{"id":"c473c740-7bd8-4f06-a1dc-b947374bfc3e","arxiv_id":"2411.17572","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sign-changed double Richardson and Schubert polynomials are covolume, and the integer cohomology ring of a variety is governed by a Lorentzian Macaulay dual generator when built from nef line bundle classes.","lead":"This paper shows that equivariant cohomology classes of torus-stable subvarieties of matrix spaces become covolume polynomials after a sign change, and that Macaulay dual generators of integer cohomology rings are denormalized Lorentzian polynomials under nefness conditions. The main combinatorial payoff is new discrete log-concavity and M-convex support results for double Schubert and Richardson polynomials.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Setup 4.3's twisted-positive condition is misprinted and, read literally, excludes the matrix action on which the main theorem depends.","rationale":"I read the paper in good faith. The central claim is Theorem 5.4(ii): sign-changed double Richardson polynomials are covolume, and its proof is: Theorem 5.4(i) identifies the equivariant class of the matrix Richardson variety with the product R_{w/u}(t,s); Corollary 4.6, derived from Theorem 4.5 via Lemma 4.4 and Theorem 3.5, converts that class into a covolume polynomial after sign change. The least secure step is the hypothesis of Theorem 4.5. Setup 4.3 as printed is not merely unclear; it is internally mis-indexed and literally fails for the weights e_i − e_j that arise from the matrix action. Without the corrected reading N^q × (−N)^{p−q}, the main application to matrix Richardson varieties does not follow. This matches the reader's weakest assumption, and the reader's CONDITIONAL verdict is appropriate. A secondary gap is Theorem 5.4(i), where the equality [D^w_u]_T = [D_u]_T · [D^w]_T is asserted without proof or reference; this is standard but should be justified. The Setup 4.3 issue is more load-bearing because it blocks the logical chain at the point where the geometric input is converted into the polynomial statement. No independent objection to Theorem 3.5 or the Section 6 results undermines the main conclusion.","tokens_in":26085,"tokens_out":13110,"duration_ms":123793,"concrete_test":"Rewrite Setup 4.3 as: for every variable x_i, deg(x_i) ∈ N^q × (−N)^{p−q}. In Lemma 4.4, define ~R by flipping the signs of the last p−q coordinates of every weight and redo the K-polynomial/substitution computation from [MS05, Claim 8.54] to confirm C(~R/I; t) = C(R/I; t_1,...,t_q,−t_{q+1},...,−t_p). Then check that for Mat_{m,n} with coordinates x_{ij} and action (g,h)·M = gMh^{−1}, each weight e_i−e_j lies in N^m × (−N)^n, so Corollary 4.6 covers matrix Richardson varieties.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem A (Theorem 5.4(ii)) passes through Corollary 4.6, which applies Theorem 4.5 to Mat_{m,n}. Theorem 4.5 is stated under Setup 4.3, but Setup 4.3 as printed reads 'we require that d_i ∈ N^p \\ {0} for all 1 ≤ i ≤ q and −d_i ∈ N^p \\ {0} for all q+1 ≤ i ≤ p', indexing weights by the variable index i while p denotes the torus dimension. This condition is ill-typed and is not satisfied by the matrix-action weights e_i − e_j ∈ Z^{m+n}, which have one positive and one negative coordinate. The intended condition must be that every weight vector lies in N^q × (−N)^{p−q}; Lemma 4.4 and Corollary 4.6 are valid only under that interpretation. This is a genuine load-bearing issue for the central application, though it is repairable as a typo rather than a conceptual gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies torus-equivariant cohomology classes of invariant subvarieties of affine spaces and of matrix spaces. It introduces double Richardson polynomials R_{w/u}(t,s) and proves (Theorem 5.4) that the sign-changed polynomial R_{w/u}(t,-s) is a covolume polynomial, hence dually Lorentzian; this yields M-convex support and discrete log-concavity for double Richardson, Richardson, double Schubert, and Schubert polynomials (Corollary 5.8). The engine is a general statement (Theorem 4.5, Corollary 4.6) that for certain \"twisted positive\" torus actions on C^n, the sign-changed equivariant class of any irreducible invariant subvariety is covolume. The second half develops Macaulay inverse systems over Z for cohomology rings and proves (Theorem 6.13) that, under flatness and positivity hypotheses, the Macaulay dual generator is a denormalized Lorentzian polynomial; a toric corollary gives a characteristic-free Khovanskii–Pukhlikov description in terms of mixed volumes.","tokens_in":26197,"tokens_out":11765,"duration_ms":102939,"significance":"If the results hold, the paper provides a large new supply of covolume polynomials — sign-changed equivariant classes under twisted positive torus actions — and unifies several previously known log-concavity results for Schubert and double Schubert polynomials while establishing the new discrete log-concavity of double Schubert and Richardson polynomials. The characteristic-free Macaulay dual generator over Z is a useful extension of the Khovanskii–Pukhlikov theorem, and the worked examples (flag variety, Grassmannian, Hirzebruch surface) make the constructions concrete. The proofs are largely self-contained and rest on established techniques: standardization for nonstandard gradings, generic local duality over Z, and the Lorentzian volume polynomial theorem of Brändén–Huh. The paper is well structured and the computations in the examples are reproducible.","major_comments":[{"comment":"The twisted positive grading condition is misprinted and, read literally, excludes the main application. The text requires d_i ∈ N^p \\ {0} for 1 ≤ i ≤ q and −d_i ∈ N^p \\ {0} for q+1 ≤ i ≤ p, which is ill-typed (i indexes variables on the left but torus coordinates on the right) and is not satisfied by the matrix-action weights e_i − e_j ∈ Z^{m+n} used in Corollary 4.6: each such weight has a positive coordinate and a negative coordinate, so it lies in neither N^p nor −N^p. The intended condition is that every variable weight d_i lies in N^q × (−N)^{p−q}; under that reading Lemma 4.4 and Theorem 4.5 are valid and Corollary 4.6 follows. Because the printed assumption does not cover the matrix action, the proof of Theorem 5.4(ii) is not valid as written. Please correct Setup 4.3, the definition of R̃ in Lemma 4.4, and the statement of Theorem 4.5 so that the condition applies coordinate-wise to all n weights.","section":"Setup 4.3, Lemma 4.4, Theorem 4.5, Corollary 4.6"},{"comment":"The theorem states that X is an arbitrary smooth complex algebraic variety of dimension d with R = ⊕ H^{2i}(X,Z) flat over Z, and defines ρ: H^{2d}(X,Z) → Z as the \"natural degree map\". For non-complete X such a map need not exist: for X = A^d_C, H^{2d}(X,Z) = 0; for X = C^*, H^2(X,Z) = 0. The statement must add that X is complete (proper); then, for connected X, H^{2d}(X,Z) ≅ Z with the usual degree map. Parts (i)–(iii) rely on Poincaré duality for R = H^*(X,Z), which also requires completeness, so the theorem as stated is false. This is load-bearing for Theorem D and Corollary 6.16. The fix is local — add the completeness hypothesis to the statements in the introduction and in Theorem 6.13 — but without it the statement overreaches.","section":"Theorem 6.13 and Theorem D (Introduction)"}],"minor_comments":[{"comment":"The phrase \"irreducible T-variety\" should be \"irreducible T-subvariety\", matching the definition in Theorem 4.5.","section":"Corollary 4.6"},{"comment":"The symbol p is overloaded: it denotes the dimension of the torus in T = (C*)^q × (C*)^{p−q} and also appears as an index bound in the conditions on d_i, where the number of variables is n. Please distinguish the torus dimension from n throughout this section.","section":"Setup 4.3 and Lemma 4.4"},{"comment":"The definition of the ring R̃ is garbled for the same reason as Setup 4.3; once the corrected twisted-positive condition is in place, R̃ should be defined on all variables by flipping the signs of the last p−q coordinates of each weight vector. The proof itself is correct after this clarification.","section":"Lemma 4.4"},{"comment":"The symbol I is used both for the annihilator ideal {g | g·N^{−1}(V) = 0} and for the Stanley–Reisner ideal I in the presentation Z[x_1,...,x_n]/I ≅ H^*(X_Σ,Z). This reuse is confusing; please rename one of the ideals.","section":"Corollary 6.16(i)"},{"comment":"The reference \"[MS05, Claim 8.54]\" appears to be to an unnumbered claim or to Proposition 8.54 in Miller–Sturmfels; please verify and cite the exact item.","section":"Proof of Lemma 4.4"}],"recommendation":"major_revision","confidential_remarks":"Both major issues are genuine and load-bearing, but both are readily repairable: Setup 4.3 is almost certainly a TeX/indexing error, and Theorem 6.13 simply needs the word \"complete\" (or \"proper\") in the hypothesis. The mathematical core of the paper is likely sound. I recommend major revision so the authors can correct the statements and re-verify the proofs of Lemma 4.4 and Theorem 6.13 under the corrected hypotheses; I do not see circularity or novelty concerns. The authors should also check that the abstract and introduction state the completeness hypothesis for Theorem D consistently."