{"id":"c11e218d-7620-4789-962b-0525369fa263","arxiv_id":"2507.13153","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every polymatroid, the cave polynomial is valuative, its support is a generalized polymatroid, and its coefficients are the Möbius values, settling the Bandari-Bayati-Herzog and Castillo-Cid-Ruiz-Mohammadi-Montano conjectures.","lead":"This paper introduces a new polynomial invariant for polymatroids, the cave polynomial, and uses it to prove two open conjectures about the algebraic structure of polymatroidal ideals. The result provides a K-theoretic description of these objects and shows their Möbius supports are always generalized polymatroids.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central identification cave_P(t)=Σ μ_P(n)t^n is imported from an unpublished preprint and used to prove both conjectures; Theorem A(ii) is not self-contained, and the support condition (b) is asserted.","rationale":"The paper's main theorem rests on two pillars: (1) cave_P equals the Möbius polynomial, and (2) the support of cave_P is a cave. The first is the more load-bearing because it is used in all major conclusions: it turns K-theoretic coefficients into Möbius values, it gives the explicit homological shift formula, and it underlies the valuativity reduction in Proposition 2.18. The provided justification is Remark 2.14, which invokes a shelling and [CCRMM22, Proposition 4.6] from an unpublished preprint sharing an author with this paper; the shelling and the coefficient identification are not proved here. If that proposition has hidden hypotheses, or if the shelling fails for non-realizable polymatroids, the equality can fail and both conjectures lose their proof. This is a missing verifiable step rather than a demonstrated error; the result may well be true. The second issue, the one-line verification of cave condition (b) in Theorem A(i), is also unresolved but secondary: even granting cave_P equals the Möbius polynomial, the proof needs an explicit verification of condition (b). The reader's conditional verdict is therefore appropriate, and I see no reason to change it.","tokens_in":10575,"tokens_out":8548,"duration_ms":101215,"concrete_test":"Independently prove Remark 2.14 from Definition 2.13: show that the lexicographic order on B(P)∩N^p gives a shelling of Δ(J_P) for every polymatroid, and that [CCRMM22, Proposition 4.6] identifies the coefficient of t^n with c_n(Y_P); then the desired equality follows from Proposition 2.10(i). As a computational cross-check, compute both sides of cave_P(t)=Σ μ_P(n)t^n for all polymatroids on [3] with cage (2,2,4) and for random polymatroids on [4] with small cages, comparing coefficients; any mismatch falsifies Theorem A(ii). In the same examples, verify [CCRMM22, Definition 5.8(b)] directly for the support of cave_P to test the asserted proof of Theorem A(i).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing bridge of the paper is Remark 2.14: the coefficients of cave_P(t) are claimed to equal the class [O_Y_P] in K(P) and hence the Möbius values μ_P(n). This equality is not derived in the present text. The sketch is: order B(P)∩N^p lexicographically, obtain a shelling of Δ(J_P) from [CCRMM22, proof of Lemma 6.8], and then invoke [CCRMM22, Proposition 4.6]. Both steps are cited from an unpublished preprint on which one of the present authors is a coauthor. If Proposition 4.6 in [CCRMM22] has hidden hypotheses, or if the lexicographic shelling is not valid for arbitrary polymatroids, the equality cave_P = Σ μ_P t^n can fail, and then both Conjecture 1.1 and Conjecture 1.2 lose their proof. The identity is also used in the valuativity argument of Proposition 2.18 and in the K-polynomial formula of Theorem A(iii). A secondary gap is in the proof of Theorem A(i): condition (b) of [CCRMM22, Definition 5.8] is checked with the sentence 'holds by construction since the cave polynomial mimics stalactites,' with no explicit verification of the condition.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10841,"tokens_out":6043,"duration_ms":78093,"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":[{"comment":"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.","section":"Remark 2.14 and Theorem A(ii)"},{"comment":"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.","section":"Proof of Theorem A(i), condition (b)"},{"comment":"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.","section":"Definition 2.13 and Remark 2.14"}],"minor_comments":[{"comment":"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.","section":"Definition 2.13"},{"comment":"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.","section":"Proposition 2.18"},{"comment":"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.","section":"Example 2.21"},{"comment":"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.","section":"Abstract and Remark 2.6"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the manuscript's dependence on the unpublished preprint [CCRMM22] for the identification cave_P = sum mu_P t^n, a step on which all main conclusions rest. Since one of the present authors is a coauthor of that preprint, it is especially important that the present paper either reproduce the argument or await/verify publication. The other gap, the unverified cave condition (b), may be fixable locally. I do not see an internal inconsistency that would force rejection; the result is plausible and significant, but the paper needs to be made self-contained at its load-bearing points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper introduces a cave polynomial for any polymatroid and uses it to prove the Bandari–Bayati–Herzog conjecture on homological shift ideals and the Castillo–Cid-Ruiz–Mohammadi–Montaño conjecture on Möbius supports. That is a real result, and the cave polynomial itself is a decent new invariant with a clean valuativity statement. The proof strategy is coherent: show the polynomial is valuative, pin its support, and read off the Möbius values from the dual.