{"id":"75def3de-0684-4586-ba70-6c0b6188a6be","arxiv_id":"2508.20364","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every skew Ferrers bipartite graph, the binomial edge ring has a quadratic Sagbi/Grobner presentation, is Koszul, Cohen Macaulay, normal, and its dimension is the number of NW and SE perimeter cells.","lead":"These authors give a Sagbi basis and a quadratic Grobner basis for binomial edge rings arising from skew Ferrers diagrams, proving the rings are Koszul, Cohen Macaulay, normal domains and counting their dimension with perimeter cells. A generalist reader may care because these rings encode algebraic statistics and determinantal geometry, and the paper turns a diagram shape into explicit algebraic structure.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem rests on an incomplete 13-case lifting analysis in §4.1: Remark 4.3 concedes omitted cases are dismissed as symmetric or standard without proof, so the Sagbi and quadratic Gröbner basis conclusions are not fully established.","rationale":"I read the paper in good faith and its main construction is plausible: the dimension formula in Theorem 5.3 has a coherent proof, and the general strategy—deform the binomial edge ring to K[LT(H)], compute a Gröbner basis there, then lift—is standard. The reader's weakest_assumption identifies exactly the point at which the proof's universal quantifier is not discharged: the lifting step in §4.1. The authors' own Remark 4.3 admits the case analysis is incomplete, so this is not an external nitpick; it is an explicit caveat inside the manuscript. The only other soft spot I noticed is Lemma 3.10, whose proof is delegated to 'roughly 25 subregions' and a figure, but that is more plausibly fillable and only affects termination of reductions, not the shape of the final theorem. The omitted-case issue, by contrast, directly underwrites the Sagbi basis and the quadratic Gröbner basis, and therefore the Koszul/Cohen–Macaulay/normality conclusions. Since the reader already chose CONDITIONAL, and my independent read does not move that verdict, I recommend UNCHANGED: the paper should be accepted only after the omitted cases in Remark 4.3 are either enumerated or machine-checked.","tokens_in":18910,"tokens_out":8283,"duration_ms":94083,"concrete_test":"Run an exhaustive computational check: for every skew Ferrers diagram with a+b ≤ 8, generate the full set F of quadratic reductions from Definition 3.5; classify every quadratic monomial in RΓ that is not standard; verify that each falls under one of the F1–F13/F' families or a stated symmetry/standard case, and that the displayed lift lies in ker(φ) with the required initial monomial. A direct algebraic version: implement the proposed generators J of §4.1 in Macaulay2/Singular, compute the Gröbner basis of ker(φ) for the binomial edge ring, and compare its initial ideal with ⟨LT(J)⟩. Equality for all small diagrams would reduce the gap to a missing proof; any failure would disprove Theorem 4.4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.4 is proved by lifting the Gröbner basis F of ker(φ*) from K[LT(H)] to ker(φ) via Conca–Herzog–Valla. Section 4.1 verifies only the 13 families F1–F13, with replacements F'_3, F'_4, F'_9, F'_12 for some skew-specific cases. Remark 4.3 explicitly states that the discussion 'appears incomplete' and that omitted quadratic monomials are 'excluded due to symmetry' or are 'already in standard form.' No complete enumeration of the possible relative positions is given, and no proof is offered that these two exclusions cover every non-listed case. This is load-bearing because lifting is a universal quantifier over the Gröbner basis of ker(φ*): one missing quadratic binomial would break Theorem 4.4(b) and with it the Koszul/normal/rational-singularity conclusions. Moreover, the two-branch reduction (III1)/(III2) in Definition 3.5 is decided by membership of T_{min{i1,i'1},j'1;i'2,max{j1,j'2}} in R, so for genuine skew diagrams the case split is strictly finer than in Higashitani's complete-bipartite setting; the paper itself notes that his lifting can involve cells outside Γ. Thus the assertion that the omitted cases are harmless is the weakest link in the central argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the