{"id":"a5b35575-e4c0-4935-aa25-3902916c4095","arxiv_id":"2412.03468","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit linear quotient orderings exist for all powers of anticycle edge ideals and for powers of any quadratic monomial ideal with linear quotients.","lead":"This paper proves that all higher powers of the edge ideals of anticycles have linear quotient orderings, even though the edge ideals themselves do not, settling an open question. It also gives an explicit recipe for ordering powers of any quadratic monomial ideal that already admits linear quotients.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof is internally consistent, and the only exposed dependency is the cited [21, Prop 3.1] used for the antipath sub-ordering, which deserves an independent check but is not evidently wrong.","rationale":"The reader's weakest assumption correctly identifies the reliance on [21, Proposition 3.1] as the most exposed point of Theorem 5.5, and I agree that this proposition is the one external input on which the proof depends most heavily. However, my own check of the surrounding argument did not reveal an internal gap: the case split in Case 3 is exhaustive, the degree-counting arguments in Subcases 3.1-3.4 are consistent, and the handling of the moved generator in Case 2 is careful. Since the cited proposition is a standard kind of external result and the paper gives a computational sanity check for a small case, I do not find a load-bearing objection that would justify changing the acceptance verdict. The lack of a full proof of [21, Prop 3.1] is best addressed by an independent verification rather than by rejecting or rewriting the paper. The constructive part of the paper, Theorem 3.11, is also supported by a clear proof and by the whisker-graph applications, so the overall central claims appear credible.","tokens_in":25225,"tokens_out":38684,"duration_ms":342967,"concrete_test":"Verify [21, Prop 3.1] independently: for n=6 and n=7, k=3 and k=4, encode the minimal generators of I(P_{n-1})^k and test whether the lexicographic order x_1 > x_2 > ... > x_{n-1} is a linear quotient ordering, using the authors' isLinear method or an independent Macaulay2/Singular implementation. Also test the full ordering O(3)_6 and O(3)_7 from Construction 5.1. If the lex order fails in any tested case, Theorem 5.5 is unsupported; if it passes, the cited dependency is validated in the regimes most likely to expose an error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I cannot identify a load-bearing concern that would change the reader's verdict. The central claim of Theorem 5.5 is supported by a long but coherent case analysis. I checked the exhaustiveness of the cases in Case 3: after fixing the first differing exponent in the lex order (x_n, x_2, ..., x_{n-1}, x_1), Subcases 3.1-3.4 cover every possibility, and the degree-counting contradictions in each subcase are valid. For example, the inequality in (5.5.3) is justified because every edge incident to x_n uses one of x_2, ..., x_{n-2}, so alpha_n <= alpha_2 + ... + alpha_{n-2}. The constructions in Case 1 and Case 2 also respect the move of (x_1 x_{n-1})^k after the distinguished generator. The only genuinely exposed point is the invocation of [21, Proposition 3.1] in Case 2, which supplies the linear quotient property for the lexicographic order on I(P_{n-1})^k for every k. This proposition is not proved in the present paper, and the included Macaulay2 verification only covers n=6, k=2, so a misquotation or a restriction to k=2 would damage Theorem 5.5. However, citing a prior result is standard practice, and I found no evidence that the cited proposition is false or misapplied. The general construction in Theorem 3.11 is independently verified by the lemmas in Section 3 and appears sound. Thus I do not regard the external dependency as a defect sufficient to change the verdict.