{"id":"3bce9c00-2652-4681-a140-c63106f37a5b","arxiv_id":"2411.16654","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The support of a dual Schubert polynomial equals the Minkowski sum of inversion intervals, and its Newton polytope vertices come from coefficient-1 monomials of a product of linear forms.","lead":"This paper pins down exactly which monomials occur in every dual Schubert polynomial, a family that encodes geometry of Schubert varieties. It gives an elementary proof of their M-convexity and a recipe for the corner points of their Newton polytopes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the support identity rests on Lemma 3.7, whose terse proof is opaque but the existence claim is easily completed by choosing a maximal cover.","rationale":"The central claim is the support identity supp(D_w)=supp(GW(w)). The inclusion supp(m_C)⊆supp(GW(w)) for every chain is proved by Lemma 3.13's dominant pairing and Lemma 3.14; I re-checked the case analysis in Lemma 3.13 and it is correct. The reverse inclusion needs a chain whose weight is exactly GW(w); Lemma 3.9 shows any greedy chain has this weight, provided Lemma 3.7 guarantees existence. The only under-specified part is Lemma 3.7's construction. However, the greedy condition only rules out replacements with strictly larger intervals, so choosing an inclusion-maximal cover (with respect to interval containment) among all covers lying in the interval [u,w] yields a valid greedy step. Such a maximal cover exists by finiteness. Adding the new lower element does not change [u,w], so previously satisfied greedy conditions persist. Consequently the proof is complete up to an easily filled detail. No circularity: the argument does not assume M-convexity. The vertex characterization via [1, Theorem 3.5] is not independently verified here, but the worked example is consistent with it, and it is not the load-bearing step for the main support theorem. I therefore agree with the reader's acceptance; the weakest assumption is a presentation gap, not a threat.","tokens_in":10822,"tokens_out":24317,"duration_ms":206874,"concrete_test":"Verify the existence of greedy chains computationally for all Bruhat intervals in S_n, n≤6, by implementing the maximal-cover construction: at each downward step from w, among covers u' of the current element with u ≤ u' < current, select one whose transposition interval [a,b] is inclusion-maximal; confirm the resulting chain satisfies Definition 3.5. If any interval fails, the reverse inclusion in Theorem 3.15 would need revisiting.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After re-examining Theorem 1.2's proof, I find no load-bearing flaw. The reverse inclusion supp(GW(w)) ⊆ supp(D_w) uses Lemma 3.7 to obtain a dominant chain. Lemma 3.7's one-sentence proof is terse, but the assertion is true: at each top-down step, among the finitely many covers u' of u_{i-1} lying in [u,w], choose one whose swapped pair (a,b) is inclusion-maximal in the partial order of intervals. If some cover w' had b'>b or a'<a, the chosen pair would not be maximal, so the greedy condition holds, and earlier edges are unaffected because [u,w] is fixed. Lemma 3.9's inversion-set analysis and Lemma 3.13's dominant pairing also check out; the example in Section 4 is consistent with Corollary 1.4's coefficient-1 characterization. Thus the reader's identified weakness is a presentation gap, not a correctness risk.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies dual Schubert polynomials D_w, defined as averages of weights of saturated chains in the Bruhat interval [id, w]. Its main result (Theorem 1.2) identifies supp(D_w) with the Minkowski sum of elementary-vector sets indexed by the inversions of w, equivalently with supp(GW(w)), where GW(w) is the product over inversion pairs (a,b) of the linear forms x_a + ... + x_{b-1}. From this support identity the authors derive an elementary proof of M-convexity (Corollary 1.3), describe Newton(D_w) as a generalized permutahedron with explicit parameters (Theorem 3.18), and characterize its vertices as the monomials of coefficient 1 in GW(w) (Corollary 1.4), together with a staircase-tiling description.","tokens_in":10905,"tokens_out":11844,"duration_ms":102586,"significance":"If correct, this is a genuine strengthening of the Huh--Matherne--Mészáros--St. Dizier M-convexity theorem: the support is made explicit, the Newton polytope is identified as a generalized permutahedron