binomial edge ring K[G(J_G)] associated to a skew Ferrers diagram Γ, i.e., the K-subalgebra of S generated by the binomials f_{i,j}=x_i y_j - p_j q_i for cells (i,j)∈Γ, together with auxiliary quadratic polynomials f_{i1,j1;i2,j2}. The main result (Theorem 4.4) states that the set H is a Sagbi basis, that the defining ideal of K[G(J_G)] has a quadratic Gröbner basis J, and consequently the ring is Koszul, Cohen–Macaulay, normal, with rational singularities in characteristic 0 and F-rational in positive characteristic. Theorem 5.3 identifies the Krull dimension with the number #P of northwest and southeast perimeter cells of Γ. The proof strategy is to analyze the initial algebra K[LT(H)], construct a quadratic Gröbner basis F for ker(φ*), and then lift F to ker(φ) via Conca–Herzog–Valla.","tokens_in":19312,"tokens_out":9443,"duration_ms":95931,"significance":"If fully correct, the paper would give a substantial and nontrivial extension of Higashitani's theorem on binomial edge rings of complete bipartite graphs to the wider class of skew Ferrers diagrams. The authors introduce a combinatorial reduction system adapted to the non-closed bipartite graph, and the dimension formula in terms of perimeter cells is elegant and computable. The potential value for the field is real: it connects Sagbi basis theory, Gröbner bases, and combinatorial commutative algebra in a concrete setting. The main weakness is the incomplete lifting verification in Section 4.1, which is load-bearing for the central theorem. The manuscript also contains several presentation issues that should be fixed in revision.","major_comments":[{"comment":"The lifting verification is incomplete. Remark 4.3 explicitly states that 'the discussion above appears incomplete' and that omitted cases are excluded either by symmetry or because the monomial is already in standard form, but no proof is given that these two exclusions cover all quadratic binomials in F whose cell sets lie in Γ. Since Theorem 4.4(b) depends on lifting every element of the Gröbner basis F of ker(φ*) to ker(φ), a single missing case would undermine the quadratic Gröbner basis conclusion and, with it, the Koszul/Cohen–Macaulay/normal/rational-singularity results. The authors should provide a complete enumeration of all possible leading monomials of F and a systematic proof that the listed cases F1–F13 and F'_i cover them.","section":"§4.1, Remark 4.3"},{"comment":"Case F13 is dismissed with the sentence 'This case is identical to the case in (xii) by symmetry with respect to the anti-diagonal line.' This symmetry is not demonstrated. For a skew Ferrers diagram, reflection about the anti-diagonal does not generally preserve Γ, and the reduction (III1)/(III2) in Definition 3.5 depends on membership of a T-variable in R, so symmetry arguments require explicit justification. Please provide a direct verification or a precise symmetry statement with proof.","section":"§4.1, case F13"},{"comment":"The proof of the validity of the reduction (III2) relies on '9 subcases' illustrated in Figure 3, but the subcases are not enumerated or checked in the text. This reduction is used in Proposition 3.11 and in Theorem 3.14 to establish that F is a Gröbner basis for ker(φ*). The geometric description is plausible, but the proof should be made rigorous by listing the 9 configurations and verifying the claimed containment of the relevant rectangular regions in Γ, or by giving an alternative formal argument.","section":"§3, Lemma 3.6 (III2)"}],"minor_comments":[{"comment":"The condition 'ui1 < j1 < j2 ≤ λi2' should read 'μ_{i1} < j1 < j2 ≤ λ_{i2}'. The same typo appears in the description of the variables of R in §3.","section":"§2, Eq. (2) and Notation 2.3"},{"comment":"In part (a), the index range 'a− b − 2' in 'For each k = 1, 2, . . . , a− b − 2' should be 'a+b−2'. As written, the range is wrong when a<b.","section":"§5, proof of Theorem 5.3"},{"comment":"The same symbol Γ is used for the original skew Ferrers diagram and for the ambient rectangular diagram. This is confusing; please use a distinct notation such as \\(\\overline{\\Gamma}\\) for the rectangular diagram.","section":"§3, Definition 3.1 and Remarks 3.3"},{"comment":"References [10] and [11] appear to be the same article: 'Binomial