Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Syzygies of polymatroidal ideals

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A new cave polynomial for polymatroids is shown to be valuative and to encode the Möbius support, settling two conjectures about syzygies of polymatroidal ideals.

desk verdict Resolves two open conjectures with a new cave polynomial, but the key identification cave = Möbius series is imported from an unpublished preprint and needs a self-contained proof. read the letter →

arxiv 2507.13153 v1 pith:4BKJIGVR submitted 2025-07-17 math.AC math.AGmath.CO

Yairon Cid-Ruiz , Jacob P. Matherne , Anna Shapiro This is my paper · ORCID
classification math.ACmath.AGmath.CO MSC 05B3505E4013D02
keywords cavepolynomialpolymatroidpolymatroidalidealshomologicalshiftMöbiussupportK-polynomialvaluativefunctiongeneralized
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 new invariant for polymatroids, the cave polynomial, and proves that it is valuative, that its homogenized support is a generalized polymatroid, and that it equals the generating series of the Möbius values of the polymatroid. These facts settle two open conjectures: every homological shift ideal of a polymatroidal ideal is again polymatroidal, and the Möbius support of a polymatroid is a generalized polymatroid. The main mechanism is K-theoretic: the cave polynomial describes the class of a polymatroid in the augmented K-ring of a multisymmetric lift, and a duality identity expresses the K-polynomial of the polymatroidal ideal through the cave polynomial of the dual polymatroid. If the claims hold, the full multigraded syzygy structure of a polymatroidal ideal is read off from the Möbius support of its dual.

What carries the argument

The cave polynomial is $$\operatorname{cave}_P(t_1,\dots,t_p) = \sum_{n\in B(P)\cap \mathbb{N}^p,\ |n|=\operatorname{rk}(P)} 1_P(n) \prod_{i=1}^{p-1}\left(1 - \max_{i<j} 1_P(n-e_i+e_j)\, $t_i^{{-1}}$\right) t^n;$$ the factors record which neighboring lattice points of the base polytope are missing. The proof's engine is the equality of the coefficients of this polynomial with the Möbius values $\mu_P(n)$, together with the duality identity $K(I_P;t) = t^m \operatorname{cave}_{P^\vee}(t^{-1})$, which turns the syzygy question into a statement about the support of one polynomial. Valuativity then extends the results from realizable polymatroids to all polymatroids.

What would settle it

Compute both sides of $\operatorname{cave}_P(t) = \sum \mu_P(n)t^n$ for every polymatroid on a small ground set with small cages, including non-realizable ones; a single polymatroid where any coefficient differs would refute Theorem A(ii) and invalidate the homological-shift formula. The paper's own example shows the computation is feasible by hand in small cases.

Watch

Extended reading notes

Core claim

Theorem A states that for every polymatroid $P$ with cage $m$, the cave polynomial satisfies $$\operatorname{cave}_P(t_1,\dots,t_p) = \sum_{n\in\mathbb{N}^p} \mu_P(n) $t_1^{{n_1}}$\cdots $t_p^{{n_p}}$,$$ so its support is the Möbius support of $P$, and that support is a generalized polymatroid. It also states that the K-polynomial of the polymatroidal ideal $I_P$ is $$K(I_P;t) = $t_1^{{m_1}}$\cdots $t_p^{{m_p}}$\, \operatorname{cave}_{P^\vee}($t_1^{{-1}}$,\dots,$t_p^{{-1}}$),$$ where $P^\vee = m - P$ is the dual polymatroid. Consequently the $i$-th homological shift ideal is generated by the monomials $x^n$ with $|n| = \operatorname{rk}(P)+i$ and $\mu_{P^\vee}(m-n) \neq 0$. The assignment $P \mapsto \operatorname{cave}_P(t)$ is valuative, which lets the authors transfer arguments from realizable polymatroids to all polymatroids.

Load-bearing premise

The load-bearing premise is that $\operatorname{cave}_P(t) = \sum \mu_P(n)t^n$, an identification imported from earlier results [CCRMM22] and [Knu09] rather than derived in this paper, together with the assertion in the proof of part (i) that one condition in the definition of a cave holds "by construction" without explicit verification.

