{"id":"42247ec9-3e1a-4897-a307-d545baa645ab","arxiv_id":"2411.16364","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Polyomino ideals from ladder, weakly closed path, simple thin, and certain thin polyominoes are Knutson (hence radical), with some classes shown to be prime and Gröbner bases computed.","lead":"The paper proves that several new families of polyominoes (ladder, closed path, weakly closed path, simple thin, and some thin shapes) have Knutson ideals, which are automatically radical. It also shows some of these ideals are prime and computes Gröbner bases, extending what is known about when polyomino ideals are radical or prime.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Simple-thin Knutson proof hinges on Lemma 3.4, whose proof is deferred to [HH23] even though the paper claims a new strengthened version; without a proof the induction in Theorem 3.6 is unsupported.","rationale":"The reader's weakest assumption identifies exactly this gap. I considered whether Theorem 3.12's case analysis is a more serious problem, but that result is independently proved by Navarra [Nav24], and the paper's figures give reasonably explicit, if terse, coverage. Lemma 3.4, by contrast, is a claim of novelty: the authors state that the original lemma from [HH23] is insufficient and that a more general version is required, but then do not prove it. The entire simple-thin section depends on it, and it is also used as a template for later König-type arguments. Since the gap is a missing proof rather than an identified falsehood, the appropriate outcome is to keep the conditional acceptance pending a complete proof or an explicit reference.","tokens_in":84,"tokens_out":17396,"duration_ms":208861,"concrete_test":"Obtain [HH23] and check whether Lemmas 3.2 and 3.3 there imply Lemma 3.4 verbatim; if not, write a complete proof of Lemma 3.4. In particular, verify that the graded lexicographic order <' can be chosen to extend < and satisfy in<'(f')=xu xd and in<'(f'')=xa xv, and that f', f'' together with f1,...,f_{r-1} generate the ideal of P'. If the cited lemmas do not cover the 'edge only in P' hypothesis, Theorem 3.6 is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.1's Theorem 3.6, asserting that every simple thin polyomino is of König type, is proved by induction that repeatedly invokes Lemma 3.4. That lemma is explicitly introduced as a 'more general version' of [HH23, Lemma 5.3/5.4] needed because Example 3.3 shows the original lemma is insufficient. Yet its proof is a single sentence: 'The proof follows the argument of [HH23, Lemma 3.2, 3.3].' No argument is given that those cited lemmas actually cover the strengthened hypotheses—in particular, the condition that {a,b} is an edge of C 'only in P' and the requirement that the new monomial order <' extend < while forcing the two new initial terms. If Lemma 3.4 is false or its hypotheses are not met in the induction, Theorem 3.6 collapses, and with it Corollary 3.7 (simple thin polyominoes are Knutson). Since the paper's own example demonstrates that the original [HH23] lemma is not up to the task, the burden is on the authors to justify the strengthened version; the current text does not do so.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the study of Knutson ideals for polyomino ideals. It proves that the polyomino ideals of simple thin, closed path, weakly closed path, ladder, and certain thin polyominoes are Knutson, hence radical; in several cases it also computes reduced Gröbner bases and proves primality. The main bridge is Proposition 3.2, which converts König-type ideals with unmixedness/radicality into Knutson ideals. The later sections develop a filtration method based on anti-diagonal vertex sets (Discussion 4.2) and apply it to ladder polyominoes, thin polyominoes, and parallelogram polyominoes with a parallelogram removed.","tokens_in":30445,"tokens_out":11777,"duration_ms":114290,"significance":"If the gaps noted below are repaired, the paper is a substantial contribution to the radicality and primality theory of polyomino ideals. It gives a clean criterion (Proposition 3.2) for passing from König type to Knutsonness, and Section 4 develops a genuinely useful anti-diagonal filtration with explicit permutation lemmas. The paper also supplies checkable consequences (explicit Gröbner bases, primality of certain thin polyominoes) and makes appropriate use of prior work by Dinu–Navarra, Herzog–Hibi, Mascia–Rinaldo–Romeo, and others. The main weaknesses are that two load-bearing proofs are deferred to figures or to [HH23] and one geometric decomposition in Section 6 is asserted without proof. These are correctness risks rather than demonstrated errors.","major_comments":[{"comment":"Lemma 3.4 is load-bearing for Theorem 3.6 and is explicitly introduced as a strengthened version of [HH23, Lemma 5.4] needed because Example 3.3 shows the original lemma is insufficient. However, its proof is only the sentence 'The proof follows the argument of [HH23, Lemma 3.2, 3.3].' The manuscript does not show that the cited lemmas cover the strengthened hypotheses, in particular the condition that {a,b} is an edge of C only in P and the specification that the new order <' extends < and forces the two displayed initial terms. Since Theorem 3.6 and Corollary 3.7 collapse