{"id":"2e952db8-3c32-4e7e-8de4-828a83b18d28","arxiv_id":"1908.10702","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"For every n and d there is a monomial ideal in n variables with |G(I)| > |G(I^i)| for all 2 ≤ i ≤ d.","lead":"This paper constructs monomial ideals whose powers are generated by fewer monomials than the ideal itself, for any number of variables and for arbitrarily many powers at once. It disproves a natural conjecture that the number of generators must grow when an ideal is squared.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the construction is sound and the only unproved supporting facts are elementary and true.","rationale":"The reader's ACCEPT with high confidence matches my assessment. I checked the main containment Lemma 2.1 and Corollary 2.2 directly: x_i^{4t}q and x_i^{2t}μq are visibly in J^2, and q^2∈J^2; the parity split in Corollary 2.2 is sound because Q is principal. The antichain conditions for the added cross-section monomials are exactly right: equal total degree rules out mutual divisibility, and all exponents in [2t,3t−1] make them incomparable with the skeleton generators. The reader's weakest assumption is accurate: the only unproved supporting facts are t-invariance of |G(J^i)| and unboundedness of central cross-section counts. Both are elementary and true. The abstract's “all i≤d” overstatement is real but purely cosmetic: the theorem is intended and proved for i≥2, and the body states it correctly. I also see no circularity or parameter fitting; t is chosen after A(n,d) is fixed. Section 3 is an independent improvement and is not needed for the main claim. Since no substantive concern survives, the verdict should remain ACCEPT and no adjustment is needed.","tokens_in":8592,"tokens_out":8584,"duration_ms":84548,"concrete_test":"Run an independent verification for n=2, d=6 and n=3, d=3 using t=1 and t=2: compute |G(J^i)| for each i≤d in a monomial ideal computer algebra system; if the counts differ between t=1 and t=2, t-invariance fails. Also enumerate the central cross-section of [2t,3t−1]^n for t=7 through 12; the counts should grow to at least A(n,d)−2n+1. If both checks pass, the construction stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central construction. Section 2 correctly reduces the problem to adding antichain monomials from Q\\J that leave all higher powers unchanged. Corollary 2.2 follows from JQ⊆J^2 and Q^2⊆J^2, and the parity split on j is valid. The added central cross-section monomials lie in [2t,3t−1]^n, hence in Q\\J, are mutually non-dividing because they share one total degree, and are incomparable with the 2n skeleton generators. Therefore G(I) has exactly 2n+s elements. The two facts flagged by the reader—t-invariance of |G(J^i)| and unbounded growth of the central cross-section count—are stated without proof, but both are correct: multiplying every exponent vector by t gives a bijection on minimal generators, and the central multinomial coefficient grows polynomially in t. The only genuine defect is verbal: the abstract's “for all i≤d” is false at i=1, while the body correctly states i≥2; this does not affect the construction or the intended theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8751,"tokens_out":17447,"duration_ms":167815,"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.","major_comments":[],"minor_comments":[{"comment":"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":"Abstract"},{"comment":"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":"Section 2"},{"comment":"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":"Section 2"},{"comment":"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":"Section 2, Corollary 2.2"},{"comment":"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":"Section 2, Lemma 2.3"},{"comment":"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.","section":"Section 3, Theorem 3.4"},{"comment":"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.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The main construction is sound and the paper is a worthwhile contribution. The referee's only substantive request is to supply the two elementary justifications in Section 2 and to fix the abstract's range of i. Once those are addressed, the paper is suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You asked about Gasanova's note. The main result is solid and genuinely new: for any n and d, there is an m-primary monomial ideal with |G(I)| > |G(I^i)| for every i = 2,...,d. That is the first construction covering all n and all finite depths at once, not just the two-variable square case from Eliahou–Herzog–Saem. The skeleton-plus-cross-section mechanism is the right idea and it works: Lemma 2.1 and Corollary 2.2 correctly show that adding monomials from Q leaves all higher powers unchanged, and the antichain check via a common total degree is clean. The worked examples in Section 2 are convincing and match the claimed counts.\n\nSection 3 is also a real improvement, not a filler. Dropping the redundant conditions from the tiny-squares theorem and proving the nine-generator conclusion from five conditions (A,B,B*,C,C*) is a strict strengthening, and the duality explanation clarifies why those five are the right ones. The three-parameter family in Example 3.5 is a useful illustration.