{"id":"6eee5527-1ad6-47ad-9987-bc43fb1571a3","arxiv_id":"2608.01199","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For forests, the localized v-number of a Lovasz-Saks-Schrijver ideal equals the number of vertices of degree at least d, and analogous combinatorial formulas hold for parity binomial edge ideals.","lead":"This paper computes the v-number, a measure of how complicated a polynomial ideal is, for graph-based ideals such as Lovasz-Saks-Schrijver ideals and parity binomial edge ideals, and proves it is bounded by another invariant called regularity for several graph families. It introduces a framework called coordinate-saturated ideals that turns these algebraic computations into graph counting problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The forest formula in Theorem 4.9 rests entirely on the unpublished irreducible decomposition [28, Thm 3.18]; if that decomposition is incomplete or needs the deleted condition (11), the main forest result and Theorem 4.14 for d≥3 do not follow.","rationale":"After checking the surrounding arguments, I did not find a concrete internal error in the parts that do not depend on [28]. The Section 3 framework is coherent; the hypergraph identification in Lemma 4.7 is correct once one uses minimal primes, since every admissible singleton is minimal and all other admissible sets lie inside V_{\\ge d}(G). The parity binomial edge ideal adaptation in Sections 6–7 is internally consistent, and the counting in Lemma 4.13 checks out. The genuinely load-bearing unresolved point is [28, Thm 3.18], an arXiv preprint by the second author that supplies the irreducible and primary decomposition for LSS ideals of forests. The paper even notes that it is modifying the admissibility definition from [28] by deleting condition (11), with a redundancy argument. Because Theorems 4.9 and 4.14 for d≥3 depend directly on that decomposition, this is the most important unverified assumption. The reader's CONDITIONAL verdict already captures this risk, so I recommend no change to the verdict. The remaining issues mentioned by the reader, such as the wording in Lemma 7.2 and the algebraically closed field hypothesis in Section 9, are minor and do not alter the assessment.","tokens_in":26909,"tokens_out":37264,"duration_ms":347643,"concrete_test":"Compute the minimal primes of L_G^K(3) for every tree on n≤8 vertices and compare them with the set {p_S(G) : S⊆[n] is G-admissible} from Definition 4.1. This can be done with a computational primary decomposition over Q and, as a cross-check, over a large finite field; any mismatch in the prime list would disprove [28, Thm 3.18] as used here. Additionally, independently compute v_{p_emptyset(G)}(L_G^K(3)) for these examples and compare it with v_3(G); if the formula fails on any small tree, Theorem 4.9 needs a corrected proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central computation for forests is not self-contained. Theorem 4.9 obtains v_{p_emptyset(G)}(L_G^K(d)) = v_d(G) by feeding Theorem 3.13 with three ingredients: (i) Corollary 4.3, the primary decomposition L_G^K(d) = ∩_S p_S(G) over G-admissible S, quoted from [28, Thm 3.18]; (ii) Lemma 4.5, which uses Corollary 4.3 to prove coordinate-saturation; (iii) Lemma 4.7, which identifies M with I_{\\ge d}. Every one of these uses the admissible-set classification of the irreducible components of OR_d(G). The paper itself modifies [28, Def. 3.15] by deleting condition (11) and gives a redundancy argument that relies on d≥3 and on forests having no K_{2,2}. If [28, Thm 3.18] needs the full definition, or if its component list is incomplete for some forest, then Theorem 4.9 and the d≥3 case of Theorem 4.14 do not follow from the arguments supplied. This is an external-validity risk rather than a detected inconsistency, but it is the least secure point in the whole chain: the main new forest theorem is exactly as reliable as that cited preprint.