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Volume polynomials

T0 review · 4 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This survey argues that volume polynomials, defined in convex and algebraic geometry, are completely characterized among quadratic forms by the Lorentzian property, and that covolume polynomials are exactly the linear operators that preserv

desk verdict A lucid survey of the volume-polynomial program whose central duality theorem rests on an unproved, preprint-level input; worth refereeing, but the referee should demand a proof or a clear status flag. read the letter →

arxiv 2601.13249 v4 pith:2OS3TFVG submitted 2026-01-19 math.AG math.CO

June Huh This is my paper · ORCID
classification math.AGmath.CO MSC 14-0652-06
keywords volumepolynomialsLorentzianmixedvolumescovolumealgebraicmatroidsrealizationproblemsAlexandrov–Fenchelinequalitypolymatroids
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 surveys the theory of volume polynomials: homogeneous polynomials that measure how the volume of Minkowski sums or intersections changes with scaling. Its central thesis is that these polynomials form a distinguished log-concave class whose realization problem — which polynomials actually arise — has a complete and surprisingly clean answer in low degree, matched by a sharp operator-theoretic characterization in general. A sympathetic reader comes away with a unified picture: quadratic volume polynomials over any field are precisely the Lorentzian forms, bivariate forms are exactly Lorentzian, and the dual notion of covolume polynomial is exactly the class of differential operators that preserve realizable volume polynomials. These results settle old questions, like Gurvits's conjecture, and tie the subject to algebraic matroids.

What carries the argument

Volume polynomials themselves — Minkowski-sum volume polynomials in convex geometry and intersection-product volume polynomials in algebraic geometry — with the Alexandrov–Fenchel inequality as the source of log-concavity. Lorentzian polynomials, defined by the Hessian condition or the M-convex support condition, form the ambient class. Covolume polynomials, defined via the action of R[∂] on R[x], are the dual objects; Theorem 4.7 is the key bridge showing that they are precisely the linear operators preserving realizable volume polynomials.

What would settle it

A concrete check: exhibit a single polynomial g for which g(∂) maps every realizable volume polynomial to a realizable volume polynomial, but g is not itself realizable as a covolume polynomial over k; Theorem 4.7 says no such g exists. Alternatively, produce a non-realizable Lorentzian quadratic form over a field, which would refute the degree-two characterization.

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Extended reading notes

Core claim

The paper's core claim is that the set of volume polynomials over a field k is a proper, well-behaved subset of Lorentzian polynomials for degree d≥3 and at least 3 variables, but coincides with the Lorentzian class in the quadratic case and in the bivariate case. The main structural result, Theorem 4.7, characterizes realizable covolume polynomials over k as those polynomials g for which g(∂) sends every realizable volume polynomial to another realizable volume polynomial; taking limits gives the analogous statement for volume polynomials. Theorem 2.11 identifies Lorentzian polynomials as exactly those homogeneous polynomials with nonnegative coefficients whose support is M-convex and whose

Load-bearing premise

The survey's load-bearing premise is that the theorems quoted from the author's recent preprints — in particular Theorem 4.7 and Theorem 2.11 — are correct as stated; since they are not proved in these notes, any error in those sources would propagate.

Editorial extensions

If this is right

  • Quadratic volume polynomials over any field are exactly the Lorentzian quadratic forms, so the realization problem in degree two is completely solved.
  • The existence of a Lorentzian cubic that is not a volume polynomial over any field disproves Gurvits's strong-log-concavity conjecture in three variables.
  • Since covolume polynomials preserve volume polynomials under differential action, every known inequality for volume polynomials can be applied to Schubert transforms to yield new mixed-volume inequalities.
  • The support of every realizable volume polynomial is an algebraic polymatroid, giving a new proof of Lindström's theorem for polymatroids and the closure of algebraic matroids under intersection.
  • The characterization of covolume polynomials gives a volume-polynomial analogue of the symbol theorem for Lorentzian operators.

