{"id":"71dd7f82-68c7-4ff8-aacf-84b4a18ad6f0","arxiv_id":"2608.03635","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In every fixed dimension, all lattice polytopes of sufficiently large lattice width have real-rooted Ehrhart h*-polynomials.","lead":"Lattice polytopes are geometric shapes whose corners lie on the integer grid. This paper proves that, in any fixed dimension, once such a shape is wide enough, the polynomial that encodes its grid-point counts has only real negative roots, so its coefficients are strictly log-concave and unimodal.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's verdict ACCEPT with high confidence is justified. The proofs are short, correct, and the only external dependency is a published lemma. I considered possible hidden failures: the volume calculations in Theorem 1.2 (hypersimplex normalized volume and affine determinant) check out; the width inequality for slices is valid; the passage from full lattice points to relative interior points is standard for large-width lattice polytopes; and the uniform threshold argument by contradiction is valid. I do not think any of these rises to a load-bearing objection. The proposed concrete test would serve as a sanity check but is not required for acceptance.","tokens_in":4647,"tokens_out":49450,"duration_ms":480276,"concrete_test":"For a non-trivial check of the full pipeline, take d=2 and a sequence of lattice triangles with width→∞; compute h*(P_n;t)/area(P_n) and compare with A_2(t)=t+t^2, and verify the roots of h*(P_n;t) are negative and real for all sufficiently large widths. This exercises Lemma 2.1, the inversion step, and Lemma 2.2 without relying on any further external claims.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is a clean corollary of the quoted lattice-point/volume approximation. I checked the internal steps: the inversion formula in §3 is standard; the normalized h*-polynomial converges coefficientwise to the shifted Eulerian polynomial, and Lemma 2.2 correctly transfers simple real-rootedness under small perturbations. For Theorem 1.2, the width inequality (2) is correct, the hypersimplex volume computation uses the affine-lattice normalized volume (not Euclidean volume), and the affine determinant factor D is consistent, so the normalization d!/D_n is right. The only nontrivial external input is Lemma 2.1 (Basu-Oertel). If it holds as stated, the proofs are sound; I found no internal inconsistency or missing hypothesis in its application. The relint point-count variant used in §4 would follow from Lemma 2.1 plus the fact that boundary lattice points are negligible for large-width lattice slices, which is standard. No load-bearing concern identified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two theorems in Ehrhart theory. Theorem 1.1 states that for each fixed dimension d there is a constant C_d such that every d-dimensional lattice polytope of lattice width exceeding C_d has an h*-polynomial with positive coefficients and d distinct negative real roots; hence its h*-vector is strictly log-concave and unimodal. Theorem 1.2 extends the conclusion to the local h*-polynomial (box polynomial) of lattice simplices of large lattice width, showing that its coefficients are positive and its roots consist of a simple zero and d-1 distinct negative reals. The proofs are short: they combine a result of Basu and Oertel on the ratio of lattice point counts to volume for convex bodies of large lattice width with the Ehrhart inversion formula, and then use the fact that the limiting polynomial is the shifted Eulerian polynomial, which has simple real roots, together with the stability of simple real-rootedness under small perturbations.","tokens_in":4829,"tokens_out":21787,"duration_ms":180641,"significance":"If correct, the results resolve a natural question posed by Averkov, Hofscheier, and Nill and establish a clean geometric sufficient condition for real-rootedness of Ehrhart h*-polynomials. The proof strategy is elegant: it reduces a global statement about arbitrary lattice polytopes to a quantitative lattice-point-count approximation. The paper also notes that the constants are explicit in principle, though very large, and correctly identifies the Eulerian polynomial as the universal limiting object. The central derivation is elementary after quoting the external Basu-Oertel lemma, which is the load-bearing input; the application of that lemma is mathematically sound. The paper is a significant contribution to Ehrhart theory and combinatorial geometry.","major_comments":[],"minor_comments":[{"comment":"The sentence 'This answers negatively Question 5 by Averkov, Hofscheier and the author in [1]' is confusing because Theorem 1.1 is a positive result; please clarify whether