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the class of coordinate-saturated ideals and proves a framework (Theorem 3.13) for computing localized v-numbers. It applies this framework to three families: Lovász–Saks–Schrijver ideals of forests (exact formula for the localized v-number at the empty-set prime, leading to v ≤ reg for all d), LSS ideals over the real numbers (an upper bound via connected non-bipartite dominating sets), parity binomial edge ideals of non-bipartite graphs (an explicit formula involving totally non-bipartite dominating sets, with applications to decorated trees and other classes), and generalized binomial edge ideals (a monotonicity statement in m and a reproof of a known localized v-number formula). The paper also proves v(J_G^k) ≤ reg(R/J_G^k) for all powers of binomial edge ideals of closed graphs. The main formulas are parameter-free and combinatorial. The forest results, however, are conditional on an unpublished preprint by a coauthor for the irreducible decomposition of the orthogonal representation variety, and the equality statement in the generalized binomial edge ideal section omits the algebraic closure hypothesis required by the framework.","tokens_in":27191,"tokens_out":18724,"duration_ms":161742,"significance":"If the cited decomposition is valid, the paper makes a substantial contribution: it gives exact, combinatorial formulas for localized v-numbers of LSS ideals and parity binomial edge ideals, and it establishes v ≤ reg for several nontrivial graph classes. The coordinate-saturated framework is a clean and potentially reusable tool, and the radicality arguments via monomial orders and hypergraph transversals are elegant. The parity binomial edge ideal sections are intricate and largely consistent. The paper's strengths are its parameter-free predictions and its reduction of algebraic invariants to explicit graph-theoretic optimization problems. The main caveats are the heavy reliance of the headline forest theorem on an unreviewed coauthor preprint and a missing hypothesis in Theorem 9.6.","major_comments":[{"comment":"The forest computation is not self-contained. Corollary 4.3 quotes [28, Theorem 3.18] for the irreducible decomposition OR_d(G) = ∪_S V_S and for the classification of G-admissible subsets, and the paper itself modifies [28, Definition 3.15] by deleting condition (11), giving a redundancy argument that uses d ≥ 3 and the absence of K_{2,2}. Since Lemma 4.5 (coordinate-saturation), Lemma 4.7 (identification of M with I_{\\ge d}), and hence Theorem 4.9 all use that decomposition, the central forest formula is exactly as reliable as the unpublished preprint [28]. Moreover, the d ≥ 3 case of Theorem 4.14 inherits this dependence through Corollary 4.11 and Lemma 4.13. Please either include a proof of the decomposition and admissibility classification, or state it explicitly as an assumption and mark the affected theorems as conditional.","section":"Section 4, Theorem 4.9 and Theorem 4.14"},{"comment":"The equality v_p(I) = γ(I) in Theorem 3.13 requires K algebraically closed, because the proof uses Proposition 3.12 to identify (I:p) with √(I+M), and Proposition 3.12 invokes Hilbert's Nullstellensatz. The proof of Theorem 3.13 cites Proposition 3.1 at that point; Proposition 3.1 only proves that I+M is radical and does not identify the colon ideal. In addition, the sentence \"In particular, the equality holds whenever I is binomial\" is not justified by the surrounding argument unless algebraic closure is assumed or a separate binomial-specific proof is supplied. Consequently, Theorem 9.6, which states v_∅(J_{K_m,G}) = γ_c(G) for a simple graph over an unspecified field and invokes Theorem 3.13, is missing the hypothesis that K be algebraically closed. This is load-bearing because Theorem 9.6 is one of the paper's stated applications.","section":"Theorem 3.13 and Theorem 9.6"},{"comment":"The proof of Lemma 7.2 is hard to verify as written. The sentence \"each j ∉ S ∪ {j_1, ..., j_k} is not nice with respect to S\" should refer to the current set S ∪ {j_1, ..., j_k} after the iterations, and the notation B_i for bipartite components is reused after vertices have been added, even though the bipartite components of G[S] and G_{S'} need not coincide. Since Corollary 7.3 and hence Proposition 7.4 and Theorem 7.5 depend on this