Editorial extensions

If this is right

  • Both open conjectures are settled: every homological shift ideal $\operatorname{HS}_i(I_P)$ is polymatroidal, and the Möbius support of a polymatroid is a generalized polymatroid.
  • The syzygies of $I_P$ admit a closed formula in terms of the dual polymatroid: $\operatorname{HS}_i(I_P) = \{ x^n : |n| = \operatorname{rk}(P)+i,\ \mu_{P^\vee}(m-n)\neq 0\}$.
  • The K-polynomial is determined by the cave polynomial of the dual, so the K-theoretic invariants of a polymatroid are valuative.
  • Because $\operatorname{cave}_P$ is valuative, any linear relation among polymatroid indicators forces the same relation among cave polynomials, allowing future computations to reduce to realizable polymatroids.
  • The support of $K(I_P;t)$ is a generalized polymatroid, so the multigraded Betti data of $I_P$ are organized as a polymatroidal family.

Reading between the lines

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

  • Since the paper verifies in its example that the homogenized sign-changed cave and K-polynomials are denormalized Lorentzian, one may conjecture that this Lorentzian property holds for all polymatroids; that would connect the cave polynomial to the Hodge-theoretic framework of Lorentzian polynomials.
  • The equality of the cave polynomial with the Möbius generating series suggests interpreting the cave polynomial as a discrete volume-like invariant whose valuativity could yield inclusion-exclusion formulas for Möbius supports of subdivisions of polymatroid base polytopes.
  • The dual formula for homological shift ideals gives an algorithmic route to Betti numbers: compute the Möbius function of the dual polymatroid instead of resolving $I_P$ directly; testing on random polymatroids would show whether this is practically faster.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript introduces the cave polynomial cave_P(t) of a polymatroid P, claims that it is valuative, that its support (after homogenization) is a generalized polymatroid, and that it encodes the Möbius values of P. Theorem A asserts that these properties settle two open conjectures: Bandari–Bayati–Herzog on homological shift ideals of polymatroidal ideals and Castillo–Cid-Ruiz–Mohammadi–Montaño on Möbius supports. The proof strategy combines K-theoretic classes of the multiprojective variety Y_P, a shelling argument from an earlier preprint, and valuative functions on polymatroids. An explicit rank-two example is computed and checked with SageMath.

Significance. If the main theorem is correct, the paper resolves two conjectures in one stroke and introduces a new invariant, the cave polynomial, with attractive valuativity and K-theoretic properties. The explicit K-polynomial formula in Theorem A(iii), the explicit description of homological shift ideals, and the use of valuative methods are valuable contributions. However, the central bridge of the proof is imported from an unpublished preprint [CCRMM22], and one cave axiom in the proof of Theorem A(i) is asserted rather than verified. The manuscript therefore is not yet self-contained for its main claims.