the answer to Question 5 is positive (for the real-rootedness question) or negative (for the IDP question discussed in the following sentence).","section":"Section 1"},{"comment":"The statement of Lemma 2.1 does not mention the condition from Oertel's thesis that the body contains a unimodular copy of a dilate of a standard cube; for the reader's confidence in the main load-bearing input, please include a sentence or a precise reference explaining how this condition implies the width-based bound.","section":"Section 2.1"},{"comment":"In Lemma 2.2, the statement 'after any sufficiently small real coefficient perturbation' should specify that the leading coefficient is assumed not to vanish (or that the degree is preserved), since a perturbation that lowers the degree can destroy the root count.","section":"Section 2.2"},{"comment":"Inequality (2) is applied to the slice S(k), which is defined as a subset of R^{d+1}; please state explicitly that S(k) is identified with its projection to the first d coordinates when computing its lattice width and applying Lemma 2.1.","section":"Section 4"},{"comment":"The assertion that |relint(K_n)∩Z^d|/vol(K_n) -> 1 follows 'as is easy to see' is used in the proof of Theorem 1.2; since this is not entirely immediate for arbitrary convex bodies, a one-sentence justification (e.g., boundary lattice points are negligible for bodies of large width) would improve the exposition.","section":"Section 2.1"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript relies on a single external result, Lemma 2.1 (Basu-Oertel), for both main theorems. I did not verify the original proof of that lemma, but its application in this paper is correct and the paper cites it properly. The disclosure that a first draft was produced with ChatGPT is unusual; the editor may wish to check the journal's policy on AI-assisted writing, though the author explicitly takes responsibility for the final version. The paper fits the scope of the journal and, modulo the local clarifications, is suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Ben, here's my take. The result is real: for fixed d, every lattice polytope of sufficiently large lattice width has an h*-polynomial with positive coefficients and d distinct negative real roots. That is new and answers a question in Averkov–Hofscheier–Nill. The previous known case was dilates of a fixed polytope, so moving to any family with width tending to infinity is a genuine step. The proof is short: Ehrhart inversion, coefficientwise convergence of h*/vol to the shifted Eulerian polynomial, then Lemma 2.2 transferring simple real roots under small perturbations. I checked the steps; they hold. Theorem 1.2 for box polynomials of simplices is a natural companion, and the slice-width inequality (2) plus the hypersimplex volume computation are correct. The external input, Basu–Oertel Lemma 2.1, is the whole engine; it is published and seems independent. If it fails, both theorems fail, but nothing in this paper suggests it is shaky.\n\nWhat I'd flag as minor: the constants C_d are left implicit with very large orders, so the result is qualitative. The paper says so itself, so no complaint. The conjecture that Theorem 1.2 extends to all polytopes is stated honestly; the obstruction (no combinatorial interpretation for local h* of general polytopes) is the real reason. The acknowledgment about ChatGPT is a bit unusual, but the author takes responsibility and the mathematics is checkable. Nothing circular in the citation pattern; the single self-citation is to the open question being answered.\n\nWho is this for? People in Ehrhart theory and unimodality. It won't change practice outside discrete geometry, but within the area it answers an open question with a short, clean argument. I would send this to a serious referee. My own verdict is accept.","headline":"Short, sound paper proving that large lattice width forces real-rooted Ehrhart h*-polynomials; the proof is a clean corollary of Basu-Oertel and deserves refereeing.","tokens_in":5310,"tokens_out":1592,"would_cite":true,"duration_ms":15559,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52B20","05A15","52C07"],"pacs":[],"model":"deepseek-v4-flash","headline":"In fixed dimension, every lattice polytope of sufficiently large lattice width has an Ehrhart $h^*$-polynomial with distinct negative real roots, hence a strictly log-concave and unimodal $h^*$-vector.","keywords":["lattice polytope","lattice width","Ehrhart h*-polynomial","real-rooted","Eulerian polynomial","local h*-polynomial","box polynomial","log-concavity"],"falsifier":"Compute the Ehrhart $h^*$-polynomials of a family of $d$-dimensional lattice polytopes whose lattice widths tend to infinity and check whether, for all