lemma, the construction should be rewritten with explicit current-component notation and a precise maximality condition on the sequence j_1, ..., j_k.","section":"Section 7, Lemma 7.2 and Proposition 7.4"}],"minor_comments":[{"comment":"The isomorphism L_G^K(2) ≅ J_G for bipartite G is used without a citation; please cite the relevant result, e.g., [5, Corollary 6.2].","section":"Proposition 4.10"},{"comment":"The phrase \"where R is field of real numbers\" should read \"where R is the field of real numbers\" or \"where K is the field of real numbers\".","section":"Abstract"},{"comment":"The sentence \"It is not difficult to show, using combinatorial arguments, that the invariants γ_{c,n}(G) and γ(L_R^G(2)) are in fact equal\" is an unsupported claim and is not needed for Theorem 5.9; either provide a proof or remove the sentence.","section":"Section 5, after Lemma 5.8"},{"comment":"The statement says G is non-bipartite, but the first case (both intervals even) is treated as bipartite; the wording should be adjusted to cover the bipartite case separately or to exclude it from the statement.","section":"Proposition 8.8"},{"comment":"The citation for the Bolognini–Macchia–Strazzanti equivalence appears as [5, Corollary 6.2] in the introduction but as [4, Corollary 6.2] in Sections 5 and 6; please make the numbering consistent.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main forest theorem is conditional on an unpublished coauthor preprint, [28]. For a journal publication I would require that this dependency be resolved: either the decomposition is proved in the paper or the theorem is explicitly stated as conditional on [28]. The algebraic closure gap in Theorem 9.6 is easily fixable by adding the hypothesis or by proving the binomial case directly. The paper is promising and the framework is novel, but the external dependence on [28] should be addressed before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: the coordinate-saturated framework is genuinely useful and the paper is worth a serious referee. Definition 3.5 and Theorem 3.13 are careful: the observation that every minimal prime of a coordinate-saturated ideal other than the unique indeterminate-free one must contain an indeterminate makes M generated by transversals, so the localized v-number becomes a hypergraph transversal number. That is a clean, reproducible idea, and the applications are mostly honest corollaries plus real combinatorial work. The parity binomial edge ideal section is the most intricate; Proposition 7.4, identifying transversals of Delta_G with totally non-bipartite dominating sets, is a solid piece of combinatorial rewriting. I checked the v <= reg consequences and they follow from the stated inputs. No fitting, no free parameters; the formulas are parameter-free and falsifiable.\n\nSoft spots, in proportion. First, Theorem 4.9 is not self-contained: it leans on the irreducible decomposition of OR_d(G) from an arXiv preprint by one of the authors, and the paper modifies a definition from that preprint. The redundancy argument for deleting condition (11) looks right to me -- I do not see a forest counterexample -- but the main forest formula is exactly as reliable as that preprint. This should be stated more prominently, and the dependency should ideally be upgraded to a peer-reviewed version or the proof included. Second, Theorem 9.6 omits the algebraically closed field hypothesis that Theorem 3.13 requires. The result may be true over any field, but as written the proof does not justify that; easy fix. Third, Lemma 7.2 has a small wording slip in the iteration sentence: 'not nice with respect to S' should be 'not nice with respect to S union {j_1,...,j_k}'. Cosmetic.\n\nThe reader's conditional verdict matches my reading. The stress-test note is right to flag the [28] dependency; it is the least secure point, but I did not find an actual mistake in the redundancy argument. This is an external-validity risk, not a detected inconsistency.