{"id":"d3b63c0e-6050-459e-aa22-7ac4a5f1437c","arxiv_id":"1908.02744","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A bipartite graph's toric edge ideal satisfies Green-Lazarsfeld N_p exactly when the graph's missing-edge complement is a small tree for p=2, the graph is complete bipartite for p=3, or the graph is K_{2,n} for p at least 4.","lead":"Mathematicians here determine exactly when the defining equations of certain graph-based rings have clean linear structures, for every level of the syzygy hierarchy. The answer is a simple graph condition: a bipartite graph satisfies the second-level property exactly when its bipartite complement is a small tree, and the paper also fixes an earlier result on polyomino shapes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"N3 theorem depends on an unreproduced Macaulay2 check claiming that among primes p<20 only p=3 makes K5,5 fail N3; if another prime also fails, Theorem 1.2(3) is false.","rationale":"The reader's stated weakest assumption is the N2 proof's reliance on Proposition 5.1 and the five-case taxonomy. I do not disagree with that concern in principle, and the N2 proof would certainly benefit from independent verification of the case analysis. However, after spot-checking the displayed syzygy identities in Case 4 of Theorem 5.2, I found the advertised decomposition of the Koszul syzygy into multiples of linear syzygies to be consistent. The more decisively load-bearing gap, in my view, is the unverified characteristic computation in Theorem 5.5. The full force of part (3) of the main theorem is that the sole exceptional characteristic is 3 for all complete bipartite graphs with both parts at least 5. The proof reduces every potential counterexample to K5,5, and then leans on a one-line Macaulay2 computation to rule out all primes below 20 except 3. Since no code or output is given, this is not reproducible from the manuscript. The rest of the reduction is clean: theorem 3.2 localizes any obstruction to an induced subgraph on at most 10 vertices, the known resolutions for determinantal ideals handle the small cases, and Hashimoto's theorem independently explains why characteristic 3 is special. Thus a single finite computation, if supplied and correct, would close the gap; if it is wrong, the classification in Theorem 1.2(3) needs modification. This reinforces the reader's CONDITIONAL verdict without changing it.","tokens_in":21294,"tokens_out":48015,"duration_ms":480528,"concrete_test":"Run a short Macaulay2 or Singular script over Q and over F_p for p=2,3,5,7,11,13,17,19 computing the graded Betti numbers of I_{K_{5,5}}. For example, in Macaulay2: S=QQ[x_(1,1)..x_(5,5)]; M=genericMatrix(S,5,5); I=minors(2,M); betti res coker gens I; repeat with S=ZZ/p[x_(1,1)..x_(5,5)]. Verify that β_{2,5}(I) (equivalently β_{3,5}(S/I)) is nonzero in characteristic 3 and zero in every other listed characteristic. If any p≠3 gives a nonzero value, then Theorem 1.2(3) must be revised.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Theorem 5.5, the classification of N3 is reduced to the single graph K5,5. After noting that the Lascoux/Pragacz–Weyman resolution is the same in characteristic 0 and in characteristic p>20, the proof states: “A quick Macaulay2 [22] calculation shows that p=3 is the only characteristic less than 20 which in which IG fails to satisfy N3.” No script, output, or invariant-theoretic argument is supplied. This computation is the only evidence that the exceptional set is exactly {3}; the surrounding argument only forces failure in characteristic 3 via Hashimoto [15] and success in characteristic 0 and p>20. Since Theorem 1.2(3) asserts failure if and only if char(k)=3 and min{m,n}≥5, a second bad prime among p=2,5,7,11,13,17,19 would falsify the theorem. The rest of the N3 proof, including the reduction of a hypothetical β2,5 obstruction to an induced K5,5 via Theorem 3.2 and [2, Main Theorem (2)], appears sound, so the unreproduced Macaulay2 check is the load-bearing step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper characterizes the Green-Lazarsfeld conditions N_p for toric