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The main results are genuinely new: discrete log-concavity of sign-changed double Schubert polynomials, plus the same for double Richardson polynomials, via covolume polynomials. The proof strategy is sound, and the Section 3 result that multidegrees of prime ideals are covolume is a solid piece of work. Second, there is a real typo in the central setup. Setup 4.3 as printed asks for weights d_i in N^p for i ≤ q and -d_i in N^p for i > q; the matrix action weights e_i - f_j, which have both positive and negative coordinates, satisfy neither. As written, Corollary 4.6 does not follow. The intended condition is clearly that every weight lies in N^q × (-N)^{p-q}, and Lemma 4.4's sign change should be coordinate-wise on the last p-q torus directions. This is repairable but it is load-bearing, so the paper needs a revision, not a desk reject.\n\nTwo more minor soft spots. Theorem 6.13 omits completeness of X; without it the degree map ρ: H^{2d}(X,Z) → Z is not canonical. The proof of Theorem 5.4(i) asserts the product formula for the Richardson class without proof; it is standard in this setting but should be cited.\n\nWhat the paper does well: the double Richardson polynomials are a useful new object, the covolume result for arbitrary irreducible T-subvarieties of matrix space is a clean theorem, and the Macaulay inverse system development over Z, including the Khovanskii-Pukhlikov extension and the worked examples, is valuable. The authors are honest about prior work; the citations are appropriate.\n\nI agree with the conditional verdict. The central ideas are correct, the flaws are in presentation and hypotheses. Send it to a serious referee; I expect it to be accepted after a minor revision.","headline":"Real new results on log-concavity for double Schubert and Richardson polynomials, but the twisted-grading setup has a load-bearing typo that needs fixing first.","tokens_in":26836,"tokens_out":13979,"would_cite":true,"duration_ms":125411,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","14C15","14C17","13H15","52B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that double Richardson polynomials become covolume polynomials after a sign change, and that sign-changed equivariant classes of torus-stable subvarieties of matrix spaces are covolume.","keywords":["equivariant cohomology","multidegrees","Richardson polynomials","Schubert polynomials","Lorentzian polynomials","covolume polynomials","log-concavity","Macaulay dual generators"],"falsifier":"Compute the sign-changed equivariant class polynomial of an irreducible $T$-subvariety of $\\operatorname{Mat}_{m,n}$ — for instance a matrix Schubert variety — and check whether its coefficient support is M-convex; a single such class with non-M-convex support or with adjacent coefficients violating $a_n^2 \\geq a_{n+e_i-e_j}a_{n-e_i+e_j}$ would refute Theorem C. Alternatively, verify directly on the action $(g,h)\\cdot M = g M h^{-1}$ that the weights $e_i - e_j$ are not all in $\\mathbb{N}^p$ and not all in $-\\mathbb{N}^p$, which shows the literal Setup 4.3 cannot be the hypothesis under which Corollary 4.6 is proven.","tokens_in":25795,"feed_emoji":"📐","tokens_out":8364,"duration_ms":71356,"temperature":0.7,"pith_summary":"This paper aims to prove that certain polynomials attached to torus-equivariant subvarieties of matrix spaces — in particular the double Richardson, Richardson, double Schubert, and Schubert polynomials of type A — are covolume polynomials after a sign change. Covolume polynomials are limits of the Chow classes of irreducible subvarieties of products of projective spaces, and they form a subfamily of dually Lorentzian polynomials. Because dually Lorentzian polynomials have