edge ideals and conditional independence statements' by Herzog, Hibi, Hreinsdóttir, Kahle, and Rauh. The citation [11] in Remark 2.2 should likely be [10].","section":"References"},{"comment":"The phrase 'the marked binomial' is used throughout, but the text does not visibly indicate which monomial is marked (e.g., by underlining or bold). Please make the leading monomial explicit in each F_i and F'_i, since the lifting argument depends on knowing which monomial is the initial term.","section":"§4.1, cases F1–F13"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's the short version: this is a real extension of Higashitani's complete-bipartite Sagbi-basis result to skew Ferrers diagrams, and the Krull dimension formula (#P perimeter cells) is genuinely nice. But the main theorem is not fully proved. The authors admit in Remark 4.3 that the lifting analysis in §4.1 is incomplete, and that gap is load-bearing.\n\nWhat I like. The reduction system in Definition 3.5, especially the III1/III2 split, is new and adapted to the skew setting. The Noetherian proof via χ and the fiber-invariance argument in Theorem 3.14 are coherent. The dimension formula in Theorem 5.3 is independent of the flawed lifting: the proof via edge-connected components and the rotation Γ'' is clean and convincing. The dimension result alone is worth having.\n\nWhere it gets soft. Section 4.1 lifts Higashitani's 13 quadratic binomials one by one. For several cases the authors supply modified polynomials (F'3, F'4, F'9, F'12). But they explicitly say that omitted cases are excluded 'due to symmetry' or because the monomial is 'already in standard form' – with no enumeration and no proof that those two exclusions cover everything. The lifting step is a universal quantifier over the Gröbner basis of ker(φ*): one missing binomial would invalidate Theorem 4.4(b) and the Koszul/normal/rational-singularity conclusions derived from it. The stress-test note is right that in skew diagrams the III1/III2 case split is strictly finer than in Higashitani's complete-bipartite setting, and Higashitani's lifted relations can involve cells outside Γ. So this is not a cosmetic gap.\n\nThe dimension formula and the reduction machinery are solid enough that the paper deserves a serious referee, but not as is. The authors need to either write out the omitted cases or verify them in a computer-assisted way. If that's done, the main theorem will likely go through.\n\nRecommendation: send to peer review, conditional on a complete case analysis or a formal (e.g., Coq/Lean) check of the lifting step. I'd bring it to reading group as a case study in how Sagbi lifting arguments can hide a universal-quantifier bug, but I wouldn't cite the main theorem yet.\n\nBest","headline":"Solid dimension formula and reduction machinery, but the main Sagbi/Gröbner theorem rests on an unverified lifting step that the authors themselves flag as incomplete.","tokens_in":19725,"tokens_out":2914,"would_cite":false,"duration_ms":31232,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F65","05E40","14M25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every skew Ferrers diagram, the binomial edge ring over any field is Koszul, Cohen–Macaulay, and normal, with a quadratic Gröbner basis and dimension equal to the number of perimeter cells.","keywords":["binomial edge ring","skew Ferrers diagram","Sagbi basis","Grobner basis","Koszul algebra","Cohen-Macaulay ring","normal domain","Krull dimension"],"falsifier":"Take a small skew Ferrers diagram, for instance λ=(3,3,3) and μ=(1,1,0), and compute the reduced Gröbner basis of the defining ideal of its binomial edge ring with a computer algebra system. If the basis contains any generator of degree at least 3, or if the S-polynomial of two listed quadratic lifts reduces to a binomial not covered by the 13 case types, then the quadratic Gröbner basis claim fails; conversely, exhaustively checking all diagrams up to, say, 4×4 cells would test the exhaustive-cover claim directly.","tokens_in":18841,"feed_emoji":"📐","tokens_out":8507,"duration_ms":90198,"temperature":0.7,"pith_summary":"Binomial edge rings are generated by the 2×2 minors that a graph picks out from a 2×n matrix; this paper shows that when the graph comes from a skew Ferrers