with a simple parameter formula, and the vertices are characterized combinatorially. A particular strength is that the proof is self-contained: it does not invoke the earlier M-convexity result, and it proceeds through an elementary greedy-chain argument and the global weight GW(w). The matroid-polytope passage in Section 3.3 is clean and uses only standard facts. The conjectures about Postnikov--Stanley polynomials are clearly labeled as such, and the computational checks in SageMath are an honest part of the evidence. Overall, the paper is a solid contribution to the Newton-polytope literature in algebraic combinatorics.","major_comments":[],"minor_comments":[{"comment":"The proof of Lemma 3.7 is a single sentence and is the only place where the reverse inclusion supp(GW(w)) ⊆ supp(D_w) is justified. The construction of a maximal cover at each top-down step should be written out, and the proof should state explicitly why the greedy condition for previously constructed edges is preserved when the interval [u,w] is fixed. The assertion is true, but as written it is too terse for a load-bearing lemma.","section":"Lemma 3.7"},{"comment":"The permutation in Example 4.2 is written as D254361 in the text and as D253641 in Step 2 and the figure caption; the two notations should be reconciled.","section":"Example 4.2 / Figure 2"},{"comment":"D_w is a polynomial in n-1 variables, but the matroid polytopes P(M_ab) in Equation (1) live in R^n. The proof should state explicitly that supports and Newton polytopes are embedded in R^n with the last coordinate equal to zero, so that the Minkowski-sum calculation is formally consistent.","section":"Theorem 3.18"},{"comment":"The procedure states that each rectangle sum is written at the bottom right corner of the rectangle, but in Figures 2 and 3 the sums appear in the bottom rows and are then read top to bottom; a sentence clarifying the reading order and how the displayed vertex coordinates are formed would improve reproducibility.","section":"Section 4, Step 4"},{"comment":"The remark asserts, based on SageMath, that D^{4231}_{1324} has SNP but not SCNP; since this is used to explain why the SCNP method does not generalize, it would be helpful to include the exact polynomial or the computation script in an ancillary file.","section":"Remark after Example 3.2"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is within the scope of the journal and the citations to the prior literature are appropriate. The only point that gave me pause is the one-sentence proof of Lemma 3.7, but the statement is true and the gap is easily filled; after the requested expansion and the small corrections listed in the minor comments, the paper should be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Best read of this: the paper actually delivers what it advertises. The new content is Theorem 1.2, the support identity supp(D_w) = Minkowski sum of inversion intervals, equivalently supp(GW(w)), and Corollary 1.4 identifying vertices of Newton(D_w) as the coefficient-one monomials of GW(w). M-convexity itself was known from Huh–Matherne–Mészáros–St. Dizier, but their proof is Lorentzian/algebraic; this one is a direct chain argument and it gives more precision, not just a rerun.\n\nThe main line is sound. Lemma 3.13 and Lemma 3.9 are the load-bearing parts; I checked the inductions and the inversion-set analysis. The inclusion supp(D_w) ⊆ supp(GW(w)) follows from the dominant pairing in Lemma 3.13 and is clean. The reverse inclusion depends on Lemma 3.7, existence of a greedy chain in every Bruhat interval. That lemma has to carry more weight than its proof does. The text gives a one-sentence top-down construction and doesn't spell out why the chosen cover exists with the required maximality, nor why all earlier edges remain greedy. As the stress-test note says, the existence claim is true — choose a cover whose swapped pair is inclusion-maximal among covers in the interval at each step — so this is a presentation gap, not a correctness flaw. But for a referee, I'd want Lemma 3.7 expanded before publication.\n\nThe generalized permutahedron formulation in Theorem 3.18 and the rectangle-tiling vertex description are useful additions. The examples check out, and the explicit vertex computation for D_{253641} is consistent with the coefficient-one characterization. No circularity: the proof never assumes M-convexity or SNP; it derives them from the chain-weight argument. The conjectures in Section 5 are marked as conjectures, appropriately.