M-convex support and discretely log-concave coefficients, the proof directly yields new log-concavity and support-shape statements for Schubert-related polynomials. The paper also develops Macaulay inverse systems over the integers for cohomology rings, proving that the Macaulay dual generator of the even cohomology ring of a smooth complex variety is a denormalized Lorentzian polynomial under nefness hypotheses, and extending the classical volume-polynomial description of toric cohomology to integer coefficients.","feed_headline":"Sign flip turns Schubert classes into log-concave polynomials","feed_subtitle":"Torus-stable matrix subvarieties yield polynomial families with M-convex support and discrete log-concavity.","key_machinery":"The load-bearing mechanism is standardization of multigraded polynomial rings: a positive $\\mathbb{N}^p$-grading is converted to a standard multigrading by replacing each variable $x_i$ of total degree $\\ell_i$ with a product $y_{i,1}\\cdots y_{i,\\ell_i}$ of standard-graded variables. This substitution preserves multidegree polynomials, Betti numbers, and primality, so the multidegree polynomial of a prime ideal becomes the covolume polynomial of an irreducible subvariety of a product of projective spaces. A sign-flip lemma (Lemma 4.4) transfers the result from positive gradings to the 'twisted positive' gradings that arise from equivariant cohomology of matrix Schubert and Richardson varieties. The Macaulay dual generator construction over $\\mathbb{Z}$ plays the analogous role for cohomology rings, replacing field-level Gorenstein duality with a relative Gorenstein statement over $\\mathbb{Z}$.","core_discovery":"On the paper's own terms, the central discovery is Theorem C: for the space $\\operatorname{Mat}_{m,n} = \\mathbb{C}^{m\\times n}$ with the torus $T = (\\mathbb{C}^*)^m \\times (\\mathbb{C}^*)^n$ acting by $(g,h)\\cdot M = g M h^{-1}$, the polynomial representing the equivariant class $[X]_T$ of any irreducible $T$-subvariety $X$ becomes a covolume polynomial after the substitution $s_i \\mapsto -s_i$. From this, Theorem A states that for permutations $u,w$ with $w \\geq u$ in Bruhat order, the double Richardson polynomial $R_{w/u}(t,s) = S_u(t,s)S_{w_0w}(t,s')$, where $s'$ reverses the $s$ variables, has the property that $R_{w/u}(t,-s)$ is covolume. Corollary B then extracts concrete combinatorial content: these sign-changed polynomials, together with ordinary Richardson and Schubert polynomials and their truncations, have M-convex support and are discretely log-concave. In the cohomology-ring half of the paper, Theorem D establishes that for a smooth complex variety with $\\mathbb{Z}$-torsion-free even cohomology, the even cohomology ring is Artinian Gorenstein over $\\mathbb{Z}$, is recovered as the annihilator of a Macaulay dual generator, and that generator's normalization is Lorentzian when the chosen generators are nef first Chern classes.","pith_inferences":["If Theorem C extends beyond matrix spaces, the same standardization-and-sign-flip mechanism should yield covolume polynomials for equivariant classes of torus-stable subvarieties in other representations whose weights are sign-separable, such as quiver representations with bipartite orientation.","The paper's truncation result suggests that the skew Schubert polynomials of Lenart and Sottile, which are normal-form representatives of Richardson classes, may be dually Lorentzian even where they do not equal truncations of Richardson polynomials; Question 5.10 is likely to have a positive answer.","The integral Macaulay dual generator formalism may provide a route to Lorentzian and log-concavity statements for cohomology rings with torsion by passing to a universal coefficient or flat-approximation statement, though the paper requires $\\mathbb{Z}$-torsion-freeness.","A concrete testable extension would be to replace cohomology by equivariant K-theory and ask whether the sign-changed K-class polynomial, after normalization, is dually Lorentzian; the paper does not address K-theory."],"forward_implications":["The sign-changed double Richardson polynomials $R_{w/u}(t,-s)$ are dually Lorentzian, hence their coefficient supports are integer points of generalized permutohedra and are discretely log-concave.","Double Schubert polynomials $S_u(t,-s)$ inherit the same properties; the