\n\nWhat's good: Theorem A(iii) gives an explicit description of every HS_i(I_P) in terms of Möbius values of the dual, and the example is concrete, with Sage Lorentzian checks. Proposition 2.18 (valuativity) is a solid argument, reducing to the realizable case via known generators of the valuative group.\n\nSoft spots. The main bridge is Remark 2.14: cave_P(t) = sum mu_P(n) t^n. It is imported from [CCRMM22], an unpublished preprint, and from Knutson's 2009 notes. The paper does not derive it. Since one of the authors is a coauthor of that preprint, this is not an independent check. If [CCRMM22, Proposition 4.6] has hidden hypotheses, or the shelling is not valid for arbitrary polymatroids, both conjectures lose their proof. A referee needs to see a self-contained proof of this identification, or at least a clear indication that no extra hypotheses are used.\n\nSecond soft spot: the proof of Theorem A(i) verifies cave condition (b) by saying it holds by construction since the cave polynomial mimics stalactites. That is a hand-wave. The condition needs an explicit verification.\n\nThese are fixable, and nothing in the paper suggests the result is wrong. The dependence on an unpublished preprint is the main reason I wouldn't cite it yet in my own work.\n\nBottom line: the paper deserves a serious referee, but the referee should be told to focus on making Remark 2.14 self-contained and expanding the proof of (b) in Theorem A(i). If those checks pass, this is a solid contribution. Take it to peer review, with the caveat.","headline":"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.","tokens_in":11365,"tokens_out":2251,"would_cite":false,"duration_ms":24674,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B35","05E40","13D02"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["cave polynomial","polymatroid","polymatroidal ideals","homological shift ideals","Möbius support","K-polynomial","valuative function","generalized polymatroid"],"falsifier":"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.","tokens_in":10388,"feed_emoji":"🧮","tokens_out":7276,"duration_ms":70750,"temperature":0.7,"pith_summary":"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.","feed_headline":"Cave polynomial settles two polymatroid conjectures","feed_subtitle":"One polynomial links Möbius values to free resolutions, resolving both open questions.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the shelling of the simplicial complex and Proposition 4.6 equating cave coefficients with K-class coefficients, the bridge used for Theorem A(ii).","marker":"[CCRMM22]"},{"why":"Gives the Möbius-inversion link used to identify the coefficients $c_n(Y_P)$ with $\\mu_P(n)$ in Proposition 2.10(i).","marker":"[Knu09]"},{"why":"Establishes valuativity of the upper-region indicator used in Proposition 2.18 to prove that the cave polynomial is valuative.","marker":"[AFR10]"},{"why":"Provides the Lorentzian criterion invoked in Lemma 2.19 to show truncations and shifted polymatroids are again polymatroids.","marker":"[BH20]"},{"why":"Extends Lorentzian preservation to the generating functions of the shifted and truncated polymatroids used in Lemma 2.19.","marker":"[RSW23]"},{"why":"Identifies the Snapper polynomial of a matroid with the Hilbert-function datum of $Y_P$, giving equality (2) used in the K-theoretic description.","marker":"[EL23]"},{"why":"Introduces the augmented K-ring of a matroid used to define the Snapper polynomial of a polymatroid.","marker":"[LLPP24]"},{"why":"Shows the valuative group of polymatroids is generated by realizable polymatroids, used in Remark 2.12 to reduce to the realizable case.","marker":"[DF10]"}],"fun_headline_variants":["Cave polynomial proves two polymatroid conjectures","Polymatroid conjectures resolved by cave polynomial","Cave polynomial reveals Möbius support of polymatroids","Valuative cave polynomial settles polymatroid conjectures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Cave polynomial proves two polymatroid conjectures","Polymatroid conjectures resolved by cave polynomial","Cave polynomial reveals Möbius support of polymatroids","Valuative cave polynomial settles polymatroid conjectures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001089,"raw_usage":{"total_tokens":4524,"prompt_tokens":890,"completion_tokens":3634,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":3567}},"tokens_in":506,"tokens_out":3634,"duration_ms":27414,"temperature":1.0,"reasoning_tokens":3567,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:29:46.615419+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}