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Tropical Ehrhart Theory and Tropical Volume

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Pith's one-line read Counting lattice points in tropical polytopes gives a genuine intrinsic volume: the tropical barycentric volume.

desk verdict Loho–Schymura builds a real tropical Ehrhart theory and an intrinsic volume with the right axioms; the alcoved-triangulation domain gap for unbounded −∞ vertices is real but fixable. read the letter →

arxiv 1908.07893 v1 pith:7OFRLZIQ submitted 2019-08-21 math.MG math.CO

Georg Loho , Matthias Schymura This is my paper · ORCID
classification math.MGmath.CO MSC 14T0552C0752A38
keywords tropicalgeometryEhrharttheorylatticepointcountingpolytopesbarycentricvolumealcovedtriangulationdequantizedrank
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 develops a tropical analogue of Ehrhart theory: for a tropical lattice polytope $P\subseteq \mathbb{T}^d$ and an integer $b\ge 2$, the function $L_b^P(k)=\#\big((k\odot P)\cap \Gamma_b^d\big)$ that counts tropical $b$-lattice points, with $\Gamma_b^d=(\log_b\mathbb{Z}_{\ge 0})^d$, in the $k$-th tropical dilation of $P$ is a polynomial in $b^k$. The leading coefficient of this polynomial is the tropical $b$-volume, and after taking the limit $\mathrm{Log}|f|=\lim_{b\to\infty}\log_b|f(b)|$ the paper obtains a parameter-free invariant, the tropical barycentric volume $\mathrm{tbvol}(P)=\max_{x\in \mathrm{Tr}_d(P)}(x_1+\cdots+x_d)$. The paper proves that this invariant satisfies the natural analogues of the Euclidean volume axioms: monotonicity, the valuation property, rotation invariance, homogeneity, non-singularity, and multiplicativity under Cartesian products. It also shows that for lattice polytopes $\mathrm{tbvol}(P)\le \mathrm{qtvol}_+(M)$, with equality exactly for pure tropical lattice polytopes. A reader should care because this gives tropical geometry a genuinely tropical metric invariant, computable by an explicit algorithm in many cases, rather than a Euclidean volume imported from an ambient space.

What carries the argument

The mechanism is the alcoved triangulation, a subdivision of a tropical lattice polytope $P$ into alcoved simplices $\Delta_\pi(a)=\mathrm{conv}\{a+e_{\pi(1)}+\cdots+e_{\pi(\ell)}:\ell=0,\dots,d\}$ with integral shift vectors $a\in\mathbb{Z}^d$; each such simplex is both tropically and classically convex. Lemma 3.11 is the hinge: the coordinatewise logarithm $\varphi(z)=(\log_b z_1,\dots,\log_b z_d)$ restricts to a bijection between the ordinary lattice points of $b^k D_b^a\mathbf{1}+(b^{k+1}-b^k)D_b^a\Delta_s(0)$ and the tropical lattice points of $k\odot\Delta_s(a)$, where $D_b^a=\mathrm{diag}(b^{a_1},\dots,b^{a_d})$. This reduces tropical lattice-point counting to classical Ehrhart counting on the diagonally transformed alcoved simplices, yielding the explicit signed-sum formula (3) for every tropical Ehrhart coefficient. The logarithmic map $\mathrm{Log}|f|=\lim_{b\to\infty}\log_b|f(b)|$ then strips the fineness parameter $b$ and gives the closed form $\mathrm{tbvol}(P)=\max\{a_1+\cdots+a_d+d:\Delta_\pi(a)\in\mathcal{T}_P\}$.

What would settle it

One decisive test: compute $L_b^P(k)$ for a small tropical lattice polygon with a known alcoved triangulation and check the result against formula (3); the deeper structural check is whether every tropical lattice polytope admits an alcoved triangulation with integral shifts, since a polytope without one would invalidate the explicit coefficient formula.

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Extended reading notes

Core claim

The central claim is that a tropical Ehrhart theorem holds in the strong sense: for every tropical lattice polytope $P\subseteq\mathbb{T}^d$, the enumerator $L_b^P(k)$ agrees with a polynomial in $b^k$, so the usual Ehrhart machinery, including leading coefficients, reciprocity, and valuation, has a tropical counterpart. The leading coefficient $c_b^d(P)$ equals the Euclidean volume of the exponentiated polytope $\exp_b(P)$, and its logarithmic asymptotics in $b$ converge to the tropical barycentric volume $\mathrm{tbvol}(P)=\max_{x\in\mathrm{Tr}_d(P)}\sum_i x_i=\max\{a_1+\cdots+a_d+d:\Delta_\pi(a)\in\mathcal{T}_P\}$. This quantity is well defined even without integrality, is monotone, idempotent, rotation invariant under scaled permutation matrices of tropical determinant zero, homogeneous, non-singular, multiplicative under products, and it is bounded above by the tropical dequantized volume $\mathrm{qtvol}_+(M)$ whenever $P=\mathrm{tconv}(M)$. Equality holds if and only if the tropical barycenter lies in the $d$-trunk, which in particular covers pure tropical lattice polytopes.