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies linear quotient orderings for powers of edge ideals. It constructs, in Construction 3.3, a revlex ordering of edge decompositions and proves in Theorem 3.11 that every power of a quadratic monomial ideal with linear quotients again admits linear quotients; this gives a constructive proof of a known result of Herzog, Hibi, and Zheng. In Section 4 the ordering is applied to whisker graphs, yielding explicit projective dimension and Betti number formulas in Theorem 4.5. The main result is Theorem 5.5, which gives an explicit ordering O_n^(k) showing that all powers I(A_n)^k with n ≥ 5 and k ≥ 2 admit linear quotients, although I(A_n) itself does not; this answers Hoefel and Whieldon's Question 5.1. Example 3.5 gives a counterexample to a conjecture of Erey et al. Section 6 provides Macaulay2 methods and verifications for the constructions.","tokens_in":25542,"tokens_out":16629,"duration_ms":142008,"significance":"If the results are correct, Theorem 5.5 resolves an open question in the affirmative and supplies the first family of gap-free graphs whose edge ideals fail to have linear quotients while all higher powers do. Theorem 3.11 is a useful constructive strengthening of the Herzog-Hibi-Zheng theorem, and the whisker graph formulas in Theorem 4.5 are explicit and checkable. The paper is notable for giving concrete, reproducible computational support: the Macaulay2 code is available in [30], and Example 6.1 verifies the anticycle ordering for A_6^2. I read the proofs as internally consistent: the case analysis in Theorem 5.5 is lengthy but the subcases are exhaustive as presented, and the counterexample in Example 3.5 is valid under the stated lex convention. The only external input that is load-bearing for the main theorem is the cited [21, Proposition 3.1]; I found no evidence that it is misquoted or inapplicable.","major_comments":[],"minor_comments":[{"comment":"The proof of Case 2 depends on the cited [21, Proposition 3.1] for the statement that the lex ordering x_1 > x_2 > ... > x_{n-1} gives a linear quotient ordering of I(P_{n-1})^k for every k. Since this is the single most delicate external dependency and the included Macaulay2 check covers only n = 6, k = 2, the authors should state the proposition explicitly, with the exact hypotheses on n and k, to make the dependency transparent.","section":"§5, Theorem 5.5, Case 2"},{"comment":"The statement of Theorem 4.5 does not specify the allowed ranges of r and ℓ. If r = ℓ = 0 is allowed, the graph W_{0,0} is a single edge and the claimed formula gives pd = 0 for k ≥ 2 instead of the correct value 1; the authors should add the standing assumption r + ℓ ≥ 1 (or r, ℓ ≥ 1, as intended).","section":"§4, Theorem 4.5"},{"comment":"The notation I(P_n) for the antipath is nonstandard and can be confused with the edge ideal of an ordinary path. The authors should use a clearer notation such as I(\\overline{P_n}) or explicitly remind the reader at every occurrence that P_n denotes the antipath.","section":"§5, opening paragraph"},{"comment":"The Macaulay2 output of getQuotients is difficult to read in print because of the line breaks and spacing; reformatting the output as a vertical list or table would improve readability and make the verification easier to follow.","section":"§6, Example 6.1"}],"recommendation":"accept","confidential_remarks":"The paper is well within the scope of the journal and the central claims appear sound. The only point I would flag for the editor is that Theorem 5.5 rests on one cited result, [21, Proposition 3.1], which is not reproduced in the manuscript; I do not consider this a blocking issue, but a statement of the cited proposition would make the paper more self-contained."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read this carefully and agree with the ACCEPT verdict. The main news is Theorem 5.5: the second and higher powers of anticycle edge ideals admit linear quotient orderings even though the edge ideals themselves do not. That answers an open question of Hoefel and Whieldon and gives the first gap-free graphs whose edge ideals have this property on all powers. The ordering in Construction 5.1 is explicit and surprisingly simple, and the Macaulay2 verification for A6^2 is a useful independent check.\n\nThe paper also does something genuinely useful in Section 3. Theorem 3.11 constructs an explicit revlex ordering on edge decompositions proving that powers of any quadratic monomial ideal with linear quotients again have linear quotients. This was known existentially, but the previous arguments went through the Rees algebra or the x-condition; here the ordering depends only on the original linear quotient ordering. Example 3.5 is a valid counterexample to [12, Conjecture 4.1] under the stated lex convention, and the whisker graph Betti formulas in Theorem 4.5 are new and follow cleanly from Lemma 4.4.