if this lemma is not available, the authors should either provide a complete proof or a precise reduction to [HH23] with every hypothesis checked.","section":"Section 3.1, Lemma 3.4"},{"comment":"The proof of Theorem 3.12 is presented as a case analysis conducted through Figures 8–11. The central assertions that the inductive labeling 'allows us to complete the labeling of all the vertices of P' and that the chosen generators f_i have pairwise coprime square-free initial terms are asserted rather than formally verified. A rigorous proof should enumerate the possible configurations, prove that the figures cover all weakly closed path shapes, and verify the stated order inequalities at every step, including the final step involving A_{n-4}, A_{n-3}, A_{n-2}, A_{n-1}. Since Corollary 3.14 depends on this theorem, the case analysis needs to be made checkable.","section":"Section 3.2, Theorem 3.12"},{"comment":"The proof of Theorem 6.2 rests on an unproved geometric decomposition of Q1: it asserts that if Q1 is not a polyomino, then Q1 is a union of polyominoes R, R_1^L,...,R_r^L, R_1^R,...,R_s^R satisfying the listed vertex-intersection conditions (1)–(3). No argument is given that this decomposition follows from the definition of Q1 in terms of the V_i filtration, nor is the analogous statement for Q2 proved. This decomposition is essential for applying Proposition 4.10 and Proposition 6.1. The authors should state and prove this decomposition as a lemma.","section":"Section 6, Theorem 6.2"},{"comment":"The primality conclusions in Corollary 5.2 and Proposition 5.4 depend on identifying the monomial order < of Discussion 4.2 with the orders <4 and <6 of [MRR22], respectively. The text only asserts these identifications. Since [MRR22, Corollary 3.3] is an external result whose hypotheses include the specific monomial order, the identifications are load-bearing and should be verified explicitly with reference to the definitions in [MRR22].","section":"Corollary 5.2 and Proposition 5.4"}],"minor_comments":[{"comment":"In the proof of Lemma 4.5, in the subcase 4.4.2 (1), the displayed equalities appear to contain a typo: they should probably read eσ(i)=σ(i) and eσ(j)=σ(j), not eσ(i)=σ(j), eσ(j)=σ(j).","section":"Lemma 4.5"},{"comment":"In the proof of Theorem 5.1, the symbol C_{i_k} is used both for the set of cells associated with V_{i_k} and for the single cell C_k obtained as C_{i_k} ∩ J. Separate notations would make the argument much easier to follow.","section":"Theorem 5.1 proof"},{"comment":"The sentence following Figure 19 is confusing: it says the displayed polyomino satisfies the hypothesis of Theorem 5.1 even though it contains a collection of cells isomorphic to Figure 18a. Since condition (1) of Theorem 5.1 is coordinate-specific, the intended convention about rotations/reflections should be spelled out.","section":"After Figure 19"},{"comment":"The proof states without justification that IP1 (resp. IP2) is a minimal prime ideal of (f1,...,f_{n-1}) (resp. (f0,...,f_{n-2})). A sentence explaining minimality, using the known height of these ideals, would remove an avoidable gap.","section":"Corollary 3.14"},{"comment":"The notation h1 = ∪_{i<b} f_i uses a union symbol where the product of polynomials is meant; the same applies to h2. This should be corrected to ∏_{i<b} f_i and ∏_{i>a} f_i.","section":"Theorem 6.2"},{"comment":"The table header 'IF it occurs ... THEN we refer to ...' is typographically awkward, and the surrounding text would be clearer if each row explicitly stated which rotations and reflections are allowed rather than relying on a single global phrase.","section":"Table 1"}],"recommendation":"major_revision","confidential_remarks":"The reader's conditional assessment is fair: the main technical gap is Lemma 3.4, whose proof is deferred to [HH23] even though the paper itself says the original version is insufficient. The other major concerns, the informal case analysis in Theorem 3.12 and the asserted decomposition in Theorem 6.2, appear fillable within the scope of the manuscript. I recommend major revision rather than rejection, because the central results are plausible and the missing arguments are localized."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read on Koley–Kotal–Veer. The paper is worth engaging with. It introduces ladder polyominoes and proves their polyomino ideals are Knutson, gives a new class of thin polyominoes whose ideals are Knutson and prime, shows weakly closed path polyominoes are of König type, and proves an extraction result for parallelogram polyominoes. The main technique—partitioning vertices into anti-diagonal sets and using determinants (Discussion 4.2)—is new and is the strongest part of the paper. Lemmas 4.7 and 4.8 do real work, and the proof of Theorem 4.1 is substantive. The paper also honestly identifies a gap in Herzog–Hibi's argument for simple thin polyominoes (Example 3.3) and notes that Navarra independently obtained the weakly closed path result. That is good scholarly practice.