Reading between the lines

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

  • The paper's conjectures suggest that the distinction between 'realizable' and 'limit of realizable' volume polynomials is the right seam: the algebraic realization problem has integrality obstructions that the convex one lacks, as the (1,1,1,1,1,3) example shows.
  • If Conjecture 3.12 is true, the algebraic geometric volume polynomials are exactly the closure of minors of convex-geometric ones, which would mean Hodge-theoretic positivity is fully generated by convex bodies.
  • A resolution of Question 5.7 would determine whether covolume polynomials carry genuinely new algebraic information; if duals of algebraic matroids are always algebraic, the support theory of covolume polynomials collapses into that of volume polynomials.
  • The triangular hyperfield appears throughout, suggesting the realization conditions may be expressible as Grassmannian Plücker relations over a tropical-style semiring.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. This paper is a survey of recent developments on volume polynomials: the homogeneous degree-d polynomials f_C(x) = (1/d!)vol(x_1C_1+...+x_nC_n) of convex bodies and their algebraic analogues f_D(x) = (1/d!)∫_Y (Σ x_i D_i)^d for semiample divisors. It reviews realization problems for projection areas (Theorems 1.1–1.2, Conjectures 1.6–1.7), the Lorentzian characterization of quadratic and bivariate forms (Theorem 2.11, Example 2.2), the strict inclusion of volume polynomials over a field in the Lorentzian cone (Section 3.2), and the covolume/symbol-theoretic duality (Theorem 4.7, Theorem 4.10). The final section applies these ideas to algebraic polymatroids, including Proposition 5.5 and Corollary 5.6, and states open problems such as Question 5.7. Most displayed theorems are quoted without proof from the author's previous works, several of which are arXiv preprints or 'in preparation'.

Significance. If the quoted results are correct, the survey gives a valuable and accessible point of entry to a fast-moving area, with well-chosen concrete examples (e.g., Example 1.4 and Example 1.8) that illustrate the distinction between convex and algebraic realization. The Lorentzian baseline (Theorem 2.11) is published [BH20], and the nontriviality of the realization problem is supported by published reverse Khovanskii–Teissier inequalities. However, the manuscript's covolume/symbol core (Theorem 4.7, Corollary 4.8, Theorem 4.10) is inherited from the preprint [GHM+], and no proof or proof sketch is supplied. Because these results are load-bearing for the survey's structural claims and for the operator-preservation applications in Section 4.3, the survey should either include a careful statement of their logical status or provide enough of a proof/argument to let a reader assess the dependence. The paper otherwise reads as a reliable overview and is likely to be useful to the intended audience.

major comments (4)
  1. [§4.2 (Theorem 4.7)] The equivalence (1)⇔(2) is quoted from [GHM+], an arXiv preprint, and is not proved or marked as conditional. This is the pivotal structural input: it is used to derive Corollary 4.8, the symbol theorem (Theorem 4.10), and the operator-preservation results in §4.3. If the preprint contains a hidden hypothesis (e.g., a support or Leibniz-term assumption, or a characteristic restriction), the survey's statements are incomplete. Please add (i) an explicit statement that Theorem 4.7 is an unpublished theorem of [GHM+], (ii) a proof sketch or at least a precise formulation of the hypotheses under which it is known, and (iii) a note in §4.3 stating which of the listed consequences depend on this theorem.
  2. [§4.1 (Definition 4.1 and Remark 4.2)] Definition 4.1 tests a covolume polynomial on a single monomial x^[μ], while Theorem 4.7(2) asserts an action on every realizable volume polynomial f. Remark 4.2 uses translation invariance to show independence of μ, but the step from one monomial to arbitrary realizable f is exactly the nontrivial content of Theorem 4.7; it is not a formal consequence of translation invariance. The manuscript should state this explicitly and not give the impression that Remark 4.2 reduces Theorem 4.7 to a routine check.
  3. [§3.2] The example after 'For example, consider the cubic polynomial' is used to establish the proper inclusion V^d_n(R,k) ⊊ L^d_n for d≥3,n≥3. The Lorentzian property is verified by listing three Hessians, but the non-realizability of f is only asserted to follow from the reverse Khovanskii–Teissier inequality; no computation shows how the inequality applies to this specific f. Since this is the paper's main demonstration that the realization problem has a nontrivial answer, please add the missing verification or give the exact location in [Huh23, Example 14] and reproduce the key inequalities.
  4. [§5 (Proposition 5.5 and its use)] Proposition 5.5 is quoted from [GHM+, Proposition 5.4] and is then used to derive Corollary 5.6 and Theorem 5.11. This is another load-bearing transfer from the same unpublished source. Please state the provenance clearly and, if the result depends on the same hypotheses as Theorem 4.7, say so. Otherwise the survey's application section rests on a result whose proof is not available to the reader.
minor comments (6)
  1. [§2.3 and §2.4] The text refers to pictures that do not appear in the arXiv version ('The above pictures show...' and the discussions of four matroids). Please add the figures or remove the references.
  2. [Example 3.6] The Young diagram is blank; the displayed polynomial suggests λ=(2,1). Please insert the intended diagram.
  3. [Definition 3.1] 'Limit of realizable volume polynomials' should specify the topology (coefficient-wise convergence, presumably) and, for non-algebraically-closed fields, the convention for semiample divisors should be clarified.
  4. [§3.2] The strict inclusion is stated for any field k, while the cited reverse Khovanskii–Teissier inequality of [JL23] is stated for algebraically closed fields; clarify whether a base-change reduction is used.
  5. [Example 2.2] The endpoint positivity convention in the log-concavity criterion should be stated, to avoid ambiguity when p_0=0 or p_d=0.
  6. [§4.3] The sentence 'If this map is induced by an irreducible correspondence Γ... then, by [GHM+, Lemma 2.1], it preserves the classes of irreducible cycles up to a rational multiple' is unclear. Please define φ_T more precisely and rephrase the geometric interpretation.