\n\nThe soft spot is the proof of Theorem 5.5 itself. It is a long case analysis, and the exhaustiveness is asserted rather than transparently organized. I checked the case split in Case 3: after fixing the first differing exponent in the lex order, Subcases 3.1–3.4 do cover every possibility, and the degree-counting contradictions are sound. The genuinely exposed point is Case 2, which relies on [21, Proposition 3.1] for the antipath powers: that the lex order x1 > x2 > ... > x_{n-1} is a linear quotient ordering of I(P_{n-1})^k for every k. That proposition is not proved in this paper, and the Macaulay2 check only covers n=6, k=2. If it were misquoted or had a hidden k restriction, Theorem 5.5 would need repair. But citing a prior result is standard, and I see no sign it is false or misapplied. So this is a 'verify at referee time' item, not a defect.\n\nThe computational package (isLinear, getQuotients, findLinearOrderings) with code on GitHub is a real asset for others working on this problem. The paper is aimed at commutative algebraists studying edge ideals, linear quotients, and resolutions of powers; it gives constructive tools and a clean new example. I would bring it to a reading group and would cite it.\n\nRecommendation: send to a serious referee. If I were the editor, I would ask the referee to check the invocation of [21, Proposition 3.1] and spot-check Subcase 3.3, but I expect the paper to pass. It is a solid advance in a specialized area, not a foundational breakthrough, and it deserves publication in a good commutative algebra journal.","headline":"Solid, publishable paper: explicit linear quotient ordering for powers of anticycle edge ideals resolves Hoefel–Whieldon, plus a clean constructive proof for all quadratic monomial ideals with linear quotients; the main proof is long but sound, with one external dependency worth an independent check.","tokens_in":26079,"tokens_out":1876,"would_cite":true,"duration_ms":16977,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D02","13F55","13P20","05E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for every anticycle graph on five or more vertices, every power of its edge ideal beyond the first admits an explicit ordering with linear quotients, even though the edge ideal itself does not.","keywords":["edge ideals","linear quotients","powers of ideals","anticycle graphs","gap-free graphs","whisker graphs","Betti numbers","projective dimension"],"falsifier":"For a fixed $n\\ge5$ and $k\\ge2$, compute the ideals $((\\text{earlier generators}):(\\text{current generator}))$ for the ordering of Construction 5.1 and inspect their minimal generators. The theorem is false as soon as any quotient contains a minimal generator that is not a single variable; an independent finite check for $A_6^3$ or $A_7^2$ would settle it. Equivalently, one pair $M_1$ preceding $M_2$ with no $M_3$ satisfying the divisibility condition of Lemma 2.3 disproves the claim.","tokens_in":25023,"feed_emoji":"📐","tokens_out":12796,"duration_ms":113191,"temperature":0.7,"pith_summary":"An ideal with linear quotients can be resolved with linear maps, and its Betti numbers can be read off from the sizes of certain quotients. This paper asks which powers of edge ideals inherit that property. The central answer is that the edge ideal of an anticycle graph—the complement of a cycle on at least five vertices—does not have linear quotients, but its second and every higher power do, via an explicit and simple generator ordering. That settles an open question and supplies the first family of gap-free graphs whose edge ideals behave this way. The same paper gives a general construction showing that whenever a quadratic monomial ideal has linear quotients, all of its powers do, and it derives closed formulas for projective dimension and Betti numbers of powers of whisker graphs.","feed_headline":"Anticycle edge ideals gain linear quotients in every higher power","feed_subtitle":"Explicit ordering answers an open question: first gap-free graphs whose powers, not their ideals, have linear quotients.","key_machinery":"The machinery is an explicit ordering of minimal generators, verified by a variable-divisibility criterion. Lemma 2.3 says an ordering has linear quotients exactly when, for every earlier/later