major comments (3)
  1. [Remark 2.14 and Theorem A(ii)] The equality cave_P(t) = sum_{n} mu_P(n) t^n is the load-bearing identification of the paper, but it is not proved here. The proof in Remark 2.14 cites [CCRMM22, proof of Lemma 6.8] for a shelling of Delta(J_P) and [CCRMM22, Proposition 4.6] for the coefficient interpretation, and Proposition 2.10(i) cites [Knu09]. Since [CCRMM22] is an unpublished arXiv preprint with overlapping authorship, and since no statement of its hypotheses or an outline of the shelling argument is included, the current manuscript does not establish this identification for an arbitrary polymatroid. This gap propagates into Theorem A(ii), A(iii), A(iv), and the proof of A(i), all of which use Remark 2.14. Please provide a self-contained proof of Remark 2.14, or state and prove the necessary shelling and coefficient identities inside this paper.
  2. [Proof of Theorem A(i), condition (b)] In the proof that the support C is a cave, condition (b) of [CCRMM22, Definition 5.8] is dismissed with the sentence 'Part (b) holds by construction since the cave polynomial mimics the notion of stalactites.' This is not a mathematical verification. Condition (b) concerns the behavior of the truncated set A = C_b with respect to the boundary of the cave, and it is essential for the conclusion that C is a cave and hence a generalized polymatroid. Please give an explicit verification of condition (b) for the set A, using the definition of the cave polynomial.
  3. [Definition 2.13 and Remark 2.14] The cave polynomial is defined using a fixed order 1 < 2 < ... < p on the variables, with the product over i = 1,...,p-1 and max over j > i. Remark 2.14 then asserts that 'by symmetry' the same polynomial is obtained for any permutation pi. This order-independence is not proved; it would be a consequence of the equality cave_P(t) = sum mu_P(n) t^n, but that equality is exactly the statement being imported from [CCRMM22]. Please prove the order-independence directly or clarify that the definition depends on the chosen order and only the stated properties are needed for the fixed order.
minor comments (4)
  1. [Definition 2.13] For p = 1 the product over i from 1 to p-1 is empty; please state explicitly that the empty product is 1, so that cave_P(t) = t^{rk(P)} in that case.
  2. [Proposition 2.18] The reduction to the valuativity of the Hilbert function of I_P is stated briefly as 'Due to Remark 2.14 and Proposition 2.10'; please expand this reduction so a reader can follow how the cave polynomial coefficients are obtained from the Hilbert function without additional external identifications.
  3. [Example 2.21] The SageMath check that the homogenized sign-changed polynomials are denormalized Lorentzian is reported without a reference or an explanation of its role; since this check is not used in the proofs, please clarify whether it is merely illustrative or is intended as evidence for a stronger property.
  4. [Abstract and Remark 2.6] The abstract says the support 'after homogenization is again a polymatroid,' while the body states the support is a generalized polymatroid; consider making the wording uniform to avoid confusion.

Circularity Check

2 steps flagged · score 6.0 of 10

The central equality cave_P(t)=Σ μ_P(n)t^n is imported from the same authors' unpublished preprint, and the cave-condition check is asserted rather than verified.

  1. self citation load bearing [Remark 2.14; used in Proof of Theorem A(i)-(iii), Section 2]
    "By ordering the points in B(P) ∩ Np with respect to the lexicographic order ... we obtain a shelling of the facets of the simplicial complex Δ(JP) associated to JP (see [CCRMM22, proof of Lemma 6.8]). Then by [CCRMM22, Proposition 4.6], we obtain that the coefficients of the cave polynomial caveP(t) describe the class [O_YP] ∈ K(P); ... Hence we have the equalities a_n(P) = c_n(YP) = μ_P(n)."

    Theorem A(ii) is asserted as 'This part follows from Remark 2.14 and part (i),' and Theorem A(i) begins by identifying μ-supp(P) with supp(caveP(t)) via the same remark. The remark is not proved in the paper: it cites a lexicographic shelling from [CCRMM22, proof of Lemma 6.8] and [CCRMM22, Proposition 4.6], an unpublished preprint whose author list includes one of the present authors. Thus the load-bearing bridge between the new cave polynomial and the Möbius function is an imported same-author result, not an independently derived consequence within this paper.

  2. ansatz smuggled in via citation [Proof of Theorem A(i), Section 2, p. 9]
    "Part (b) holds by construction since the cave polynomial mimics the notion of stalactites."

    The only verification given for condition (b) of [CCRMM22, Definition 5.8] is an appeal to informal mimicry. Whether the support of caveP satisfies the cave condition is exactly what must be checked to conclude it is a generalized polymatroid via [CCRMM22, Theorem 5.18]. No explicit inequalities or set-theoretic verification are supplied, so the desired structural conclusion is being assumed through the imported 'stalactites' intuition rather than derived.

full rationale

The paper introduces a genuinely new object, the cave polynomial, and proves valuativity using Hilbert-function regions and [AFR10, Corollary 4.3]; that portion has independent content. However, the central identification cave_P(t)=Σ μ_P(n)t^n — the bridge that makes Theorem A(ii) true and that underlies Theorem A(i) and the K-polynomial formula in Theorem A(iii) — is not derived in this paper. It is imported from [CCRMM22, proof of Lemma 6.8] and [CCRMM22, Proposition 4.6], an unpublished preprint with overlapping authorship. The paper also checks the crucial cave condition (b) only by the sentence 'holds by construction since the cave polynomial mimics the notion of stalactites,' without a verification. Because the main theorem's conclusion about Möbius support and homological shift ideals rests on this same-author citation chain and on an asserted structural condition, the circularity is partial but significant; it is not a total definitional circle since the cave polynomial definition, the valuativity argument, and the polarization argument are genuinely new and independent.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No numeric parameters are fitted; the ledger records the external results the proof imports, mainly from the unpublished preprint [CCRMM22] and from [Knu09], [HH11], and [AFR10]. The only genuinely new invented object is the cave polynomial, which is the paper's main contribution rather than an unexplained postulate.