\n\nWho this is for: algebraic combinatorialists working on Schubert polynomials, Newton polytopes, and M-convexity. It is a strengthening of a known result rather than a breakthrough, but it gives a genuinely simpler proof and a sharper statement. I would send this to a serious referee; the missing details in Lemma 3.7 are fixable and everything else holds up.","headline":"New support and vertex characterizations for dual Schubert polynomials, with a clean elementary proof that is mostly solid; Lemma 3.7 needs a fuller proof but the gap is fillable.","tokens_in":11536,"tokens_out":1552,"would_cite":true,"duration_ms":14475,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E14","52B12","05A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Inversions determine every dual Schubert polynomial's support.","keywords":["dual Schubert polynomials","Newton polytopes","M-convexity","saturated Newton polytope","Bruhat order","greedy chains","generalized permutahedra","staircase Young diagrams"],"falsifier":"Compute $D_w$ and $\\operatorname{GW}(w)$ explicitly for all $w$ in $S_6$ and compare supports; any monomial in one support but not the other would refute Theorem 1.2. Alternatively, exhibit a Bruhat interval with no greedy chain, directly contradicting Lemma 3.7.","tokens_in":22,"feed_emoji":"🧩","tokens_out":6227,"duration_ms":173715,"temperature":0.7,"pith_summary":"The paper proves that the support of a dual Schubert polynomial $D_w$—the set of exponent vectors appearing in its monomials—is completely determined by the inversion pairs of the permutation $w$. Concretely, $\\operatorname{supp}(D_w)$ is the Minkowski sum, over all inversions $(a,b)$ of $w$, of the intervals $\\{e_a,\\ldots,e_{b-1}\\}$, which is exactly the support of the product $\\operatorname{GW}(w)$ of linear forms $x_a+\\cdots+x_{b-1}$. This yields an elementary proof that $D_w$ is M-convex and that its Newton polytope is a generalized permutahedron, and it identifies the vertices of that polytope as the monomials with coefficient $1$ in $\\operatorname{GW}(w)$. Because dual Schubert polynomials encode degrees of Schubert varieties and pair naturally with Schubert polynomials, knowing their Newton polytopes exactly makes those invariants directly readable from the inversion set.","feed_headline":"Inversions determine every dual Schubert polynomial's support","feed_subtitle":"The vertices of the Newton polytope are exactly the coefficient-one monomials of a simple product.","key_machinery":"The load-bearing object is the greedy chain: a saturated chain $\\mathrm{id}=w_0\\lessdot w_1\\lessdot\\cdots\\lessdot w_\\ell=w$ in the strong Bruhat order such that each transposition $w_i=w_{i-1}t_{ab}$ cannot be replaced by a transposition with a wider interval $[a',b']$ while staying inside $[\\mathrm{id},w]$. The paper defines the global weight $\\operatorname{GW}(w)$ as the product over inversion pairs $(a,b)$ of $x_a+x_{a+1}+\\cdots+x_{b-1}$. It proves that every greedy chain has weight $\\operatorname{GW}(w)$ and that every Bruhat interval contains at least one greedy chain; together with the inclusion that every chain's support is contained in $\\operatorname{supp}(\\operatorname{GW}(w))$, this single-chain comparison equates $\\operatorname{supp}(D_w)$ with $\\operatorname{supp}(\\operatorname{GW}(w))$.","core_discovery":"On the paper's own terms, the central discovery is that the averaging over saturated chains that defines $D_w$ can be replaced by a single product: for the greedy chains introduced here, the weight of every greedy chain in $[\\mathrm{id}, w]$ is exactly $\\operatorname{GW}(w)$, and every Bruhat interval contains at least one greedy chain. Since every chain's weight has support contained in $\\operatorname{supp}(\\operatorname{GW}(w))$, the support of the averaged polynomial $D_w$ coincides with $\\operatorname{supp}(\\operatorname{GW}(w))$, giving Theorem 1.2. A further lemma then shows that the vertices of $\\operatorname{Newton}(D_w)$ are precisely the monomials of $\\operatorname{GW}(w)$ with coefficient $1$, so the polytope's extremal structure is a purely combinatorial artifact of the inversion set.","pith_inferences":["If the greedy-chain existence lemma could be strengthened to intervals $[u,w]$ with an arbitrary lower bound, the same argument would give support formulas for Postnikov–Stanley polynomials, which the paper leaves