\n\nWho is this for: commutative algebraists working on v-numbers, binomial edge ideals, and combinatorial invariants. It deserves peer review, with referees asked to check the use of [28] and the field hypothesis in Section 9; otherwise the arguments hang together.","headline":"A solid, useful framework for localized v-numbers, with the main forest formula riding on an unpublished preprint and a missing field hypothesis in Section 9.","tokens_in":27689,"tokens_out":3880,"would_cite":true,"duration_ms":35387,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F20","05E40","05E99","13D45"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every forest graph, the localized v-number of the LSS ideal at the variable-free prime equals the count of vertices of degree at least d, and this equality powers a general v&le;reg inequality.","keywords":["v-number","localized v-number","Lovász–Saks–Schrijver ideals","coordinate-saturated ideals","parity binomial edge ideals","binomial edge ideals","Castelnuovo–Mumford regularity","forest graphs"],"falsifier":"Take the star \\(K_{1,3}\\) with d=3 over an algebraically closed field; Theorem 4.9 predicts \\(v_{p_{\\emptyset}}(L_{K_{1,3}}^{\\mathbb{K}}(3))=1\\), since only the center has degree at least 3. Computing directly the least degree of a homogeneous f with \\((L_{K_{1,3}}^{\\mathbb{K}}(3):f)=p_{\\emptyset}\\), using the radical primary decomposition from the cited work, would settle the claim for this case, and the same check can be run on any small forest.","tokens_in":1620,"feed_emoji":"🌲","tokens_out":3318,"duration_ms":71091,"temperature":0.7,"pith_summary":"The paper introduces a framework for computing localized v-numbers of coordinate-saturated ideals, a class that includes Lovász–Saks–Schrijver (LSS) ideals of forests, LSS ideals over the reals, generalized binomial edge ideals, and related families. Its flagship result is an exact formula: for a forest G and d≥ 3, the localized v-number of the LSS ideal $L_G^{\\mathbb{K}}(d)$ at the minimal prime $p_{\\emptyset}(G)$ equals $v_d(G)$, the number of vertices of G of degree at least d. From this the authors derive that $v(L_G^{\\mathbb{K}}(d)) \\leq \\mathrm{reg}(R/L_G^{\\mathbb{K}}(d))$ for every d ≥ 1. The same framework gives a combinatorial formula for the localized v-number of parity binomial edge ideals of non-bipartite graphs, and yields $v(\\mathcal{I}_G) \\leq \\mathrm{reg}(R/\\mathcal{I}_G)$ for decorated trees and several related graph classes.","feed_headline":"Counting high-degree vertices fixes the v-number on forests","feed_subtitle":"New framework yields exact localized v-numbers for LSS ideals and v≤regularity on trees.","key_machinery":"The central object is the class of coordinate-saturated ideals, together with the hypergraph of coordinate subsets \\(S(q)\\) attached to the minimal primes q of I other than the distinguished variable-free prime p. The monomial ideal M is generated by monomials whose support meets every such \\(S(q)\\), and \\(\\gamma(I)\\) is the minimum size of such a transversal. Theorem 3.13 converts the localized v-number \\(v_p(I)\\) into this transversal number, with equality when \\(I+M\\) is radical, and the paper verifies that condition for the relevant families using monomial maps and Gröbner basis arguments showing \\(\\operatorname{in}_{\\prec}(I+M)=\\operatorname{in}_{\\prec}(I)+M\\). This machinery is what turns the v-number into a counting problem on the graph.","core_discovery":"The paper proves that for a forest graph G, an algebraically closed field \\(\\mathbb{K}\\), and d≥ 3, the localized v-number \\(v_{p_{\\emptyset}(G)}($L_G^{{\\mathbb{K}}$}(d))\\) coincides with \\(v_d(G)\\), the number of vertices whose degree in G is at least d. The argument runs through a new class of ideals: a radical ideal I is coordinate-saturated when every minimal prime has the form \\(P_S(I)=((I+(x_i:i\\in S)):\\prod_{i\\notin S}x_i^\\infty)\\) and a natural saturation compatibility holds. For such an ideal with a unique minimal prime p containing no indeterminates, Theorem 3.13 gives \\(v_p(I)\\leq \\gamma(I)\\), with equality whenever \\(I+M\\) is radical, where M is the monomial ideal generated by monomials in the intersection of the other minimal primes, and \\(\\gamma(I)\\) is the minimum degree of a