assumptions (4)
  • domain assumption The cave theory of [CCRMM22]: Definition of caves, Proposition 4.6 identifying cave coefficients with K-theoretic constants, Theorem 5.18 that caves are generalized polymatroids, and Proposition 7.15 on realizable polymatroids.
    These results are not proved in this paper and are load-bearing for Theorem A(i)-(iii). Since [CCRMM22] is a preprint coauthored by Cid-Ruiz, this is a significant external dependency.
  • domain assumption The equality c_n(Y_P) = mu_P(n) from [Knu09] and the shelling statement from [CCRMM22, proof of Lemma 6.8].
    Used in Remark 2.14 and Proposition 2.10(i) to identify cave coefficients with Möbius values, a central step in settling Conjecture 1.2.
  • domain assumption The valuativity result [AFR10, Corollary 4.3] for matroids extends verbatim to polymatroids.
    The proof of Proposition 2.18 states 'the same proof holds for polymatroids' without giving the adaptation; Lemma 2.20 also depends on this extension.
  • standard math Polymatroidal ideals have linear resolutions [HH11, Theorem 12.6.2], so the support of the K-polynomial can be read as the union of homological shift supports.
    Used in the proof of Theorem A(iii) to pass from support of K(I_P;t) to the generators of HS_i(I_P).
invented entities (1)
  • Cave polynomial of a polymatroid
    purpose: Encodes the Möbius values and K-theoretic constants of a polymatroid; its support is a generalized polymatroid and it is valuative.
    The paper introduces this definition and proves internal properties about it. There is no external empirical handle, and its usefulness will be judged by future applications.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Syzygies of polymatroidal ideals." pith.science (2026). https://pith.science/paper/4BKJIGVR

@misc{pith2026250713153,
  author       = {Pith},
  title        = {Pith review of: Syzygies of polymatroidal ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4BKJIGVR}},
  note         = {Machine review of arXiv:2507.13153}
}
abstract

We introduce the cave polynomial of a polymatroid and show that it yields a valuative function on polymatroids. The support of this polynomial after homogenization is again a polymatroid. The cave polynomial gives a $K$-theoretic description of a polymatroid in the augmented $K$-ring of a multisymmetric lift. As applications, we settle two conjectures: one by Bandari, Bayati, and Herzog regarding polymatroidal ideals, and another by Castillo, Cid-Ruiz, Mohammadi, and Monta\~no regarding the M\"obius support of a polymatroid.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Three Combinatorial Algorithms for the Cave Polynomial of a Polymatroid

    math.CO 2026-01 accept novelty 5.0 of 10

    Four polynomials of a polymatroid—cave, stalactite, box, and Möbius—are shown to coincide by purely combinatorial coefficient comparisons.

Reference graph

Works this paper leans on

24 extracted references · 23 canonical work pages · cited by 1 Pith paper

  1. [1]

    Federico Ardila, Alex Fink, and Felipe Rinc\'on, Valuations for matroid polytope subdivisions, Canad. J. Math. 62 (2010), no. 6, 1228--1245. 2760656

    work page 2010

  2. [2]

    Shamila Bayati, Multigraded shifts of matroidal ideals, Archiv der Mathematik 111 (2018), 239--246

    work page 2018

  3. [3]

    Petter Br\" a nd\' e n and June Huh, Lorentzian polynomials, Ann. of Math. (2) 192 (2020), no. 3, 821--891. 4172622

    work page 2020

  4. [4]

    Math., vol

    Michel Brion, Multiplicity-free subvarieties of flag varieties, Commutative algebra ( G renoble/ L yon, 2001), Contemp. Math., vol. 331, Amer. Math. Soc., Providence, RI, 2003, pp. 13--23. 2011763

    work page 2001

  5. [5]