diagram, those rings are exceptionally well behaved. The authors construct an explicit set H of generators for the subalgebra and prove it is a Sagbi basis: its leading monomials generate a toric ring with a quadratic Gröbner basis, and every relation can be lifted back to the original ring. It follows that the binomial edge ring is Koszul, Cohen–Macaulay, and normal, with rational singularities in characteristic zero and F-rational in positive characteristic. The paper also counts the Krull dimension: it equals the number of cells on the northwest and southeast perimeter of the diagram. This extends a known result for complete bipartite graphs and gives a combinatorial handle on rings that arise in algebraic statistics.","feed_headline":"Every skew Ferrers edge ring is Koszul and normal","feed_subtitle":"A Sagbi basis yields a quadratic Grobner basis, proving Koszul, Cohen-Macaulay, and normal, plus an exact dimension count.","key_machinery":"The central object is the set H: it consists of the defining binomials f_{i,j} together with the mixed products f_{i1,j1;i2,j2}, designed so that their leading monomials are x_i y_j for a cell and x_{i1} p_{j1} q_{i2} y_{j2} for a legitimate anti-diagonal pair. The machinery is the finite reduction system F on the monomials of the presentation ring R for K[LT(H)]—quadratic rewrites of types (I), (II), (III1), and (III2)—together with the χ-measure and a graded reverse lexicographic refinement that prove every reduction sequence terminates. The lifting principle from Sagbi theory then transfers the quadratic Gröbner basis from the initial algebra to the original binomial edge ring, and the pe","core_discovery":"Let Γ=λ/μ be a skew Ferrers diagram and G=G_{λ/μ} the bipartite graph built from its cells. The binomial edge ring K[G(J_G)] is generated by the binomials f_{i,j}=x_i y_j−p_j q_i. The paper adds the auxiliary quadrics f_{i1,j1;i2,j2}=f_{i1,j1}f_{i2,j2}−f_{i1,j2}f_{i2,j1} and collects them in a set H. With the lexicographic order, H is a Sagbi basis: the leading terms of H generate the initial algebra K[LT(H)], and the defining ideal of K[LT(H)] has a quadratic Gröbner basis consisting of reductions of types (I), (II), (III1), and (III2). The paper then verifies case by case that every one of those quadratic binomials can be lifted to a quadratic binomial in the kernel presenting K[G(J_G)]; t","pith_inferences":["If the 13-case lifting taxonomy is exhaustive, the same proof strategy should apply to any bipartite graph whose cells satisfy the same NE–SW closure properties; testing this on 'almost Ferrers' diagrams would be a natural next step.","The explicit quadratic Gröbner basis could support algorithmic work on the conditional-independence models associated to skew Ferrers diagrams, for example by supplying a normal-form algorithm or a Markov basis.","The perimeter-cell dimension formula suggests that the dimension of Sagbi-degenerate binomial edge rings can be read from boundary paths; this may generalize to bipartite graphs with several connected components by summing boundary contributions."],"forward_implications":["K[G(J_G)] is Koszul, Cohen–Macaulay, and normal for every skew Ferrers diagram; in characteristic 0 it has rational singularities, and in positive characteristic it is F-rational.","The defining ideal of K[G(J_G)] is generated in degree 2 and admits the explicitly listed quadratic Gröbner basis J, so homological and combinatorial invariants can in principle be read off from standard monomials.","The Krull dimension is #P, and for the rectangular complete bipartite case this becomes 2(a+b−2), recovering the known formula.","The standard monomials of the quadratic Gröbner basis give an explicit monomial basis of the quotient, allowing Hilbert series and Betti numbers to be computed directly from the diagram."],"supporting_citations":[{"why":"supplies the reduction-based Gröbner basis criterion (Theorem 3.12) and the normality criterion (Proposition 13.15) for the initial toric algebra.","marker":"[16]"},{"why":"gives the Sagbi lifting criterion and the corollaries that transfer Gröbner basis and singularity properties from K[LT(H)] to K[G(J_G)].","marker":"[4]"},{"why":"treats the complete-bipartite/rectangular case and provides the 13 lifted binomial