\n\nSoft spots, in proportion: First, the abstract's \"for all i ≤ d\" is false at i = 1; the body correctly restricts to i ≥ 2. That is a verbal slip, but it should be fixed. Second, two load-bearing facts are stated without proof: that |G(J^i)| is independent of t, and that the central cross-section of [2t,3t−1]^n contains arbitrarily many lattice points as t grows. Both are true and elementary, so this is not a soundness problem—but for a self-contained paper they deserve a sentence or a short lemma. Third, the statement that \"we expect |G(I^2)| > |G(I)|\" is disproven is slightly misleading: the expectation was already disproven in [1] for n=2; this paper generalizes that counterexample. The novelty is the generalization, which the text makes clear enough.\n\nCitation pattern is fine: [1] and [3] are the right prior work, [2] is standard. No circularity, no fitted parameters. The construction is explicit and reproducible. This is exactly the kind of short, verifiable counterexample paper that deserves a referee. I would accept after minor revisions.\n\nRecommendation: send to peer review. It is not a desk reject.","headline":"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.","tokens_in":9339,"tokens_out":998,"would_cite":true,"duration_ms":11485,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F20","13A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any number of variables, a monomial ideal can have more generators than its first d powers","keywords":["monomial ideals","powers of ideals","minimal generating sets","tiny squares","m-primary ideals","analytic spread","integer cross-sections","generator counts"],"falsifier":"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.","tokens_in":8325,"feed_emoji":"📉","tokens_out":4610,"duration_ms":49392,"temperature":0.7,"pith_summary":"The paper disproves the natural expectation that a non-principal monomial ideal should have strictly more minimal generators in its square than in itself, and that powers should keep growing. It proves that for any number of variables $n \\geq 2$ and any finite depth $d \\geq 2$, there exists an $\\mathfrak m$-primary monomial ideal $I$ (an ideal containing a power of every variable) whose minimal generating set is larger than the minimal generating sets of $I^2, I^3, \\ldots, I^d$. The construction is explicit: start with a fixed 'skeleton' ideal whose power-generator counts do not depend on a scaling parameter $t$, then add many monomials from a carefully chosen middle slice of a large cube without changing any power $I^i$ for $i \\geq 2$. The paper also improves a known two-variable criterion for when an ideal's square has exactly nine generators, reducing the required divisibility conditions from nine to five.","feed_headline":"Monomial ideals can shrink under their first d powers","feed_subtitle":"In every number of variables n and every depth d, there is an ideal whose first d powers each have fewer minimal generators than it does.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the two-variable construction and the nine-condition tiny-squares theorem that Section 2 generalizes to any number of variables and Section 3 improves.","marker":"[1]"},{"why":"Provides the asymptotic polynomial growth result for $|G(I^i)|$ that frames the question and explains why the failure occurs only for small powers.","marker":"[2]"},{"why":"Cited as prior exploration of how small $|G(I^i)|$ can be relative to $|G(I)|$, motivating the expectation that the paper disproves.","marker":"[3]"}],"fun_headline_variants":["Powers with fewer generators than the ideal itself","Monomial powers can shrink generator sets","For any depth d, an ideal with shrinking powers","Generator count drops in every power up to d","New ideals: each power has fewer minimal generators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Powers with fewer generators than the ideal itself","Monomial powers can shrink generator sets","For any depth d, an ideal with shrinking powers","Generator count drops in every power up to d","New ideals: each power has fewer minimal generators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000197,"raw_usage":{"total_tokens":1404,"prompt_tokens":1028,"completion_tokens":376,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":644,"completion_tokens_details":{"reasoning_tokens":305}},"tokens_in":644,"tokens_out":376,"duration_ms":4330,"temperature":1.0,"reasoning_tokens":305,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:36:56.488211+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Eliahou, J","cited_arxiv_id":null,"evidence_quote":"Supplies the two-variable construction and the nine-condition tiny-squares theorem that Section 2 generalizes to any number of variables and Section 3 improves."},{"cited_title":"Herzog and T","cited_arxiv_id":null,"evidence_quote":"Provides the asymptotic polynomial growth result for $|G(I^i)|$ that frames the question and explains why the failure occurs only for small powers."},{"cited_title":"Herzog, M","cited_arxiv_id":null,"evidence_quote":"Cited as prior exploration of how small $|G(I^i)|$ can be relative to $|G(I)|$, motivating the expectation that the paper disproves."}],"review_version":1}