pair $(M_1,M_2)$, some earlier generator $M_3$ has $M_3/\\gcd(M_3,M_2)$ a variable dividing $M_1/\\gcd(M_1,M_2)$. Against that criterion, the paper sets Construction 3.3 for quadratic ideals—formal products of $k$ edges ordered by reverse lexicographic order on their exponent vectors, with duplicates removed—and Construction 5.1 for anticycle powers, a two-block ordering whose second block reuses the antipath power ordering and whose one transposition (moving $(x_1x_{n-1})^k$ after $D$) repairs the only failure of the naive concatenation. Lemma 3.10 partitions all pairs of generators into three cases, and Theorem 5.5's proof checks each case to show every quotient is variable-generated.","core_discovery":"The paper proves two theorems. Theorem 5.5: for $n\\ge5$ and $k\\ge2$, ordering the minimal generators of $I(A_n)^k$ by Construction 5.1—first the generators divisible by $x_n$ in lex order with $x_n>x_2>\\cdots>x_{n-1}>x_1$, then the rest in lex order $x_1>x_2>\\cdots>x_{n-1}$, with $(x_1x_{n-1})^k$ moved directly after the distinguished generator $D=(x_1x_{n-1})^{k-1}(x_2x_{n-1})$—yields a linear quotient ordering. Because $I(A_n)$ itself is known not to have linear quotients, this answers the open question in the affirmative and provides the first gap-free graphs whose edge ideal powers have linear quotients while the ideal itself does not. Theorem 3.11: if a quadratic monomial ideal $I(G)$ has a linear quotient ordering, then the reverse-lexicographic ordering of the formal edge products $m_1^{\\alpha_1}\\cdots m_r^{\\alpha_r}$, with repeated vertex monomials deleted, is a linear quotient ordering of $I(G)^k$ for every $k$. As a consequence the paper computes explicit projective dimension and Betti numbers for powers of whisker graph edge ideals.","pith_inferences":["The block-ordering strategy may extend to other graphs whose edge ideals are nearly antipath-like: the only place the anticycle combinatorics enters is the verification that the two blocks $F$ and $S$ interact, so any graph whose ideal quotients match those blocks would inherit the theorem.","The revlex formal-edge ordering in Theorem 3.11 is algorithmic; applying it to other tree or chordal families would yield projective dimension and Betti number formulas analogous to the whisker ones, simply by counting quotient sizes.","The counterexample to Conjecture 4.1 suggests that the efficient-ordering notion in that conjecture is too rigid; a quotient-aware invariant, rather than the ordering alone, may be the right object to track across powers.","Since an anticycle edge ideal is far from linear quotients while its powers are not, failure of linear quotients in an edge ideal may be a poor predictor of failure in its powers, so conjectures phrased only in terms of the ideal itself may need reformulation in terms of the powers."],"forward_implications":["For every $n\\ge5$ and $k\\ge2$, the anticycle power $I(A_n)^k$ has a linear resolution, not merely a linear quotient ordering, because linear quotients feed the mapping-cone resolution.","Anticycles become the first known gap-free graphs whose edge ideals fail linear quotients while all second and higher powers have them, making them a test case for the conjecture that high powers of gap-free edge ideals have linear resolutions.","Any quadratic monomial ideal that admits linear quotients has all of its powers admitting linear quotients, with an explicit ordering that depends only on the original ordering; this makes the Betti numbers of such powers computable from quotient counts.","For whisker graphs $W_{r,\\ell}$, the projective dimension of $I(W_{r,\\ell})^k$ is $r+\\ell$ for $k\\ge2$ and $\\max(r,\\ell)$ for $k=1$, with closed binomial formulas for all Betti numbers.","The paper's Example 3.5 gives a quadratic ideal whose square's lexicographic ordering fails the linear quotient test, disproving Conjecture 4.1 of a related preprint about efficient orderings and powers."],"supporting_citations":[{"why":"Establishes the square-power case for anticycles, poses the question this paper answers, and supplies the antipath ordering result used in Case 2 of the main proof.","marker":"[21]"},{"why":"Proves the equivalence between linear resolutions and linear quotients for quadratic monomial ideals, the backdrop for asking which powers inherit linear quotients.","marker":"[19]"},{"why":"Provides