    Alessio Caminata, Yairon Cid-Ruiz, and Aldo Conca, Multidegrees, prime ideals, and non-standard gradings, Adv. Math. 435 (2023), Paper No. 109361, 46. 4658828

    work page 2023

  6. [6]

    Federico Castillo, Yairon Cid-Ruiz, Fatemeh Mohammadi, and Jonathan Monta \ n o, K-polynomials of multiplicity-free varieties, arXiv preprint arXiv:2212.13091 (2022)

    arXiv 2022

  7. [7]

    Colin Crowley, June Huh, Matt Larson, Connor Simpson, and Botong Wang, The B ergman fan of a polymatroid , arXiv preprint arXiv:2207.08764 (2022), to appear in Transactions of the American Mathematical Society

    work page Pith review arXiv 2022

  8. [8]

    Harm Derksen and Alex Fink, Valuative invariants for polymatroids, Adv. Math. 225 (2010), no. 4, 1840--1892. 2680193

    work page 2010

Show all 24 references
  1. [9]

    Christopher Eur and Matt Larson, K-theoretic positivity for matroids, arXiv preprint arXiv:2311.11996 (2023), to appear in Journal of Algebraic Geometry

    2023 arXiv

  2. [10]

    , Intersection theory of polymatroids, Int. Math. Res. Not. IMRN (2024), no. 5, 4207--4241. 4714340

    2024

  3. [11]

    Antonino Ficarra and J \"u rgen Herzog, Dirac’s theorem and multigraded syzygies, Mediterranean Journal of Mathematics 20 (2023), no. 3, 134

    2023

  4. [12]

    Antonino Ficarra, Homological shifts of polymatroidal ideals, arXiv preprint arXiv:2205.04163 (2022)

    2022 arXiv

  5. [13]

    260, Springer-Verlag London, Ltd., London, 2011

    J\" u rgen Herzog and Takayuki Hibi, Monomial ideals, Graduate Texts in Mathematics, vol. 260, Springer-Verlag London, Ltd., London, 2011. 2724673

    2011

  6. [14]

    J \"u rgen Herzog, Somayeh Moradi, Masoomeh Rahimbeigi, and Guangjun Zhu, Homological shift ideals, Collectanea Mathematica 72 (2021), 157--174

    2021

  7. [15]

    3, 1245--1318

    Allen Knutson and Ezra Miller, Gr\" o bner geometry of S chubert polynomials , Annals of Mathematics 161 (2005), no. 3, 1245--1318

    2005

  8. [16]

    Allen Knutson, Frobenius splitting and M \"obius inversion , arXiv preprint arXiv:0902.1930 (2009)

    2009 arXiv

  9. [17]

    Matt Larson, Shiyue Li, Sam Payne, and Nicholas Proudfoot, K-rings of wonderful varieties and matroids, Advances in Mathematics 441 (2024), 109554

    2024

  10. [18]

    227, Springer-Verlag, New York, 2005

    Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005

    2005

  11. [19]

    Kazuo Murota, Discrete convex analysis, SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003

    2003

  12. [20]

    Isabella Novik, Alexander Postnikov, and Bernd Sturmfels, Syzygies of oriented matroids, Duke Math. J. 111 (2002), no. 2, 287--317. 1882136

    2002

  13. [21]

    Roberto Pagaria and Gian Marco Pezzoli, Hodge theory for polymatroids, Int. Math. Res. Not. IMRN (2023), no. 23, 20118--20168. 4675068

    2023

  14. [22]

    Julius Ross , Hendrik S \"u , and Thomas Wannerer , Dually Lorentzian Polynomials , arXiv e-prints (2023), arXiv:2304.08399

    2023 arXiv

  15. [23]

    Sage Developers , S agemath, the S age M athematics S oftware S ystem ( V ersion 10.6 v2.6.0) , 2025, https://www.sagemath.org

    2025

  16. [24]

    P olyhedra and efficiency

    Alexander Schrijver, Combinatorial optimization. P olyhedra and efficiency. V ol. B , Algorithms and Combinatorics, vol. 24, Springer-Verlag, Berlin, 2003, Matroids, trees, stable sets, Chapters 39--69

    2003

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.