edge ideals of bipartite graphs with minimum degree at least 2. The main theorem gives a complete combinatorial description: N_1 is Ohsugi-Hibi chordality of cycles of length at least 6; N_2 holds exactly when the bipartite complement is essentially a tree of diameter at most 3; N_3 holds exactly for complete bipartite graphs except in characteristic 3 when both parts have size at least 5; and N_p for p at least 4 holds exactly for K_{2,n}. The proofs combine multigraded Betti-number obstructions and induced-subgraph reductions (Theorem 3.2), a graph-theoretic classification of the obstructions (Theorem 4.2), a Koszul-algebra reduction to linear and Koszul syzygies (Proposition 5.1 and the five-configuration analysis in Theorem 5.2), and known resolutions of Segre and determinantal ideals. The paper also translates the results to convex polyomino ideals, corrects a characterization in [10], and applies the N_2 result to a regularity question of Constantinescu, Kahle, and Varbaro.","tokens_in":21491,"tokens_out":13216,"duration_ms":138076,"significance":"If the results hold, they provide a complete and elegant combinatorial description of all Green-Lazarsfeld conditions for toric edge ideals of bipartite graphs, interpolating between Ohsugi-Hibi's quadratic-generation theorem and Ohsugi-Hibi's linear-resolution theorem. The N_2 proof supplies explicit syzygy decompositions for all five possible configurations of pairs of 4-cycles, and the global statement is checked against external benchmarks (Lascoux, Pragacz-Weyman, Hashimoto, Eagon-Northcott) rather than assumed. The polyomino application identifies an error in [10] and replaces it with a criterion that is consistent with the graph-theoretic classification. The regularity corollary for the Constantinescu-Kahle-Varbaro question is a clean consequence of the N_2 characterization. The main weakness is a load-bearing, unreproduced Macaulay2 computation in the proof of Theorem 5.5.","major_comments":[{"comment":"The assertion that p=3 is the only characteristic less than 20 in which I_{K_{5,5}} fails to satisfy N_3 is load-bearing for Theorem 1.2(3), but the only evidence supplied is 'A quick Macaulay2 [22] calculation shows...' with no script, log, or reproducible output. The surrounding argument only proves failure in characteristic 3 and success in characteristic 0 and p>20; therefore, a second bad prime among 2,5,7,11,13,17,19 would falsify the theorem. Please provide the Macaulay2 code and output, or replace the computation with an invariant-theoretic or representation-theoretic argument. The sentence describing the p>20 reduction ('the number of boxes in the last partition...') is also too terse as written; the integral Lascoux complex should be shown explicitly to be a resolution over the relevant localized integer ring.","section":"§5, proof of Theorem 5.5"},{"comment":"The reduction of N_2 to checking five configurations of two distinct 4-cycles rests on Proposition 5.1, quoted from [21, Proposition 2.8], but the proposition is not stated in the paper and the term 'Koszul syzygy' is not defined. Because this proposition is load-bearing for the N_2 classification, the paper should state the proposition precisely, define the Koszul syzygies being considered, and justify that the five-configuration taxonomy together with the additional edges imposed by Theorem 4.2 exhausts all minimal Koszul syzygies in the relevant graphs. As written, a reader must take on faith both the validity of the quoted proposition in this setting and the exhaustiveness of the case analysis.","section":"§5, proof of Theorem 5.2 (Proposition 5.1)"}],"minor_comments":[{"comment":"The abstract states that N_2 holds if and only if the bipartite complement of G is a tree of diameter at most 3, while Theorem 1.2 and the main body state the condition as 'essentially a tree of diameter at most 3'; these statements should be aligned.","section":"Abstract vs. Theorem 1.2"},{"comment":"There are several typographical errors: the title contains 'BIP ART ITE', Proposition 6.2 says 'every chord with length ≥ 6 has a chord' where 'cycle' is meant, and 