$\mathrm{v}$-number of Lov\'asz-Saks-Schrijver Ideals and (parity) binomial edge ideals of graphs

Emiliano Liwski; Manohar Kumar

REVIEW 3 major objections 5 minor 35 references

$\mathrm{v}$-number of Lov\'asz-Saks-Schrijver Ideals and (parity) binomial edge ideals of graphs

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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≤reg inequality.

desk verdict A solid, useful framework for localized v-numbers, with the main forest formula riding on an unpublished preprint and a missing field hypothesis in Section 9. read the letter →

arxiv 2608.01199 v1 pith:KCXFQTBF submitted 2026-08-02 math.AC

Manohar Kumar , Emiliano Liwski This is my paper · ORCID

classification math.AC MSC 13F2005E4005E9913D45

keywords v-numberlocalizedLovász–Saks–Schrijveridealscoordinate-saturatedparitybinomialedgeCastelnuovo–Mumfordregularityforestgraphs

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

Soft 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.

The 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.

Who 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Section 4, Theorem 4.9 and Theorem 4.14] 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.
[Theorem 3.13 and Theorem 9.6] 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 7, Lemma 7.2 and Proposition 7.4] 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.

minor comments (5)

[Proposition 4.10] 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].
[Abstract] 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 5, after Lemma 5.8] 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.
[Proposition 8.8] 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.
[References] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the localized v-number formulas are parameter-free consequences of independent decomposition theorems and combinatorial lemmas, not reductions to their own inputs.

full rationale

The derivation chain contains no fitted parameters and no quantity that is defined in terms of the predicted invariant. Theorem 3.13 derives v_p(I) = gamma(I) from the minimal-prime data of a radical ideal; in the applications, gamma(I) is then identified with graph-theoretic quantities (v_d(G), gamma_n(G), gamma_{c,n}(G), gamma_c(G)) by combinatorial classification of the relevant admissible sets or transversals, not by construction. The forest section depends on [28, Theorem 3.18] for the irreducible decomposition of OR_d(G) and the resulting primary decomposition; [28] is by coauthor Liwski, but it is a parameter-free structural input whose stated assumptions (forest, d >= 3) do not include the target v-number formula, so under the review rules it counts as independent support rather than circularity. The same applies to [29, Lemma 3.7] in Theorem 9.6. The paper's modification of [28, Definition 3.15] by omitting condition (11) is an external-validity risk: if the redundancy argument or the cited decomposition were faulty, Theorem 4.9 and the d >= 3 case of Theorem 4.14 would not follow from the supplied arguments. That is a correctness/verification concern, not a circularity, and no equation in the paper is shown to be equivalent to its own input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard Nullstellensatz arguments and on structural theorems taken from prior literature, several authored or coauthored by the present authors. None of the results are numerical fits; the only inputs are the graph G and the integer d. New definitions such as coordinate-saturated ideals are mathematical notions, not additional postulated entities.

assumptions (6)

standard math Hilbert's Nullstellensatz over algebraically closed fields.
Used in Proposition 3.12 and Proposition 6.12 to prove ideal containments by comparing varieties.
domain assumption Irreducible decomposition OR_d(G) = union of V_S over G-admissible subsets for forests ([28, Theorem 3.18]).
Corollary 4.3 and Lemma 4.5 depend on this decomposition to identify the minimal primes of L_G^K(d), and Lemma 4.7 uses the resulting admissible subsets to identify the hypergraph. This source is a preprint by a coauthor.
domain assumption Primary decomposition of L_R_G(2) over the real numbers ([17, Theorems 1.1 and 5.2]).
Theorem 5.9 and Lemma 5.6 use the minimal primes Q_S(G) and the unique prime I_K_n containing no indeterminates.
domain assumption Primary decomposition and radicality of parity binomial edge ideals ([24, Theorems 4.15 and 5.5]).
Theorems 6.15 and 7.5 rely on the description q = m_S + p, sign-split primes, and the formula q(-) = S for disconnectors.
domain assumption Primary decomposition of generalized binomial edge ideals ([32, Corollary 4 and Theorem 7]).
Lemma 9.4 and Theorem 9.6 use the minimal primes P_T(K_m,G) indexed by cut sets of G.
domain assumption Quoted v-number and regularity bounds from prior literature ([21], [23], [30], [12], [27], [3], [33]).
The v at most reg applications in Sections 4, 8, and 10 chain bounds such as reg(R/L_G^K(d)) at least t(G), v(J_G) at most |iv(G)|, and the strong persistence property; these are cited, not proved.

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Pith. "Pith review of $\mathrm{v}$-number of Lov\'asz-Saks-Schrijver Ideals and (parity) binomial edge ideals of graphs." pith.science (2026). https://pith.science/paper/KCXFQTBF

@misc{pith2026260801199,
  author       = {Pith},
  title        = {Pith review of: $\mathrmv$-number of Lov\'asz-Saks-Schrijver Ideals and (parity) binomial edge ideals of graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCXFQTBF}},
  note         = {Machine review of arXiv:2608.01199}
}

abstract

In this paper, we introduce a new framework for computing certain localized $\mathrm{v}$-numbers of a class of ideals called coordinate-saturated ideals, which includes certain classes of Lov\'asz-Saks-Schrijver (LSS) ideals and (generalized) binomial edge ideals associated with graphs. For a forest graph $G$, we derive an explicit formula for the localized $\mathrm{v}$-number of the LSS ideal $L_G^{\mathbb{K}}(d)$, denoted by $\mathrm{v}_{\mathfrak{p}_{\emptyset}(G)}(L_G^{\mathbb{K}}(d))$, for all $d \geq 2$, where $\mathbb{K}$ is an algebraically closed field. As a consequence, we prove that $\mathrm{v}(L_G^{\mathbb{K}}(d)) \leq \mathrm{reg}(R/L_G^{\mathbb{K}}(d)),$ where $\mathrm{v}(L_G^{\mathbb{K}}(d))$ and $\mathrm{reg}(R/L_G^{\mathbb{K}}(d))$ denote the $\mathrm{v}$-number of $L_G^{\mathbb{K}}(d)$ and the Castelnuovo-Mumford regularity of $R/L_G^{\mathbb{K}}(d)$, respectively. Also, we give an upper bound for $\mathrm{v}_{I_{K_{n}}}(L_{G}^{\mathbb{R}}(2))$, where $\mathbb{R}$ is field of real numbers. Furthermore, we provide combinatorial descriptions of the localized $\mathrm{v}$-number of parity binomial edge ideals, denoted by $\mathrm{v}_{\mathfrak{p}^{+}(G)}(\mathcal{I}_{G})$, and, as an application, show that $\mathrm{v}(\mathcal{I}_G) \leq \mathrm{reg}(R/\mathcal{I}_G)$ for several classes of non-bipartite graphs. Finally, we prove that $\mathrm{v}(J_G^k)\leq \mathrm{reg}(R/J_G^k)$ for all powers of binomial edge ideals of closed graphs $G$.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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E. Liwski. Lov´ asz–saks–schrijver ideals and the irreducible components of the variety of orthogonal representations of a graph.arXiv:2512.22954, 2025

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work page 2024

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S. Beyarslan, H. T. H` a, and T. N. Trung. Regularity of powers of forests and cycles.J. Algebraic Combin., 42(4):1077–1095, 2015

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