the quotient criterion used in Lemma 2.3 and the earlier existence result for powers that Theorem 3.11 makes constructive.","marker":"[18]"},{"why":"Gives the mapping-cone construction that turns a linear quotient ordering into an explicit resolution and Betti numbers.","marker":"[20]"},{"why":"Supplies the formula for projective dimension and Betti numbers of an ideal with linear quotients, applied to whisker graph powers.","marker":"[29]"},{"why":"Gives the star-graph formulas that the whisker graph theorem generalizes.","marker":"[13]"},{"why":"States the gap-free conjecture that makes anticycle powers a significant test case.","marker":"[26]"},{"why":"Contains the conjecture that Example 3.5 disproves and is discussed as related work.","marker":"[12]"}],"fun_headline_variants":["Anticycle edge ideal powers finally get linear quotient orderings","First gap-free graphs with linear quotients in edge ideal powers","Answering Hoefel-Whieldon: powers of edge ideals have linear quotients","Explicit orderings show powers of anticycle edge ideals have linear quotients"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The main theorem inherits the previously proved fact that every power of the antipath's edge ideal has linear quotients in lexicographic order; if that cited fact is wrong or misquoted, the verification of the second block of the anticycle ordering collapses.","fun_headline_variants_meta":{"raw":{"variants":["Anticycle edge ideal powers finally get linear quotient orderings","First gap-free graphs with linear quotients in edge ideal powers","Answering Hoefel-Whieldon: powers of edge ideals have linear quotients","Explicit orderings show powers of anticycle edge ideals have linear quotients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000779,"raw_usage":{"total_tokens":3437,"prompt_tokens":936,"completion_tokens":2501,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":552,"completion_tokens_details":{"reasoning_tokens":2421}},"tokens_in":552,"tokens_out":2501,"duration_ms":17214,"temperature":1.0,"reasoning_tokens":2421,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:23:53.539629+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed $n\\ge5$ and $k\\ge2$, compute the ideals $((\\text{earlier generators}):(\\text{current generator}))$ for the ordering of Construction 5.1 and inspect their minimal generators. The theorem is false as soon as any quotient contains a minimal generator that is not a single variable; an independent finite check for $A_6^3$ or $A_7^2$ would settle it. Equivalently, one pair $M_1$ preceding $M_2$ with no $M_3$ satisfying the divisibility condition of Lemma 2.3 disproves the claim.","supporting_citations":[{"cited_title":"Linear Quotients of the Square of the Edge Ideal of the Anticycle","cited_arxiv_id":"1106.2348","evidence_quote":"Establishes the square-power case for anticycles, poses the question this paper answers, and supplies the antipath ordering result used in Case 2 of the main proof."},{"cited_title":"Herzog, T","cited_arxiv_id":null,"evidence_quote":"Proves the equivalence between linear resolutions and linear quotients for quadratic monomial ideals, the backdrop for asking which powers inherit linear quotients."},{"cited_title":"Herzog and T","cited_arxiv_id":null,"evidence_quote":"Provides the quotient criterion used in Lemma 2.3 and the earlier existence result for powers that Theorem 3.11 makes constructive."},{"cited_title":"Herzog and Y","cited_arxiv_id":null,"evidence_quote":"Gives the mapping-cone construction that turns a linear quotient ordering into an explicit resolution and Betti numbers."},{"cited_title":"Sharifan and M","cited_arxiv_id":null,"evidence_quote":"Supplies the formula for projective dimension and Betti numbers of an ideal with linear quotients, applied to whisker graph powers."},{"cited_title":"Ferr`o, M","cited_arxiv_id":null,"evidence_quote":"Gives the star-graph formulas that the whisker graph theorem generalizes."},{"cited_title":"Nevo and I","cited_arxiv_id":null,"evidence_quote":"States the gap-free conjecture that makes anticycle powers a significant test case."},{"cited_title":"Gapfree graphs and powers of edge ideals with linear quotients","cited_arxiv_id":"2412.06467","evidence_quote":"Contains the conjecture that Example 3.5 disproves and is discussed as related work."}],"review_version":1}