REVIEW 3 major objections 4 minor 7 references
Examples of IDP lattice polytopes with non-log-concave $h^*$-vector
T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read Two explicitly listed lattice polytopes with the integer decomposition property have $h^*$-vectors that are not log-concave, answering the stronger form of the Ehrhart unimodality question.
desk verdict Explicit IDP polytopes with non-log-concave h*-vectors, answering an open question, but the load-bearing IDP claim is only asserted via software; the explicit data makes this worth refereeing with a request for certificates. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The key object is the $h^*$-polynomial: the numerator in $\sum_{k\ge 0} E_P(k)t^k = h^*_P(t)/(1-t)^{d+1}$, where $E_P(k)$ counts lattice points in the $k$-th dilation of a lattice polytope $P$ of dimension $d$. The hypothesis that makes the question sharp is IDP, the integer decomposition property, which says every lattice point of $kP$ is a sum of $k$ lattice points of $P$. The load-bearing examples are two explicit convex hulls with $d+3$ vertices each, chosen by an ongoing machine-learning-guided search; the paper's verification consists of computing their Ehrhart data and checking the coefficient inequality. For the 12-dimensional example, an additional mechanism is the claim that all triangulations are unimodular, which places the polytope in a very restrictive class and rules out explanations based on triangulation pathology.
What would settle it
Run an independent exact-arithmetic computation on the two displayed vertex sets, checking whether every lattice point of $kP$ is a sum of $k$ lattice points of $P$ for every $k$, and recompute the coefficients of $h^*_P(t)$; any failure of IDP or any coefficient different from the displayed $h^*$-vectors would settle the central claim false.
Extended reading notes
Core claim
The paper's central claim is that the two vertex sets displayed in Theorems 1.2 and 1.3 are actual IDP polytopes whose Ehrhart-series numerators are not log-concave. In dimension 7 the $h^*$-polynomial is $h^*(t)=1+2t+3t^2+4t^3+5t^4+3t^5+2t^6+t^7$, and at $i=5$ the inequality $h^*_4h^*_6\le (h^*_5)^2$ reads $10\le 9$, which fails. In dimension 12 the same initial coefficient pattern appears, followed by five zeros, for a 0/1-polytope with 15 vertices that is claimed to have every triangulation unimodular. That second polytope is also described as the arc polytope of a small bipartite directed graph, meaning it is a convex hull of columns of an incidence matrix. The paper states that all these properties can be verified with computational software and says the examples answer a question posed in a recent survey; this version is explicitly described as work in progress.
Load-bearing premise
The load-bearing premise is that the software computations certifying the integer decomposition property (and, in the 12-dimensional case, the unimodularity of every triangulation) are correct, since the paper provides no code, logs, or independent certificate for that assertion.
Editorial extensions
If this is right
- Question 3.9(b) of the cited survey has a negative answer: IDP polytopes need not have log-concave $h^*$-vectors.
- The original unimodality conjecture is not disproved: both displayed $h^*$-sequences are unimodal, so the paper kills only the stronger log-concavity version.
- The 12-dimensional example shows that even a 0/1-polytope whose every triangulation is unimodular can have a non-log-concave $h^*$-vector.
- Both examples have $d+3$ vertices, connecting the failure to the low-vertex-count family of 9 and 15 vertices that the authors' search converged to.
Reading between the lines
- Beyond the paper's results, one can test whether the recurring coefficients $1,2,3,4,5,3,2,1$ appear for IDP polytopes in dimensions below 7; finding one would show that dimension 7 is not the threshold.
- Because the 12-dimensional example is an arc polytope of a bipartite graph, arc polytopes of other small bipartite graphs form a natural search space for further counterexamples, including possible non-unimodal $h^*$-vectors.
- The paper's machine-learning-guided search converged to $d+3$ vertices but does not claim this vertex count is necessary; a reader might investigate whether this count is sufficient in some family of lattice polytopes.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper announces two explicit lattice polytopes whose Ehrhart h*-vectors are not log-concave: a 7-dimensional polytope in R^7 with 9 vertices and h*-vector (1,2,3,4,5,3,2,1), and a 12-dimensional 0/1-polytope in R^12 with 15 vertices and h*-vector (1,2,3,4,5,3,2,1,0,0,0,0,0). The authors assert that both polytopes have the integer decomposition property (IDP), and that the 12-dimensional example is unimodular (every triangulation is unimodular) and has no quadratic triangulation. If these assertions are correct, the examples answer Question 3.9(b) of Ferroni and Higashitani, which asks whether IDP polytopes must have log-concave h*-vectors. The paper also states, without proof, that the 12-dimensional example is unimodularly equivalent to the arc polytope of a directed bipartite graph.
