all v1 ≤ v' ≤ v2.
- standard math For the ideals in Theorem 3.4, |G(I^2)| ≥ 9.
Cite this review
Pith. "Pith review of Monomial ideals with arbitrarily high tiny powers in any number of variables." pith.science (2026). https://pith.science/paper/MH65KOCB
@misc{pith2026190810702,
author = {Pith},
title = {Pith review of: Monomial ideals with arbitrarily high tiny powers in any number of variables},
year = {2026},
howpublished = {\url{https://pith.science/paper/MH65KOCB}},
note = {Machine review of arXiv:1908.10702}
}
abstract
Powers of (monomial) ideals is a subject that still calls attraction in various ways. Let $I\subset \mathbb K[x_1,\ldots,x_n]$ be a monomial ideal and let $G(I)$ denote the (unique) minimal monomial generating set of $I$. How small can $|G(I^i)|$ be in terms of $|G(I)|$? We expect that the inequality $|G(I^2)|>|G(I)|$ should hold and that $|G(I^i)|$, $i\ge 2$, grows further whenever $|G(I)|\ge 2$. In this paper we will disprove this expectation and show that for any $n$ and $d$ there is an $\mathfrak m$-primary monomial ideal $I\subset \mathbb K[x_1,\ldots,x_n]$ such that $|G(I)|>|G(I^i)|$ for all $i\le d$.
Figures
Reference graph
Works this paper leans on
Reviewed August 14, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.