sufficiently large widths, the polynomial has $d$ distinct negative real roots and all coefficients positive; a single sequence with a complex-conjugate pair of roots at arbitrarily large width would disprove Theorem 1.1. In dimension 2 this means finding arbitrarily wide lattice polygons whose quadratic $h^*$-polynomial has discriminant $a^2 - 4b < 0$; in dimension 3, a cubic with only one real root.","tokens_in":4478,"feed_emoji":"📐","tokens_out":14552,"duration_ms":109662,"temperature":0.7,"pith_summary":"In fixed dimension, every lattice polytope whose lattice width exceeds a constant depending only on the dimension has an Ehrhart $h^*$-polynomial with positive coefficients and $d$ distinct negative real roots. Because a real-rooted polynomial with positive coefficients is strictly log-concave and unimodal, the $h^*$-vectors of all sufficiently wide lattice polytopes inherit these strong coefficient properties. The same conclusion holds for the local $h^*$-polynomial, or box polynomial, of lattice simplices. The paper answers a question posed in [1] by showing that large width, while not forcing the integer decomposition property, does force real-rootedness.","feed_headline":"Large width forces real roots in Ehrhart h*-polynomials","feed_subtitle":"In fixed dimension, h*-vectors of sufficiently wide lattice polytopes become strictly log-concave and unimodal.","key_machinery":"The load-bearing input is Lemma 2.1, taken from [2]: there is a constant $N_d$ such that any convex body $K \\subset \\mathbb{R}^d$ with lattice width greater than $cN_d$ satisfies $e^{-1/c} \\le |K \\cap \\mathbb{Z}^d|/\\mathrm{vol}(K) \\le e^{1/c}$. Combined with the Ehrhart-series inversion formula, this makes each normalized coefficient $h^*_j(P_n)/\\mathrm{vol}(P_n)$ converge to the $j$-th Eulerian number, so $h^*(P_n;t)/\\mathrm{vol}(P_n) \\to A_d(t)$ coefficientwise. Since $A_d(t)$ has $d$ simple real roots, one at $0$ and the rest negative, and simple real roots survive sufficiently small coefficient perturbations, eventual real-rootedness follows. For the simplex theorem, the $k$-slice volume computation reduces to the hypersimplex volume $\\langle {d \\atop k-1}\\rangle / d!$, and the same perturbation argument applies.","core_discovery":"The paper proves two theorems. Theorem 1.1: for every $d \\ge 1$ there is a constant $C_d > 0$ such that every $d$-dimensional lattice polytope with lattice width greater than $C_d$ has an $h^*$-polynomial with positive coefficients and $d$ distinct negative real roots; consequently its $h^*$-vector is strictly log-concave and unimodal. Theorem 1.2: for every $d \\ge 1$ there is a constant $C_d^\\square > 0$ such that the local $h^*$-polynomial of every $d$-dimensional lattice simplex of lattice width greater than $C_d^\\square$ has a simple root at $0$, positive coefficients, and $d-1$ distinct negative real roots. The proof shows that as width grows, the normalized $h^*$-polynomial $h^*(P;t)/\\mathrm{vol}(P)$ converges coefficientwise to the Eulerian polynomial $A_d(t)$, whose roots are known to be simple, real, and nonpositive; a standard perturbation lemma then preserves that simplicity.","pith_inferences":["Because the convergence to the Eulerian polynomial is coefficientwise, one could plausibly derive quantitative bounds on the $h^*$-coefficients of wide polytopes, or a central-limit-type statement for their distribution; the paper does not pursue these.","The same volume-approximation mechanism suggests that Conjecture 1.3 might be approachable by finding any combinatorial interpretation of local $h^*$-coefficients of arbitrary lattice polytopes as lattice-point counts in slices; this is not done here.","The lemma from [2] applies to all convex bodies, not only polytopes, so a natural testable extension is whether an analogous real-rootedness statement survives for rational polytopes or for sequences with Ehrhart-like counting series, although the $h^*$-polynomial is not classically defined there.","The constants obtained from the proof are far larger than the likely optimal thresholds, so computational searches on families of wide polytopes could reveal the true dimension-dependent cutoff."],"forward_implications":["Any infinite family of $d$-dimensional lattice polytopes whose lattice widths go to infinity eventually has real-rooted $h^*$-polynomials, hence strictly log-concave and unimodal $h^*$-vectors.","The normalized $h^*$-polynomial $h^*(P;t)/\\mathrm{vol}(P)$ converges coefficientwise to the Eulerian polynomial $A_d(t)$ as width grows, so Eulerian numbers describe the asymptotic coefficient distribution of wide lattice polytopes.","For every sufficiently wide lattice simplex, the box polynomial has positive coefficients, a simple root at $0$, and $d-1$ distinct negative real roots, giving strict log-concavity and unimodality for the local $h^*$-vector.","Question 5 from [1] is answered negatively: large lattice width does not force the integer decomposition property, but it does force the strong coefficient behavior that IDP is often used to establish."],"supporting_citations":[{"why":"It supplies Lemma 2.1, the width-to-volume lattice-point approximation that drives both theorems.","marker":"[2]"},{"why":"It contains the earlier thesis version of the approximation inequality cited as the source of Lemma 2.1.","marker":"[17]"},{"why":"It provides the fact that the Eulerian polynomial has simple real roots, one zero and the rest negative.","marker":"[9]"},{"why":"It gives the perturbation principle that simple real roots survive small coefficient changes.","marker":"[16]"},{"why":"It computes the volume of hypersimplices, used to evaluate slice volumes in the simplex proof.","marker":"[15]"},{"why":"It contains the earlier result on eventual real-rootedness of dilates of a fixed polytope, which the present theorem extends to arbitrary width-unbounded families.","marker":"[12]"},{"why":"It contains the question that Theorem 1.1 answers negatively.","marker":"[1]"},{"why":"It is part of the standard Ehrhart theory underlying the inversion formula that converts point-count convergence to coefficient convergence.","marker":"[19]"}],"fun_headline_variants":["Sufficiently wide polytopes have real-rooted h* polynomials","Width threshold forces real roots in h* polynomials","Past a width threshold, h* polynomials are real-rooted","For wide lattice polytopes, h* roots are real"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the lemma from [2] that once a convex body is wide enough, its lattice-point count is within a factor arbitrarily close to 1 of its volume; if that uniform approximation fails for some family of unbounded width, the coefficientwise convergence to the Eulerian polynomial and the real-rootedness conclusion do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Sufficiently wide polytopes have real-rooted h* polynomials","Width threshold forces real roots in h* polynomials","Past a width threshold, h* polynomials are real-rooted","For wide lattice polytopes, h* roots are real"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001561,"raw_usage":{"total_tokens":6207,"prompt_tokens":891,"completion_tokens":5316,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":507,"completion_tokens_details":{"reasoning_tokens":5258}},"tokens_in":507,"tokens_out":5316,"duration_ms":40030,"temperature":1.0,"reasoning_tokens":5258,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:50:52.225475+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Ehrhart $h^*$-polynomials of a family of $d$-dimensional lattice polytopes whose lattice widths tend to infinity and check whether, for all sufficiently large widths, the polynomial has $d$ distinct negative real roots and all coefficients positive; a single sequence with a complex-conjugate pair of roots at arbitrarily large width would disprove Theorem 1.1. In dimension 2 this means finding arbitrarily wide lattice polygons whose quadratic $h^*$-polynomial has discriminant $a^2 - 4b < 0$; in dimension 3, a cubic with only one real root.","supporting_citations":[{"cited_title":"Optim.27(2017), no","cited_arxiv_id":null,"evidence_quote":"It supplies Lemma 2.1, the width-to-volume lattice-point approximation that drives both theorems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It contains the earlier thesis version of the approximation inequality cited as the source of Lemma 2.1."},{"cited_title":"138, Springer-Verlag, Berlin-New York, 1970","cited_arxiv_id":null,"evidence_quote":"It provides the fact that the Eulerian polynomial has simple real roots, one zero and the rest negative."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the perturbation principle that simple real roots survive small coefficient changes."},{"cited_title":"Geom.48(2012), no","cited_arxiv_id":null,"evidence_quote":"It computes the volume of hypersimplices, used to evaluate slice volumes in the simplex proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It contains the earlier result on eventual real-rootedness of dilates of a fixed polytope, which the present theorem extends to arbitrary width-unbounded families."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It contains the question that Theorem 1.1 answers negatively."},{"cited_title":"Stanley,Decompositions of rational convex polytopes, Ann","cited_arxiv_id":null,"evidence_quote":"It is part of the standard Ehrhart theory underlying the inversion formula that converts point-count convergence to coefficient convergence."}],"review_version":2}