monomial in M. In the forest LSS case, M is generated by the variables \\(x_{i,j}\\) with \\(i\\in V_{\\geq d}(G)\\), so \\(\\gamma(I)=v_d(G)\\), and the paper shows \\(I+M\\) is radical, giving the exact formula. The same machinery also yields an upper bound for LSS ideals over the real numbers and the equality \\(v_{p_+(G)}(\\mathcal{I}_G)=\\gamma_n(G)\\) for parity binomial edge ideals of non-bipartite graphs.","pith_inferences":["If Theorem 4.9 is correct, computing the localized v-number for forests is a pure degree-counting problem; a natural testable extension is whether the formula persists for graphs with cycles that admit an analogous decomposition of the orthogonal-representation variety, such as complete graphs or complete bipartite graphs.","The coordinate-saturated condition is checkable from the minimal primes alone, so one could scan other families of radical binomial ideals to find new instances where the localized v-number is a transversal number and where v≤reg follows automatically.","The paper's v≤reg results for decorated trees and special non-bipartite graphs suggest a broader conjecture that \\(v(\\mathcal{I}_G)\\leq \\mathrm{reg}(R/\\mathcal{I}_G)\\) for all connected non-bipartite graphs; testing chordal or cactus graphs would be a direct next step.","Since the equality \\(v_p(I)=\\gamma(I)\\) holds whenever \\(I+M\\) is radical, the most useful future direction may be locating more classes of ideals where that radicality can be certified combinatorially."],"forward_implications":["For every forest G and every d≥ 1, the inequality \\(v(L_G^{\\mathbb{K}}(d))\\leq \\mathrm{reg}(R/L_G^{\\mathbb{K}}(d))\\) holds; the exact localized formula at d≥ 3 is the engine for the d≥ 3 case.","For every non-bipartite simple graph G, the localized v-number \\(v_{p_+(G)}(\\mathcal{I}_G)\\) equals \\(\\gamma_n(G)\\), the minimum size of a dominating set S such that G[S] is totally non-bipartite.","For decorated trees and for several concrete non-bipartite families, \\(v(\\mathcal{I}_G)\\leq \\mathrm{reg}(R/\\mathcal{I}_G)\\), so the v-number is controlled by a standard homological invariant of the quotient ring.","For generalized binomial edge ideals, the sequence \\(v(J_G)\\geq v(J_{\\mathbb{K}_3,G})\\geq v(J_{\\mathbb{K}_4,G})\\geq \\cdots\\) is weakly decreasing, and for every closed graph G one has \\(v(J_G^k)\\leq \\mathrm{reg}(R/J_G^k)\\) for all k≥ 1.","The framework gives a unified route to previously separate v-number results for binomial edge ideals, parity binomial edge ideals, and LSS ideals."],"supporting_citations":[{"why":"Supplies the irreducible decomposition of the orthogonal-representation variety for forests and the primary decomposition of the LSS ideal, on which the coordinate-saturation proof and Theorem 4.9 rest.","marker":"[28]"},{"why":"Provides radicality of LSS ideals and the Cartwright–Sturmfels property used to show that the initial ideal plus the monomial ideal is radical (Proposition 4.8).","marker":"[8]"},{"why":"Gives the minimal prime decomposition of LSS ideals over the real numbers, which is the basis for the coordinate-saturation argument and the bound in Theorem 5.9.","marker":"[17]"},{"why":"Supplies the primary decomposition of parity binomial edge ideals, the notion of disconnector and sign-split primes, and radicality, which underpin the characterization of transversals and Theorem 7.5.","marker":"[24]"},{"why":"Gives the radicality and minimal prime decomposition of generalized binomial edge ideals, used to prove they are coordinate-saturated and to compute the localized v-number.","marker":"[32]"},{"why":"Defines the invariant t(G) and proves the regularity lower bound for LSS ideals of forests, which together with Lemma 4.13 yields the v≤reg result for d≥ 3.","marker":"[30]"},{"why":"Provides the regularity bound for binomial edge ideals of block graphs, used in the d=2 case of Theorem 4.14.","marker":"[23]"},{"why":"Gives the result for the localized v-number of binomial edge ideals used in Proposition 4.10 and in the bipartite case of the chorded even cycle result.","marker":"[21]"},{"why":"Establishes the strong persistence property