Load-bearing premise

The load-bearing premise is that every tropical lattice polytope can be cut into the special simplex pieces with integer shifts that the whole coefficient formula uses; the paper takes this structural fact as given.

Editorial extensions

If this is right

  • The tropical Ehrhart enumerator $L_b^P(k)$ is polynomial in $b^k$, so tropical lattice-point counting inherits Ehrhart reciprocity and coefficient valuations; for pure tropical lattice polytopes reciprocity holds in the form $c_b^i(\overset{\circ}{P})=(-1)^{d-i}c_b^i(P)$.
  • The leading coefficient $c_b^d(P)=\mathrm{vol}(\exp_b(P))$ is the tropical $b$-volume, and its $b$-asymptotic limit is $\mathrm{tbvol}(P)$; hence every tropical lattice polytope has a well-defined parameter-free volume.
  • $\mathrm{tbvol}(\cdot)$ satisfies the volume axioms of monotonicity, valuation or idempotency, rotation invariance, homogeneity, non-singularity, and multiplicativity, making it a genuine intrinsic volume for tropical polytopes rather than an ad hoc measure.
  • For $P=\mathrm{tconv}(M)$ with integer data, $\mathrm{tbvol}(P)\le \mathrm{qtvol}_+(M)$, and equality holds exactly when the tropical barycenter lies in the $d$-trunk, in particular for all pure tropical lattice polytopes, linking the new volume to maximal tropical determinants.
  • The lower barycentric $i$-volumes satisfy $\mathrm{tbvol}_i^-(P)\le \mathrm{tmi}(M)$, and the paper conjectures $\mathrm{Log}|c_b^i(P)|\le \mathrm{tmi}(M)$; computing $\mathrm{tbvol}$ reduces to evaluating tropical simplices in $O\big(\binom{m}{d+1}d^3\big)$, and deciding non-vanishing lies in NP $\cap$ coNP via tropical linear programming.

Reading between the lines

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

  • Because $\mathrm{tbvol}(P)$ is the coordinate sum of the tropical barycenter of the $d$-trunk, it measures the full-dimensional core of $P$ and ignores lower-dimensional tentacles; this makes it robust to small perturbations of vertices, in contrast to the dequantized volume, which is governed by an extremal tropical determinant.
  • The multiplicativity of $\mathrm{tbvol}$ under Cartesian products, a property the dequantized volume lacks, suggests that $\mathrm{tbvol}$ is the more natural candidate for factorization and isoperimetric statements in tropical convex geometry; the isoperimetric questions raised in the paper could be tested first on products of tropical simplices.
  • The conjecture $\mathrm{Log}|c_b^i(P)|\le \mathrm{tmi}(M)$ is computationally checkable on random 0/1 tropical matrices; if it holds, the tropical Ehrhart coefficients become functions of tropical minors, tying Ehrhart theory to tropical rank and to the hardness results of Section 6.
  • The definition of tropical integers as $\log_b(\mathbb{Z}_{\ge 0})$ is one choice in a family; replacing $b$ by other multiplicative bases or by algebraic integers would give a parametrized family of tropical Ehrhart theories whose $b\to\infty$ limit is presumably the same $\mathrm{tbvol}$, providing a robustness check for the whole construction.
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Referee Report

2 major / 3 minor

Summary. The paper develops a tropical analogue of Ehrhart theory. It introduces the tropical b-lattice Gamma_b^d = (log_b(Z_{\ge 0}))^d and defines tropical lattice polytopes as tropical polytopes whose vertices lie in Gamma_b^d, with a distinguished class of canonical tropical lattice polytopes with vertices in (Z_{\ge 0} \cup \{-\infty\})^d. The main structural result, Theorem 3.6, shows that the tropical lattice point enumerator L_b^P(k) = #((k \odot P) \cap Gamma_b^d) is a polynomial in b^k, obtained by applying the semiring isomorphism log_b to a classical Ehrhart result for polytopes over the (max, \cdot)-semiring. The paper then refines this via the covector decomposition and alcoved triangulations, giving explicit formulas for the tropical Ehrhart coefficients (Theorem 3.14). The leading coefficient leads to the tropical b-volume, and taking the b-to-infinity logarithmic limit yields the tropical barycentric volume tbvol(P) = max_{x \in Tr_d(P)} (x_1 + \cdots + x_d). The paper proves monotonicity, valuation, rotation invariance, homogeneity, non-singularity, and multiplicativity of tbvol, compares it with the tropical dequantized volume qtvol_+, introduces lower-dimensional barycentric i-volumes, and discusses algorithms and complexity, including an O(\binom{m}{d+1} d^3) algorithm for computing tbvol of a tropical d-polytope with m vertices.