'polynomo' appears in Proposition 6.2.","section":"Title and Section 6"},{"comment":"The phrase 'the only characteristic less than 20 which in which IG fails to satisfy N3' contains a duplicated 'which in'.","section":"Proof of Theorem 5.5"},{"comment":"Proposition 4.1 as stated is not true for graphs with isolated vertices under the convention used in its proof, where a graph with no edges is counted as essentially connected; for example, a graph consisting of one edge plus an isolated vertex has every induced subgraph essentially connected and acyclic but is not a tree. The proposition is applied only to graphs without isolated vertices, so this does not affect the main theorem, but the statement should be restricted or the convention clarified.","section":"Proposition 4.1"},{"comment":"The sentence 'The ideal I_{H^{(1)}} ... is clearly not associated to any convex polyomino, which cannot have exactly 2 minimal generators' is ambiguous: 'which' appears to refer to a convex polyomino ideal rather than to a polyomino. Please rephrase for clarity.","section":"Section 6, after Remark 6.1"}],"recommendation":"major_revision","confidential_remarks":"The most serious issue is the unreproduced Macaulay2 computation in Theorem 5.5; it is the only evidence ruling out additional bad primes in the N_3 classification. If the authors can supply the script and output, or another check, the paper is likely publishable. I would not recommend rejection on the current evidence, because the rest of the argument is detailed and internally coherent, but the exceptional-characteristic statement cannot be accepted on faith."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper completes the N_p story for toric edge ideals of bipartite graphs, and the main theorem is a genuine classification: N2 iff the bipartite complement is essentially a tree of diameter at most 3, N3 iff complete bipartite except for the characteristic-3 K_{m,n} (min>=5) exception, and N4+ iff K_{2,n}. The N2 proof is the heart, and it holds up: the five configurations of two distinct 4-cycles are handled with explicit Koszul syzygy decompositions, and the graph-theoretic reduction in Theorem 4.2 is long but coherent. I don't see a missing case.\n\nThe paper also corrects a published result on linearly presented polyomino ideals [10], which is a nice bonus if it checks out.\n\nThe soft spot is real and it's the N3 theorem. Theorem 5.5 reduces to K_{5,5} and then asserts that 'a quick Macaulay2 calculation' shows p=3 is the only characteristic <20 where N3 fails. No script, no output, no alternative argument. Theorem 1.2(3) depends on exactly that computation. For p>20 the Lascoux-resolution argument is fine, and the characteristic-3 failure is backed by Hashimoto's result, but the exclusion of p=2,5,7,11,13,17,19 is the unreproduced part. That has to be fixed before I'd trust the statement as a theorem.\n\nThe N2 proof uses Proposition 5.1 from Mastroeni's Koszul paper. I found that application reasonable. The 'subscript inconsistency' the reader flagged looks like a labeling issue, not a mathematical gap: the two 4-cycles sharing two vertices are described with shared vertices on the Y side, so it's K_{4,2}, which is K_{2,4} up to swapping sides.\n\nThis is a paper I'd want to see in the literature. I'd send it to a referee with a request for the Macaulay2 script and the finite-field verification. It's not a desk reject; it's a solid result with one missing reproducibility check.","headline":"Completes the N_p classification for toric edge ideals of bipartite graphs; N2 proof is solid, but the N3 exceptional-characteristic claim rests on an unreproduced Macaulay2 computation.","tokens_in":22039,"tokens_out":5801,"would_cite":true,"duration_ms":53715,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D02","15A75"],"pacs":[],"model":"deepseek-v4-flash","headline":"A bipartite graph's toric edge ideal is linearly presented exactly when its bipartite complement is essentially a tree of diameter at most 3, and the whole $\\mathbf{N}_p$ ladder is classified.","keywords":["Green-Lazarsfeld condition","toric