discrete log-concavity of double Schubert polynomials is new, while the M-convexity recovers earlier results.","Ordinary Schubert and Richardson polynomials, along with their truncations, have M-convex support and are discretely log-concave.","For smooth complex varieties with torsion-free even cohomology, the cohomology ring over $\\mathbb{Z}$ is determined by a Macaulay dual generator; when the ring generators are nef first Chern classes, the normalization of that generator is Lorentzian.","For smooth complete toric varieties, the Macaulay dual generator is exactly the mixed-volume polynomial of the polytopes associated to nef torus-invariant divisors, giving a characteristic-free extension of the classical toric volume-polynomial description."],"supporting_citations":[{"why":"Defines covolume polynomials and provides the stability properties used to identify sign-changed equivariant classes as covolume.","marker":"[Alu24]"},{"why":"Introduces Lorentzian polynomials, proves volume polynomials are Lorentzian, and shows normalization preserves the Lorentzian property.","marker":"[BH20]"},{"why":"Introduces dually Lorentzian polynomials and establishes the bridge from covolume polynomials to M-convex support and discrete log-concavity.","marker":"[RSW23]"},{"why":"Supplies the multidegree formalism and the identification of equivariant classes of matrix Schubert varieties with multidegree polynomials.","marker":"[KM05]"},{"why":"Provides the geometric interpretation of double Schubert polynomials as equivariant classes of matrix Schubert varieties used in Theorem 5.1.","marker":"[FR03]"},{"why":"Gives the equivariant cohomology setup and the pullback computation under the involution used in Lemma 5.2.","marker":"[AF24]"},{"why":"Establishes that standardization preserves multidegree polynomials, Betti numbers, and primality in non-standard gradings.","marker":"[CCRC23]"},{"why":"Supplies the positive-grading flip lemma and the standardization context for double Schubert polynomials.","marker":"[CCRMMn23]"},{"why":"The classical volume-polynomial description of toric cohomology that the paper extends to integer coefficients.","marker":"[KP92]"},{"why":"Provides the toric variety cohomology presentation and the mixed-volume identity used in Corollary 6.16.","marker":"[CLS11]"}],"fun_headline_variants":["Covolume polynomials from torus-equivariant classes","Sign-changed Richardson polynomials are discretely log-concave","Equivariant cohomology yields log-concave polynomial families","Macaulay dual generators become Lorentzian in toric cases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the torus weights can be split into a nonnegative group and a nonpositive group so that flipping signs gives a positive grading, a condition that the matrix-action weights satisfy only under the intended $\\mathbb{N}^q \\times (-\\mathbb{N})^{p-q}$ reading rather than the literal printed Setup 4.3.","fun_headline_variants_meta":{"raw":{"variants":["Covolume polynomials from torus-equivariant classes","Sign-changed Richardson polynomials are discretely log-concave","Equivariant cohomology yields log-concave polynomial families","Macaulay dual generators become Lorentzian in toric cases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000953,"raw_usage":{"total_tokens":4094,"prompt_tokens":1006,"completion_tokens":3088,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":622,"completion_tokens_details":{"reasoning_tokens":3016}},"tokens_in":622,"tokens_out":3088,"duration_ms":21926,"temperature":1.0,"reasoning_tokens":3016,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:01:55.769325+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the sign-changed equivariant class polynomial of an irreducible $T$-subvariety of $\\operatorname{Mat}_{m,n}$ — for instance a matrix Schubert variety — and check whether its coefficient support is M-convex; a single such class with non-M-convex support or with adjacent coefficients violating $a_n^2 \\geq a_{n+e_i-e_j}a_{n-e_i+e_j}$ would refute Theorem C. Alternatively, verify directly on the action $(g,h)\\cdot M = g M h^{-1}$ that the weights $e_i - e_j$ are not all in $\\mathbb{N}^p$ and not all in $-\\mathbb{N}^p$, which shows the literal Setup 4.3 cannot be the hypothesis under which Corollary 4.6 is proven.","supporting_citations":[],"review_version":1}