\n\nThe soft spots are real, though I do not think the main claims are wrong. The biggest issue is Lemma 3.4. The paper states it as a strengthened version of [HH23, Lemma 5.4], explicitly argues that the original lemma is insufficient, and then gives a one-sentence proof: 'The proof follows the argument of [HH23, Lemma 3.2, 3.3].' No argument shows that the cited lemmas actually cover the strengthened hypotheses—the 'only in P' edge condition and the extension of the monomial order. Theorem 3.6 and Corollary 3.7 depend on this lemma, so the claim that all simple thin polyominoes are Knutson is not fully supported as written. The authors should either prove Lemma 3.4 or point to a source that does.\n\nTwo other gaps are minor by comparison. Theorem 3.12's proof is a case analysis via Figures 8–11 with several subcases dismissed as 'one can conclude'; it is plausible but not fully formal. Theorem 6.2 asserts a geometric decomposition of Q1 without proof. These are presentation gaps, not fatal flaws.\n\nOn citation practice, the paper uses external published results appropriately. The reliance on [CNV24] for unmixedness is fine because that is an independent published result. No circularity or parameter fitting.\n\nWho should read this: people working on polyomino ideals, binomial ideals, and Knutson theory. It deserves a serious referee, but the referee should push for a complete proof of Lemma 3.4 and a more formal treatment of the case analyses. I would send it to peer review with a request for major revision, not desk-reject.","headline":"Real progress on Knutson and prime polyomino ideals, but a load-bearing lemma for simple thin polyominoes is left unproved, so the paper needs revision before acceptance.","tokens_in":30981,"tokens_out":2635,"would_cite":true,"duration_ms":26168,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E40","05B50","13P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the polyomino ideals of closed path, weakly closed path, simple thin, and ladder polyominoes are Knutson ideals, and that a restricted class of thin polyominoes is prime with an explicit reduced Gröbner basis.","keywords":["polyomino ideals","Knutson ideals","radical ideals","ideals of König type","Gröbner bases","prime ideals","thin polyominoes","ladder polyominoes"],"falsifier":"Run the induction of Theorem 3.6 on the polyomino of Figure 2: if, when the final cell is added, the asserted new generators do not have squarefree pairwise-coprime initial terms under the constructed order, the single-cell lemma and the theorem collapse. Separately, compute all S-pairs of the inner-interval binomials for the thin polyomino of Figure 19 under the stated reverse lexicographic order; any S-pair that fails to reduce to zero would refute the claimed reduced Gröbner basis and primality.","tokens_in":29971,"feed_emoji":"🧩","tokens_out":10518,"duration_ms":90733,"temperature":0.7,"pith_summary":"The paper sets out to prove radicality and primality of polyomino ideals by bringing Knutson ideals into polyomino combinatorics. It establishes that the polyomino ideal of every closed path, weakly closed path, simple thin, and ladder polyomino is Knutson, meaning it has a squarefree initial ideal and is therefore radical. For a restricted class of thin collections of cells it proves more: the ideal is prime and the reduced Gröbner basis consists exactly of the binomials attached to inner intervals. A separate theorem shows that when a parallelogram polyomino is cut out of another parallelogram polyomino, the leftover collection is Knutson under a non-crossing condition, with a Gröbner basis computed. The payoff is a structural certificate of radicality that also produces explicit Gröbner bases, aligning these binomial ideals with the behavior already known for determinantal and ladder ideals.","feed_headline":"Polyomino ideals of five families are Knutson, hence radical","feed_subtitle":"A binomial-ideal proof gives squarefree initial ideals, and explicit Gröbner bases for a prime thin class.","key_machinery":"The central objects are the polyomino ideal $I_P$, the binomial ideal generated by $x_a x_b - x_c x_d$ for every inner interval $[a,b]$ with anti-diagonal vertices $c,d$; the Knutson ideal, an ideal obtainable from a principal ideal $(f)$ with squarefree leading term through colons, sums, and intersections; and the ideal of König type, a height-$h$ ideal generated by $h$ homogeneous generators whose initial terms form a regular sequence. Proposition 3.2 is the bridge: for an unmixed radical ideal, König type with squarefree initial terms implies Knutson. In the ladder section the paper adds a structural tool: vertices are partitioned into anti-diagonal sets $V_k$, each $V_k$ yields a determinant $f_k$ of a matrix of variables, and the product $f = \\prod_k f_k$ has squarefree initial monomial equal to the product of all vertex variables. A transposition-pairing lemma shows each $f_k$ lies in the ideal of the sub-polyomino with one added cell, driving an induction that builds $I_P$ from Knutson pieces.","core_discovery":"On the paper's own terms, the discovery is that radicality of several classes of polyomino ideals can be certified by exhibiting them as Knutson ideals rather than by ad hoc initial-ideal computations. The proof chain is: certain polyomino ideals are ideals of König type; an unmixed radical ideal of König type whose chosen generators have squarefree initial terms is Knutson (Proposition 3.2); and Knutson ideals have squarefree initial ideals, hence are radical. Along the way the paper identifies