Lattice polytopes of large width have real-rooted Ehrhart $h^*$-polynomials

Benjamin Nill

Lattice polytopes of large width have real-rooted Ehrhart $h^*$-polynomials

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2608.03635 v1 pith:V5LUHQ76 submitted 2026-08-04 math.CO

Benjamin Nill This is my paper · ORCID

classification math.CO MSC 52B2005A1552C07

keywords latticepolytopewidthEhrharth*-polynomialreal-rootedEulerianpolynomiallocalboxlog-concavity

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What 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.

Who 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.

Referee Report

0 major / 5 minor

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.

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.

minor comments (5)

[Section 1] 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 2.1] 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 2.2] 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 4] 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 2.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof relies on the external Basu-Oertel lemma and standard facts, with no fitted inputs or self-citation load-bearing steps.

full rationale

The paper's derivation is not circular. Theorem 1.1 is proved by combining Lemma 2.1, a lattice-point-counting approximation result of Basu and Oertel, with the standard Ehrhart inversion formula and the classical fact that the shifted Eulerian polynomial has simple real roots. The normalized h*-polynomial converges coefficientwise to that Eulerian polynomial, and Lemma 2.2 transfers simple real-rootedness under small perturbations. Nothing in this chain assumes the target conclusion. Theorem 1.2 similarly follows from the width inequality for slices, the standard hypersimplex volume formula, and the same external approximation lemma. The only self-citation, reference [1], is used solely to identify the question being answered, not as an ingredient of the proof. No parameter is fitted and renamed a prediction, no uniqueness theorem is imported from prior work by the author, and no ansatz is smuggled in via citation. The result is therefore a genuine corollary of an independent external lemma.

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

The central claim rests on the Basu-Oertel lattice-point-counting approximation (Lemma 2.1); the other ingredients are standard facts about Ehrhart series, Eulerian polynomials, and hypersimplex volumes. There are no fitted parameters or new entities.

assumptions (6)

domain assumption Lemma 2.1 (Basu-Oertel): for each d there is a constant N_d such that if a convex body K in R^d has width(K) > c N_d, then |K∩Z^d|/vol(K) is between e^{-1/c} and e^{1/c}.
Quoted from [2, Lemma 3.8] and [17, Lemma 4.1.11]; not proved in the paper. Both Theorem 1.1 and Theorem 1.2 use it to replace lattice point counts with volumes.
standard math The Eulerian polynomial A_d(t) has d simple real roots, one equal to 0 and the others negative.
Used in Sections 3 and 4 as the limit polynomial; cited to [9].
standard math If a real polynomial has only simple real roots, then this remains true after any sufficiently small real coefficient perturbation (Lemma 2.2).
Used to transfer simple real-rootedness from A_d to the normalized h*-polynomials; cited to [16].
standard math The volume of the hypersimplex H_k with respect to the affine lattice is ⟨d, k-1⟩/d!.
Used in Section 4 to compute vol(S(k)); cited to [15].
standard math For lattice polytopes, the Ehrhart series inversion formula h*_j = sum_{i=0}^j (-1)^{j-i} binom(d+1, j-i) |i P ∩ Z^d| holds.
Standard fact in Ehrhart theory, used in Section 3.
domain assumption V_n = vol(P_n) tends to infinity for a sequence of d-dimensional lattice polytopes with width(P_n) -> infinity.
Used in Section 3 to normalize h*; cited to [14].

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Pith. "Pith review of Lattice polytopes of large width have real-rooted Ehrhart $h^*$-polynomials." pith.science (2026). https://pith.science/paper/V5LUHQ76

@misc{pith2026260803635,
  author       = {Pith},
  title        = {Pith review of: Lattice polytopes of large width have real-rooted Ehrhart $h^*$-polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V5LUHQ76}},
  note         = {Machine review of arXiv:2608.03635}
}

abstract

In this note we prove that in fixed dimension the Ehrhart $h^*$-polynomial of a lattice polytope of sufficiently large lattice width is real-rooted. In particular, this implies strict log-concavity and unimodality of the $h^*$-vector and answers a question of Averkov, Hofscheier and the author. For a lattice simplex we prove the analogous statement for its local $h^*$-polynomial, also called box polynomial. The proofs were found using ChatGPT 5.6 Sol and follow essentially directly from a result by Basu and Oertel that for large enough lattice width counting lattice points approximates the volume.

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Works this paper leans on

20 extracted references · 19 canonical work pages

[1]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Gennadiy Averkov, Johannes Hofscheier, and Benjamin Nill,Generalized flatness constants, spanning lattice polytopes, and the Gromov width, Manuscripta Math.170(2023), 147–165

work page 2023

[2] [2]

Optim.27(2017), no

Amitabh Basu and Timm Oertel,Centerpoints: a link between optimization and convex geometry, SIAM J. Optim.27(2017), no. 2, 866–889

work page 2017

[3] [3]

Z.264(2010), no

Matthias Beck and Alan Stapledon,On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series, Math. Z.264(2010), no. 1, 195–207

work page 2010

[4] [4]

Math.99(1985), no

Ulrich Betke and Peter McMullen,Lattice points in lattice polytopes, Monatsh. Math.99(1985), no. 4, 253–265

work page 1985

[5] [5]

Benjamin Braun,Unimodality problems in Ehrhart theory, Recent trends in combinatorics, IMA Vol. Math. Appl., vol. 159, Springer, [Cham], 2016, pp. 687–711

work page 2016

[6] [6]

Francesco Brenti and Volkmar Welker,The Veronese construction for formal power series and graded algebras, Adv. in Appl. Math.42(2009), no. 4, 545–556

work page 2009

[7] [7]

Eugène Ehrhart,Sur les polyèdres rationnels homothétiques àn dimensions, C. R. Acad. Sci. Paris254 (1962), 616–618. WIDE LATTICE POLYTOPES 5

work page 1962

[8] [8]

Luis Ferroni and Akihiro Higashitani,Examples and counterexamples in Ehrhart theory, preprint arXiv:2307.10852, to appear in EMS Surv. Math. Sci. (2024)

arXiv 2024