types whose verification is adapted here.","marker":"[12]"},{"why":"provides the theorem that a quadratic Gröbner basis for the defining ideal implies the ring is Koszul.","marker":"[8]"},{"why":"provides the dimension comparison between an algebra and its initial algebra used in the Krull dimension proof.","marker":"[2]"},{"why":"establishes that normal semigroup rings are Cohen–Macaulay, used for the Cohen–Macaulay conclusion.","marker":"[13]"},{"why":"shows the monomial-edge-ring analogue for skew Ferrers diagrams is Koszul, Cohen–Macaulay, and normal, motivating the binomial extension.","marker":"[6]"}],"fun_headline_variants":["Skew Ferrers edge rings: Koszul, Cohen-Macaulay, normal","Sagbi basis gives quadratic Grobner for skew Ferrers","Skew Ferrers binomial edge rings proven Koszul and normal","Exact Krull dimension found for skew Ferrers edge rings","Quadratic Grobner from Sagbi basis for skew Ferrers rings"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof that the quadratic lifted binomials generate the full defining ideal rests on the claim that the 13 cases listed in Section 4.1, together with cases set aside by symmetry or because the start monomial is already standard, exhaust every quadratic reduction that can occur; if one case is missing, the quadratic Gröbner basis conclusion—and with it the Koszul, Cohen–Macaulay, and normality conclusions—would have no support.","fun_headline_variants_meta":{"raw":{"variants":["Skew Ferrers edge rings: Koszul, Cohen-Macaulay, normal","Sagbi basis gives quadratic Grobner for skew Ferrers","Skew Ferrers binomial edge rings proven Koszul and normal","Exact Krull dimension found for skew Ferrers edge rings","Quadratic Grobner from Sagbi basis for skew Ferrers rings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000223,"raw_usage":{"total_tokens":1250,"prompt_tokens":653,"completion_tokens":597,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":397,"completion_tokens_details":{"reasoning_tokens":501}},"tokens_in":397,"tokens_out":597,"duration_ms":5938,"temperature":1.0,"reasoning_tokens":501,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:05:38.861644+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small skew Ferrers diagram, for instance λ=(3,3,3) and μ=(1,1,0), and compute the reduced Gröbner basis of the defining ideal of its binomial edge ring with a computer algebra system. If the basis contains any generator of degree at least 3, or if the S-polynomial of two listed quadratic lifts reduces to a binomial not covered by the 13 case types, then the quadratic Gröbner basis claim fails; conversely, exhaustively checking all diagrams up to, say, 4×4 cells would test the exhaustive-cover claim directly.","supporting_citations":[{"cited_title":"Sturmfels, Gr¨ obner bases and convex polytopes, University Lecture Series, vol","cited_arxiv_id":null,"evidence_quote":"supplies the reduction-based Gröbner basis criterion (Theorem 3.12) and the normality criterion (Proposition 13.15) for the initial toric algebra."},{"cited_title":"Conca, J","cited_arxiv_id":null,"evidence_quote":"gives the Sagbi lifting criterion and the corollaries that transfer Gröbner basis and singularity properties from K[LT(H)] to K[G(J_G)]."},{"cited_title":"Binomial edge rings of complete bipartite graphs","cited_arxiv_id":"2411.07812","evidence_quote":"treats the complete-bipartite/rectangular case and provides the 13 lifted binomial types whose verification is adapted here."},{"cited_title":"Ene and J","cited_arxiv_id":null,"evidence_quote":"provides the theorem that a quadratic Gröbner basis for the defining ideal implies the ring is Koszul."},{"cited_title":"Bruns and J","cited_arxiv_id":null,"evidence_quote":"provides the dimension comparison between an algebra and its initial algebra used in the Krull dimension proof."},{"cited_title":"Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes , Ann","cited_arxiv_id":null,"evidence_quote":"establishes that normal semigroup rings are Cohen–Macaulay, used for the Cohen–Macaulay conclusion."},{"cited_title":"Corso, U","cited_arxiv_id":null,"evidence_quote":"shows the monomial-edge-ring analogue for skew Ferrers diagrams is Koszul, Cohen–Macaulay, and normal, motivating the binomial extension."}],"review_version":1}