edge ideal","bipartite graph","linear syzygies","Koszul algebra","bipartite complement","polyomino ideal","graded Betti numbers"],"falsifier":"Compute the graded Betti table, over a field of characteristic not $3$, of the toric edge ideal of $K_{3,3}$ with one edge deleted; the paper's $\\mathbf{N}_3$ argument predicts $\\beta_{2,5}=1$, so a computation returning $\\beta_{2,5}=0$ would refute the classification.","tokens_in":21062,"feed_emoji":"🌳","tokens_out":14876,"duration_ms":150221,"temperature":0.7,"pith_summary":"Toric edge ideals are the defining ideals of edge rings, and the Green-Lazarsfeld condition $\\mathbf{N}_p$ asks that the first $p$ steps of a minimal free resolution be linear. This paper completes the classification for bipartite graphs: each $\\mathbf{N}_p$ becomes a purely graph-theoretic statement, from the known chord condition at $\\mathbf{N}_1$ up to the single family $K_{2,n}$ at $\\mathbf{N}_4$. The reason to care is that the proof turns a homological question into a finite list of forbidden induced subgraphs, and the classification has direct consequences for the regularity of these ideals and for polyomino ideals.","feed_headline":"Only tree complements give linear toric syzygies","feed_subtitle":"Completes the N1–N4 ladder for bipartite toric edge ideals, ending at the only fully linear case, K2,n.","key_machinery":"The load-bearing objects are the bipartite complement $\\overline{G}$ (same bipartition, edges reversed) and the notion of being essentially a tree (a tree after deleting isolated vertices). The engine is a local-to-global Betti obstruction: $\\beta_{i,\\alpha}(I_G)=\\dim_k \\tilde H_i(\\Gamma(\\alpha);k)$, where $\\Gamma(\\alpha)$ is the simplicial complex of monomials of multidegree $\\alpha$; consequently every nonzero $\\beta_{i,j}$ is witnessed by an induced subgraph on at most $2j$ vertices. This reduces each $\\mathbf{N}_p$ failure to a finite forbidden-subgraph check. For $\\mathbf{N}_2$, Proposition 5.1 — first syzygies in a Koszul ring are generated by linear syzygies and Koszul syzygies, the latter being the ones coming directly from pairs of quadratic relations — permits a five-case check of pairs of $4$-cycles, and Theorem 4.2 guarantees that whenever the bipartite complement is not essentially a tree of diameter at most $3$, one of the eight graphs $H^{(1)},\\dots,H^{(8)}$ appears as an induced subgraph.","core_discovery":"The central claim is Theorem 1.2. For a bipartite graph $G$ with minimum vertex degree at least $2$ over a field $k$, the toric edge ideal $I_G$ satisfies $\\mathbf{N}_1$ exactly when every cycle of length at least $6$ has a chord; $\\mathbf{N}_2$ exactly when the bipartite complement of $G$ is essentially a tree of diameter at most $3$; $\\mathbf{N}_3$ exactly when $G$ is complete bipartite, except that $K_{m,n}$ with $\\min\\{m,n\\}\\ge 5$ fails $\\mathbf{N}_3$ in characteristic $3$; and $\\mathbf{N}_p$ for any $p\\ge 4$ exactly when $G=K_{2,n}$. The proof route is: Theorem 3.2 turns a nonzero graded Betti number of $I_G$ into an induced subgraph on at most $2j$ vertices, Lemmas 3.3–3.6 exhibit the obstructions to $\\mathbf{N}_2,\\mathbf{N}_3,\\mathbf{N}_4$, Theorem 4.2 converts the absence of eight forbidden induced subgraphs into the tree-complement statement, and Proposition 5.1 lets the $\\mathbf{N}_2$ case be checked by examining the five configurations of two distinct $4$-cycles. The classification is then translated to convex polyomino ideals and used to rule out a family of linearly presented bipartite toric edge ideals with slow regularity growth.","pith_inferences":["The same local-obstruction method suggests a finite decision procedure for $\\mathbf{N}_p$ in any toric ring generated by quadrics: list the finitely many multidegrees of total degree $2j$ and check the reduced homology of their fibers; for bipartite graphs the list collapses to eight forbidden subgraphs, while for other normal toric rings it will be longer but still finite.","The $\\mathbf{N}_2$ characterization doubles as a generator of examples: start with any tree of diameter at most $3$ on a fixed bipartition, take