a gap in the prior argument that simple thin polyominoes are of König type and supplies a generalized single-cell extension lemma to complete the induction. For the thin class satisfying three exclusions, it shows the inner-interval binomials themselves form the reduced Gröbner basis and the ideal is prime.","pith_inferences":["My inference: the anti-diagonal determinant construction suggests a general recipe for Knutson certificates on any polyomino admitting a vertex ordering with squarefree pairwise behavior, so the ladder argument may extend beyond ladders.","My inference: the transposition-pairing lemma behind the ladder proof resembles a straightening law for 2-minors; if it generalizes, the same induction could apply to ideals generated by $k$-minors of polyomino-shaped matrices.","My inference: the condition $I_{Q_1} + I_{Q_2} = I_P$ in Section 6 is plausibly equivalent to the cut between the removed parallelogram and the remainder having no interleaving cells, which computer search over small parallelogram extractions could test directly.","My inference: because Knutson ideals behave well under sums, the proofs give constructive radicality certificates; for the thin class the certificate is explicit enough to feed directly into a Gröbner basis computation."],"forward_implications":["Every closed path, weakly closed path, simple thin, and ladder polyomino ideal is radical, because Knutson ideals have squarefree initial ideals.","For the thin class of Theorem 5.1, the polyomino ideal is prime and its quotient ring is a domain, with the inner-interval binomials as an explicit reduced Gröbner basis.","For weakly closed path polyominoes there is an explicit regular sequence of $n$ initial terms, one per cell, proving König type and hence Knutson membership.","For ladder polyominoes, the construction exhibits a polynomial $f$ whose leading term is the product of all vertex variables, and the proof realizes $I_P$ as a sum of Knutson ideals step by step.","When a parallelogram polyomino is extracted from another and no inner interval crosses both removed and kept layers, the leftover collection is Knutson and its quadratic binomials form a Gröbner basis."],"supporting_citations":[{"why":"introduces Knutson ideals and the squarefree-initial-ideal consequence used throughout.","marker":"[CV20]"},{"why":"defines ideals of König type, the property from which Knutson membership is derived in Proposition 3.2.","marker":"[HHM22]"},{"why":"supplies the earlier simple-thin König-type theorem and the lemma the paper generalizes as Lemma 3.4.","marker":"[HH23]"},{"why":"shows closed path polyominoes are of König type, the input for Proposition 3.11.","marker":"[DN22]"},{"why":"independently outlines König type for weakly closed path polyominoes, the same line followed in Theorem 3.12.","marker":"[Nav24]"},{"why":"establishes that simple polyominoes are prime, giving primality of the intermediate sub-polyominoes in the ladder induction.","marker":"[HSM14]"},{"why":"supplies the Gröbner-basis criterion for primality used in the thin-class results.","marker":"[MRR22]"},{"why":"provides the reduced Gröbner basis result for thin polyominoes used in Proposition 5.4.","marker":"[JN24]"},{"why":"provides Buchberger's criterion and distributive-lattice Gröbner basis facts used in the basis proofs of Sections 4 through 6.","marker":"[HHO18]"}],"fun_headline_variants":["Knutson ideals certify radicality for five polyomino classes","Five polyomino families turn Knutson, proving radical ideals","Thin polyomino ideals proven prime via Knutson","Knutson certificate proves radical polyomino ideals","Polyomino ideals: Knutson implies radicality in five families"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that adding a single end cell to a polyomino that already has the special generator property preserves that property; the paper cites two earlier lemmas for this step instead of proving it, and the whole induction for simple thin polyominoes depends on it.","fun_headline_variants_meta":{"raw":{"variants":["Knutson ideals certify radicality for five polyomino classes","Five polyomino families turn Knutson, proving radical ideals","Thin polyomino ideals proven prime via Knutson","Knutson certificate proves radical polyomino ideals","Polyomino ideals: Knutson implies radicality in five families"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001528,"raw_usage":{"total_tokens":6107,"prompt_tokens":922,"completion_tokens":5185,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":5097}},"tokens_in":538,"tokens_out":5185,"duration_ms":34542,"temperature":1.0,"reasoning_tokens":5097,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:12:25.262117+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the induction of Theorem 3.6 on the polyomino of Figure 2: if, when the final cell is added, the asserted new generators do not have squarefree pairwise-coprime initial terms under the constructed order, the single-cell lemma and the theorem collapse. Separately, compute all S-pairs of the inner-interval binomials for the thin polyomino of Figure 19 under the stated reverse lexicographic order; any S-pair that fails to reduce to zero would refute the claimed reduced Gröbner basis and primality.","supporting_citations":[],"review_version":1}