Significance. The claimed result is significant: it would disprove a natural log-concavity strengthening of the unimodality conjecture for IDP polytopes, and the 0/1-polytope example in dimension 12 is particularly striking because 0/1-polytopes form a very restricted class. The explicit vertex data and displayed h*-vectors make non-log-concavity directly checkable by hand for the vectors shown. However, the central relevance of the examples depends entirely on the IDP assertion (and for Theorem 1.3, on the stronger unimodularity assertion), and the paper provides no proof or reproducible computational certificate for these properties. The only support is the closing sentence stating that the properties can be verified with software. As a result, the paper is best read as a research announcement rather than a complete proof. Independent verification is feasible because the polytopes are given explicitly, so the gap is a matter of missing evidence rather than a perceived internal inconsistency.
major comments (3)
- [Theorem 1.2(1) and Theorem 1.3(1)–(2)] The central claims that the displayed polytopes have the integer decomposition property (and that the 12-dimensional polytope has the stronger property that every triangulation is unimodular) are asserted without proof. The only support is the final sentence: 'All properties in these results can be verified using software packages like Polymake, Normaliz, Magma, or SageMath.' For a mathematical theorem, this is insufficient: no scripts, logs, or certificates are provided, and the connection to Question 3.9(b) collapses if either polytope fails IDP. The authors should either give a human-readable proof (for example, exhibit a unimodular triangulation, or give an explicit Hilbert-basis argument, or in the 0/1 case provide a regular unimodular triangulation) or make the computation fully reproducible by supplying the exact inputs, outputs, and version information for one of the named software packages.
- [Proposition 1.4] The assertion that the polytope in Theorem 1.3 is unimodularly equivalent to the arc polytope of the directed bipartite graph in Figure 1 is stated without proof and without a precise correspondence between the 15 listed vertices and the edges of the graph. Since this proposition is used to claim the example 'comes from a directed graph', it should be either proved explicitly or clearly marked as an unproved observation that is not needed for the main theorem.
- [Theorem 1.3(3)] The claim that the 12-dimensional polytope has no quadratic triangulation is not substantiated anywhere in the text. This property is part of the theorem statement, so it requires either a proof or a reproducible computational check. If it is not yet verified, it should be removed from the theorem and deferred to the promised future update.
minor comments (4)
- [Abstract and title] The paper is explicitly labeled 'preliminary report on research in progress' and states that it 'will be updated', yet Theorems 1.2 and 1.3 are presented as finished results. This tension should be resolved, for example by adding a clear 'research announcement' banner or moving the caveat to the statement of the theorems.
- [Theorem 1.3(5)] The h*-vector is written as (1,2,3,4,5,3,2,1,0,0,0,0,0). While this is a valid 13-term vector for a 12-dimensional polytope, the trailing zeros may confuse readers into thinking the h*-polynomial has degree 7. Consider writing it explicitly as 1+2t+3t^2+4t^3+5t^4+3t^5+2t^6+t^7 or adding a sentence clarifying that h*_8 = ... = h*_12 = 0.
- [Figure 1] The directed bipartite graph in Figure 1 is not described in the text; the vertex labels 1 through 14 are drawn, but the edges are not listed. A short list of edges or an explicit mapping to the three non-standard vertices of Theorem 1.3 would make Proposition 1.4 meaningful to the reader.
- [Title] The title and opening line of the full text contain typographical spacing errors ('LA TTICE POL YTOPES', 'VADYM' is fine, but 'POL YTOPES' should be 'POLYTOPES'). These are likely artifacts of the TeX source and should be corrected.