its bipartite complement, and the resulting graph has a linearly presented toric edge ideal, giving a large family on which regularity bounds can be tested.","The isolated characteristic-$3$ failure at $\\mathbf{N}_3$ for large complete bipartite graphs hints that modular representation theory of symmetric groups controls higher linear syzygies of Segre varieties; the present classification shows such effects cannot survive past $\\mathbf{N}_4$ for bipartite graphs, but they may appear in other toric ideals.","The paper's correction to the polyomino criterion illustrates a general caution: when a graph-theoretic theorem is transported through a dictionary such as convex polyominoes to bipartite graphs, configurations can be silently dropped, so the image of the dictionary should be rechecked against the full graph classification."],"forward_implications":["If $I_G$ satisfies $\\mathbf{N}_2$, linear presentation can be read off by deleting isolated vertices from the bipartite complement and checking whether the rest is a tree of diameter at most $3$.","A failure of $\\mathbf{N}_3$ for a non-complete bipartite graph shows up as a nonzero second syzygy in degree $5$, so only complete bipartite graphs can reach $\\mathbf{N}_3$.","Once $\\mathbf{N}_4$ holds, the graph is $K_{2,n}$ and the resolution is fully linear via the Eagon–Northcott complex, so $\\mathbf{N}_4$ is the same as having a linear free resolution for these ideals.","For convex polyominoes, the translated criterion says $\\mathbf{N}_2$ holds exactly when all missing cells lie in the first row or first column, $\\mathbf{N}_3$ when the polyomino is an interval (with the same characteristic-$3$ caveat for large intervals), and $\\mathbf{N}_4$ when the interval has one side of length $2$.","No family of bipartite graphs with linearly presented toric edge ideals can give a positive answer to the open question about linearly presented ideals with positive regularity-to-projective-dimension ratio: if the projective dimension tends to infinity, the ratio tends to $0$."],"supporting_citations":[{"why":"Establishes the $\\mathbf{N}_1$ starting point: $I_G$ is generated by quadrics and the edge ring is Koszul exactly when every cycle of length at least $6$ has a chord, supplying the quadratic Gröbner basis used throughout.","marker":"[25]"},{"why":"Supplies Proposition 5.1, the hinge of the $\\mathbf{N}_2$ proof, that in a Koszul algebra the first syzygies are minimally generated by linear syzygies and Koszul syzygies.","marker":"[21]"},{"why":"Gives the identity $\\beta_{i,\\alpha}(I_G)=\\dim_k \\tilde H_i(\\Gamma(\\alpha);k)$ that turns multigraded Betti numbers into reduced homology of monomial fibers, the source of the local obstructions.","marker":"[28]"},{"why":"Shows that $I_G$ has a linear free resolution exactly for $G=K_{2,n}$, the endpoint against which the $\\mathbf{N}_4$ characterization is compared.","marker":"[24]"},{"why":"Supplies the resolution of the Segre embedding for determinantal ideals, used to establish $\\mathbf{N}_3$ for complete bipartite graphs in characteristic $0$.","marker":"[20]"},{"why":"Supplies the companion resolution of the same Segre embeddings, the other half of the characteristic-$0$ $\\mathbf{N}_3$ argument.","marker":"[29]"},{"why":"Shows that the graded Betti numbers of $n$-minors of $(n+2)$-square matrices are characteristic-independent, restricting the characteristic dependence to the large complete bipartite cases.","marker":"[16]"},{"why":"Shows that third Betti numbers of determinantal ideals can be larger in characteristic $3$, which the paper isolates as the only $\\mathbf{N}_3$ exception.","marker":"[15]"},{"why":"Gives vanishing of higher linear Betti numbers for Koszul algebras, reducing a failure of $\\mathbf{N}_3$ to a single Betti number $\\beta_{2,5}$.","marker":"[2]"},{"why":"Bounds the regularity of toric edge ideals, used in Corollary 7.3 to show the regularity-to-projective-dimension ratio tends to $0$.","marker":"[3]"}],"fun_headline_variants":["Tree complements give linear toric presentations","N2 equals bipartite complement being a diameter-3 