as a conjecture about M-convexity.","The vertex characterization suggests that linear optimization over $\\operatorname{Newton}(D_w)$ is governed by a greedy rectangle-tiling rule, which may yield a fast algorithm for evaluating Schubert-degree-type quantities.","The paper's matroidal decomposition of $\\operatorname{Newton}(D_w)$ as a Minkowski sum of rank-one matroid polytopes might transfer to other families of interval-defined polynomials, where a similar support identity could be tested."],"forward_implications":["$D_w$ is M-convex, so its Newton polytope is a generalized permutahedron; the paper gives explicit inequality data $z_I=\\sum_{(a,b)\\in\\operatorname{Inv}(w)} \\mathbf{1}_{I\\supseteq[a,b)}$ for that polytope.","The vertices of $\\operatorname{Newton}(D_w)$ can be listed combinatorially by rectangle tilings of a staircase Young diagram labelled by inversion pairs, giving a purely diagrammatic enumeration.","Every monomial of $D_w$ appears in $\\operatorname{GW}(w)$, so support comparisons between different permutations can be studied through inversion-set inclusion rather than chain enumeration.","The proof of M-convexity is elementary and does not rely on the algebraic-geometric machinery of the earlier proof."],"supporting_citations":[{"why":"Supplies the saturated-chain formula that defines dual Schubert polynomials and Postnikov–Stanley polynomials.","marker":"[15]"},{"why":"Gives the SNP property for products of nonnegative linear forms and the vertex characterization used in Corollary 1.4.","marker":"[1]"},{"why":"Provides the description of matroid polytopes as generalized permutahedra used to compute the explicit inequality data of Newton($D_w$).","marker":"[2]"},{"why":"Supplies the SNP definition and the additive/multiplicative Newton polytope operations used throughout.","marker":"[12]"},{"why":"Provides the generalized permutahedron framework and Corollary 8.2, which turns rectangle tilings into vertices.","marker":"[14]"},{"why":"Gives the M-convexity equivalence used to conclude that $D_w$ is M-convex.","marker":"[13]"},{"why":"Is the earlier proof of M-convexity that this paper reproves and strengthens.","marker":"[11]"}],"fun_headline_variants":["Greedy chains fix support and vertices of dual Schubert","One product yields all Newton vertices of dual Schubert","Inversion sets encode dual Schubert polytope vertices","Greedy chains make dual Schubert polytope explicit"],"cache_read_input_tokens":13696,"weakest_assumption_plain":"For the equality of supports, the proof needs the claim that every Bruhat interval contains a chain whose transpositions are all as wide as possible; this is asserted with only a one-sentence construction, and if some interval lacks such a chain the identity could fail.","fun_headline_variants_meta":{"raw":{"variants":["Greedy chains fix support and vertices of dual Schubert","One product yields all Newton vertices of dual Schubert","Inversion sets encode dual Schubert polytope vertices","Greedy chains make dual Schubert polytope explicit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000358,"raw_usage":{"total_tokens":1853,"prompt_tokens":769,"completion_tokens":1084,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":385,"completion_tokens_details":{"reasoning_tokens":1016}},"tokens_in":385,"tokens_out":1084,"duration_ms":8535,"temperature":1.0,"reasoning_tokens":1016,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:53:40.598019+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $D_w$ and $\\operatorname{GW}(w)$ explicitly for all $w$ in $S_6$ and compare supports; any monomial in one support but not the other would refute Theorem 1.2. Alternatively, exhibit a Bruhat interval with no greedy chain, directly contradicting Lemma 3.7.","supporting_citations":[{"cited_title":"On minkowski sums of simplices","cited_arxiv_id":null,"evidence_quote":"Gives the SNP property for products of nonnegative linear forms and the vertex characterization used in Corollary 1.4."},{"cited_title":"Matroid p olytopes and their volumes","cited_arxiv_id":null,"evidence_quote":"Provides the description of matroid polytopes as generalized permutahedra used to compute the explicit inequality data of Newton($D_w$)."},{"cited_title":"Permutohedra, associahedra, and bey ond","cited_arxiv_id":null,"evidence_quote":"Provides the generalized permutahedron framework and Corollary 8.2, which turns rectangle tilings into vertices."}],"review_version":1}