Circularity Check
No circularity: explicit polytopes have computed h*-vectors; reliance on software checks is a reproducibility gap, not a circular reduction.
full rationale
The paper presents two explicitly defined lattice polytopes and lists their h*-vectors as computed invariants from the Ehrhart-series definition. The non-log-concavity of those listed vectors is a direct arithmetic fact about the displayed coefficient sequences, and the examples were found by search heuristics rather than by constructing a polytope so as to force the target h*-vector by definition. No parameter is fitted to a subset of data and then renamed a prediction, and none of the paper's load-bearing claims reduces to a self-citation chain or to a uniqueness theorem imported from the authors' prior work. The only substantive concern is that the IDP property in Theorem 1.2 and the unimodular-triangulation property in Theorem 1.3 are asserted rather than proved, with support limited to the closing sentence that they can be verified using Polymake, Normaliz, Magma, or SageMath. That is an omitted computational verification and a reproducibility gap, but it is not circular: the external software checks do not presuppose the target conclusion, and the explicit vertex descriptions make the checks independently feasible. The citation to prior work on non-log-concave h-vectors in the commutative-algebra setting is contextual and not load-bearing for the present examples. The Ferroni-Higashitani question would be answered only if the asserted computational properties are correct, but that condition is an external correctness check, not an equivalence of inputs and outputs. Accordingly, no circular step is identified.
Assumptions & free parameters
assumptions (2)
- domain assumption The integer decomposition property and h*-vector are computed via standard algorithms implemented in Polymake, Normaliz, Magma, or SageMath.
- standard math Standard definitions of Ehrhart series, h*-polynomial, IDP, unimodality, and log-concavity from Beck-Robins and Braun are used throughout.
Cite this review
Pith. "Pith review of Examples of IDP lattice polytopes with non-log-concave $h^*$-vector." pith.science (2026). https://pith.science/paper/OPJEPGDV
@misc{pith2026250518896,
author = {Pith},
title = {Pith review of: Examples of IDP lattice polytopes with non-log-concave $h^*$-vector},
year = {2026},
howpublished = {\url{https://pith.science/paper/OPJEPGDV}},
note = {Machine review of arXiv:2505.18896}
}
abstract
Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i.e., any lattice point in a $k$th dilation is a sum of $k$ lattice points in the polytope. It is a long-standing conjecture whether the numerator of the Ehrhart series of an IDP polytope, called the $h^*$-polynomial, has a unimodal coefficient vector. In this preliminary report on research in progress we present examples showing that $h^*$-vectors of IDP polytopes do not have to be log-concave. This answers a question of Luis Ferroni and Akihiro Higashitani. As this is an ongoing project, this paper will be updated with more details and examples in the near future.
Figures
Reference graph
Works this paper leans on
-
[1]
Computing the continuous discretely
Matthias Beck and Sinai Robins. Computing the continuous discretely. Integer-point enumeration in polyhedra.Undergraduate Texts Math. New York, NY: Springer, 2nd edition edition, 2015
work page 2015
-
[2]
Unimodality problems in Ehrhart theory
Benjamin Braun. Unimodality problems in Ehrhart theory. In Recent trends in com- binatorics, volume 159 of IMA Vol. Math. Appl., pages 687–711. Springer, [Cham], 2016
work page 2016
-
[3]
Examples and counterexamples in Ehrhart theory
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
-
[4]
Piechnik, and Francisco Santos
Christian Haase, Andreas Paffenholz, Lindsey C. Piechnik, and Francisco Santos. Ex- istence of unimodular triangulations – positive results, volume 1321 of Mem. Am. Math. Soc. Providence, RI: American Mathematical Society (AMS), 2021
work page 2021
-
[5]
Disproving Hibi’s conjecture with CoCoA; or: Projective curves with bad Hilbert functions
Gianfranco Niesi and Lorenzo Robbiano. Disproving Hibi’s conjecture with CoCoA; or: Projective curves with bad Hilbert functions. In Computational algebraic geometry. Papers from a conference, held in Nice, France, April 21–25, 1992, pages 195–201. Boston: Birkh¨ auser, 1993
work page 1992
-
[6]
Benjamin Nill. Unimodular polytopes and a new Heller-type bound on totally un- imodular matrices via Seymour’s decomposition theorem. preprint arXiv:2405.13431, 2024
work page Pith review arXiv 2024
-
[7]
Unimodality questions for integrally closed lattice polytopes
Jan Schepers and Leen Van Langenhoven. Unimodality questions for integrally closed lattice polytopes. Ann. Comb., 17(3):571–589, 2013. (J. Hofscheier) School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK Email address : johannes.hofscheier@nottingham.ac.uk (V. Kurylenko, B. Nill) F aculty of Mathematics, Otto-von-Guericke-Uni...
work page 2013
Reviewed August 7, 2026 · model on record in the stance chip above.
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