tree","Syzygy ladder for bipartite toric edge ideals completed","All N_p for p>=4 hold only for K_{2,n}","From chorded cycles to K_{2,n}: toric syzygies"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The $\\mathbf{N}_2$ half of the theorem rests on the assumption that in a Koszul ring every first syzygy is generated by linear syzygies together with the syzygies coming directly from pairs of quadratic relations; if that decomposition misses a syzygy type, the proof's five-case check of pairs of $4$-cycles is incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Tree complements give linear toric presentations","N2 equals bipartite complement being a diameter-3 tree","Syzygy ladder for bipartite toric edge ideals completed","All N_p for p>=4 hold only for K_{2,n}","From chorded cycles to K_{2,n}: toric syzygies"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000278,"raw_usage":{"total_tokens":1698,"prompt_tokens":1036,"completion_tokens":662,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":652,"completion_tokens_details":{"reasoning_tokens":575}},"tokens_in":652,"tokens_out":662,"duration_ms":6416,"temperature":1.0,"reasoning_tokens":575,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:37:54.557396+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the graded Betti table, over a field of characteristic not $3$, of the toric edge ideal of $K_{3,3}$ with one edge deleted; the paper's $\\mathbf{N}_3$ argument predicts $\\beta_{2,5}=1$, so a computation returning $\\beta_{2,5}=0$ would refute the classification.","supporting_citations":[{"cited_title":"Ohsugi, T","cited_arxiv_id":null,"evidence_quote":"Establishes the $\\mathbf{N}_1$ starting point: $I_G$ is generated by quadrics and the edge ring is Koszul exactly when every cycle of length at least $6$ has a chord, supplying the quadratic Gröbner basis used throughout."},{"cited_title":"Mastroeni, Koszul almost complete intersections","cited_arxiv_id":null,"evidence_quote":"Supplies Proposition 5.1, the hinge of the $\\mathbf{N}_2$ proof, that in a Koszul algebra the first syzygies are minimally generated by linear syzygies and Koszul syzygies."},{"cited_title":"Peeva, Graded syzygies","cited_arxiv_id":null,"evidence_quote":"Gives the identity $\\beta_{i,\\alpha}(I_G)=\\dim_k \\tilde H_i(\\Gamma(\\alpha);k)$ that turns multigraded Betti numbers into reduced homology of monomial fibers, the source of the local obstructions."},{"cited_title":"Ohsugi, T","cited_arxiv_id":null,"evidence_quote":"Shows that $I_G$ has a linear free resolution exactly for $G=K_{2,n}$, the endpoint against which the $\\mathbf{N}_4$ characterization is compared."},{"cited_title":"Lascoux, Syzygies des vari´ et´ es determinantales,Adv","cited_arxiv_id":null,"evidence_quote":"Supplies the resolution of the Segre embedding for determinantal ideals, used to establish $\\mathbf{N}_3$ for complete bipartite graphs in characteristic $0$."},{"cited_title":"Pragacz, J","cited_arxiv_id":null,"evidence_quote":"Supplies the companion resolution of the same Segre embeddings, the other half of the characteristic-$0$ $\\mathbf{N}_3$ argument."},{"cited_title":"Hashimoto and K","cited_arxiv_id":null,"evidence_quote":"Shows that the graded Betti numbers of $n$-minors of $(n+2)$-square matrices are characteristic-independent, restricting the characteristic dependence to the large complete bipartite cases."},{"cited_title":"Hashimoto, Determinantal ideals without minimal fr ee resolutions, Nagoya Math","cited_arxiv_id":null,"evidence_quote":"Shows that third Betti numbers of determinantal ideals can be larger in characteristic $3$, which the paper isolates as the only $\\mathbf{N}_3$ exception."},{"cited_title":"Avramov, A","cited_arxiv_id":null,"evidence_quote":"Gives vanishing of higher linear Betti numbers for Koszul algebras, reducing a failure of $\\mathbf{N}_3$ to a single Betti number $\\beta_{2,5}$."},{"cited_title":"Biermann, A","cited_arxiv_id":null,"evidence_quote":"Bounds the regularity of toric edge ideals, used in Corollary 7.3 to show the regularity-to-projective-dimension ratio tends to $0$."}],"review_version":1}