and the equality of initial ideals of powers for closed graphs, used in the proof of Proposition 10.1.","marker":"[12]"}],"fun_headline_variants":["Forest LSS ideals: v-number equals vertex degree count","Exact localized v-numbers for LSS ideals on forests","v ≤ regularity for LSS and parity binomial edge ideals","Counting degree-≥d vertices yields forest LSS v-number","New framework gives exact v-numbers for forest LSS ideals"],"cache_read_input_tokens":29824,"weakest_assumption_plain":"The forest results rest on a cited decomposition theorem, currently appearing only as an unpublished preprint, asserting that the variety of orthogonal representations of a forest is the union of the closures of the strata \\(V_S\\) over all G-admissible subsets; if that decomposition fails for some forest, the localized formula and the d≥ 3 part of the regularity bound do not follow from the arguments given.","fun_headline_variants_meta":{"raw":{"variants":["Forest LSS ideals: v-number equals vertex degree count","Exact localized v-numbers for LSS ideals on forests","v ≤ regularity for LSS and parity binomial edge ideals","Counting degree-≥d vertices yields forest LSS v-number","New framework gives exact v-numbers for forest LSS ideals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000674,"raw_usage":{"total_tokens":3212,"prompt_tokens":1231,"completion_tokens":1981,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":847,"completion_tokens_details":{"reasoning_tokens":1899}},"tokens_in":847,"tokens_out":1981,"duration_ms":15146,"temperature":1.0,"reasoning_tokens":1899,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:13:12.940449+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the star \\(K_{1,3}\\) with d=3 over an algebraically closed field; Theorem 4.9 predicts \\(v_{p_{\\emptyset}}(L_{K_{1,3}}^{\\mathbb{K}}(3))=1\\), since only the center has degree at least 3. Computing directly the least degree of a homogeneous f with \\((L_{K_{1,3}}^{\\mathbb{K}}(3):f)=p_{\\emptyset}\\), using the radical primary decomposition from the cited work, would settle the claim for this case, and the same check can be run on any small forest.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the irreducible decomposition of the orthogonal-representation variety for forests and the primary decomposition of the LSS ideal, on which the coordinate-saturation proof and Theorem 4.9 rest."},{"cited_title":"Conca and V","cited_arxiv_id":null,"evidence_quote":"Provides radicality of LSS ideals and the Cartwright–Sturmfels property used to show that the initial ideal plus the monomial ideal is radical (Proposition 4.8)."},{"cited_title":"Herzog, A","cited_arxiv_id":null,"evidence_quote":"Gives the minimal prime decomposition of LSS ideals over the real numbers, which is the basis for the coordinate-saturation argument and the bound in Theorem 5.9."},{"cited_title":"Kahle, C","cited_arxiv_id":null,"evidence_quote":"Supplies the primary decomposition of parity binomial edge ideals, the notion of disconnector and sign-split primes, and radicality, which underpin the characterization of transversals and Theorem 7.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the radicality and minimal prime decomposition of generalized binomial edge ideals, used to prove they are coordinate-saturated and to compute the localized v-number."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the invariant t(G) and proves the regularity lower bound for LSS ideals of forests, which together with Lemma 4.13 yields the v≤reg result for d≥ 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the regularity bound for binomial edge ideals of block graphs, used in the d=2 case of Theorem 4.14."},{"cited_title":"Jaramillo-Velez and L","cited_arxiv_id":null,"evidence_quote":"Gives the result for the localized v-number of binomial edge ideals used in Proposition 4.10 and in the bipartite case of the chorded even cycle result."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the strong persistence property and the equality of initial ideals of powers for closed graphs, used in the proof of Proposition 10.1."}],"review_version":2}