Circularity Check

0 steps flagged · score 0.0 of 10

Survey attributes external theorems; no circular derivation chain.

full rationale

This paper is an expository survey. Its load-bearing statements, including Theorem 2.11 (L_n^d = Lorentzian polynomials, cited to [BH20]), Theorem 4.7 (realizable covolume iff universal action on realizable volume polynomials, cited to [GHM+]), and Corollary 4.8, are explicitly attributed to prior work rather than derived in the manuscript. Definition 4.1 introduces covolume polynomials via g(∂)∘x[μ] being a realizable volume polynomial; Theorem 4.7 characterizes the same g by the stronger property that g(∂)∘f is realizable volume for every realizable f. Since (2) implies (1) by taking f = x[μ] (a realizable volume polynomial), the theorem is not a tautology and is not forced by the definition. No fitted parameter is later called a prediction, no known result is merely renamed, and no ansatz is smuggled in through self-citation: the cited results (published [BH20], or preprints attributed as such) are independent inputs to the survey. The lack of proof sketches for quoted preprints is a correctness risk, not circularity. Therefore there is no circular step.

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

The survey relies on a set of external theorems that it does not prove. These are listed as axioms. No free parameters or invented entities appear; the paper is a mathematical survey.

assumptions (6)
  • domain assumption Alexandrov–Fenchel inequality for mixed volumes holds for convex bodies in R^d.
    Invoked in Section 1.2 as the forward implication for Conjectures 1.6 and 1.7, and in Section 2.1 as a cornerstone.
  • domain assumption [BH20, Theorem 2.25]: L_n^d is the set of Lorentzian polynomials in H_n^d.
    Quoted as Theorem 2.11 in Section 2.3 and used throughout to identify Lorentzian polynomials.
  • domain assumption [HHM+, Theorem 1.4]: Projection-area realizability in R^4 is equivalent to triangle inequalities on square-root products.
    Quoted as Theorem 1.1 in Section 1.1; no proof given in this paper.
  • domain assumption [HHM+, Theorem 1.6]: Rational homology classes in CH((P^1)^4) are realizable iff triangle inequalities hold.
    Quoted as Theorem 1.9 in Section 1.3.
  • domain assumption [GHM+, Theorem 1.5/4.7]: Realizable covolume polynomials over k are characterized by the operator condition.
    Quoted as Theorem 4.7 in Section 4.2; central to the operator-theoretic results.
  • domain assumption Reverse Khovanskii–Teissier inequality holds for nef divisors on projective varieties over any algebraically closed field.
    Used in Section 3.2 to show the cubic polynomial f is not a volume polynomial; cited from [LX17, JL23].

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Cite this review

Pith. "Pith review of Volume polynomials." pith.science (2026). https://pith.science/paper/2OS3TFVG

@misc{pith2026260113249,
  author       = {Pith},
  title        = {Pith review of: Volume polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2OS3TFVG}},
  note         = {Machine review of arXiv:2601.13249}
}
read the original abstract

Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss applications to the combinatorics of algebraic matroids. These notes are based on lectures given at the 2025 Summer Research Institute in Algebraic Geometry at Colorado State University.

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Works this paper leans on

2 extracted references · 1 linked inside Pith

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