Significance. If the results stand, the paper provides the first systematic intrinsic volume concept for tropical polytopes that is genuinely tied to a tropical lattice point count, with properties strongly analogous to classical Euclidean volume. The approach is conceptually clean: it reduces tropical Ehrhart theory to classical Ehrhart theory through the semiring isomorphism log_b, and the key Lemma 3.11 gives an explicit, fully proved bijection between tropical lattice points in scaled alcoved simplices and classical lattice points in diagonally transformed alcoved simplices. The paper also contributes useful structural results about higher trunks, a comparison with qtvol_+, and a complexity discussion connecting tbvol to tropical linear programming and mean-payoff games. These are substantial contributions. The main caveat is that several central statements are formulated for all tropical lattice polytopes, while the alcoved-triangulation machinery used to prove them is only established for a narrower class; this is a fixable but load-bearing gap.

major comments (2)
  1. [§2.1, Definition 2.5, Definition 4.3, Theorem 3.14]
  2. [§3.2, Theorem 3.14]
minor comments (3)
  1. [Definition 4.3]
  2. [Theorem 5.7]
  3. [§4.1]

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the tropical Ehrhart/volume derivation reduces to external classical Ehrhart theory and alcoved triangulations, not to its own outputs.

full rationale

The paper's central chain is self-contained against external results rather than circular. Theorem 3.6 obtains polynomiality of the tropical lattice point enumerator by applying the semiring isomorphism log_b to classical Ehrhart theory for lattice polytopes over (S_max,·), citing Beck–Robins and standard semiring facts; this is an external reduction, not a re-derivation of its conclusion from its conclusion. Theorem 3.14 derives the explicit coefficient formula from the alcoved triangulation imported from Lam–Postnikov and Develin–Sturmfels, combined with classical Ehrhart reciprocity; again the structural input is external and does not presuppose the tropical Ehrhart coefficients being computed. The tropical barycentric volume tbvol(P) is independently defined in Definition 4.5 as max_{x in Tr_d(P)} 1^T x, and Proposition 4.4 proves the equality with the leading-coefficient limit / max over alcoved simplices, rather than defining the quantity through that equality. The comparison with qtvol+ in Theorem 4.13 rests on the cited external identity qtvol+(M) = max tropical minor from Depersin–Gaubert–Joswig, and the paper explicitly contrasts tbvol with that concept. There are no fitted parameters, no 'prediction' that is statistically forced by a prior fit, and no load-bearing uniqueness theorem imported from the authors' own prior work. The reviewer-flagged concern about the alcoved triangulation for unbounded tropical lattice polytopes such as [−∞,0]^d is a possible domain/correctness gap in the stated class, not a circularity: the final max-over-trunk definition and the explicit examples of limiting leading coefficients do not reduce to the triangulation as their own input. Accordingly, the derivation chain does not exhibit any step where a claimed output is equivalent to an input by construction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The central claim rests on standard results in Ehrhart theory and tropical convexity (covector decomposition, alcoved triangulation, classical Ehrhart polynomiality), plus a new convention for tropical integers (log_b lattice). There are no fitted constants. The only 'free' parameter b is eliminated in the final tbvol definition via the b-to-infinity limit.

free parameters (1)
  • b (base of tropical b-lattice) = arbitrary integer >= 2
    The tropical lattice Gamma_b^d is defined via log_b(Z_{\ge0}); b is chosen by hand. The final tropical barycentric volume is the b-to-infinity limit and is independent of b, so the parameter is eliminated from the main object; it remains a parameter of the intermediate tropical b-volume and of the Ehrhart coefficient representation.
assumptions (5)
  • standard math Classical Ehrhart theorem for complexes of lattice polytopes (Beck and Sanyal [8, Cor. 5.6.1])
    Used in Theorem 3.2 to get polynomiality of #(kP intersect Z^d) for lattice polytopes over Smax,cdot, and hence, via log_b, for tropical lattice polytopes (Theorem 3.6).
  • domain assumption Develin and Sturmfels covector decomposition and Lam and Postnikov alcoved triangulation of tropical polytopes with integer vertices
    Imported in Section 2.1; underpins Definition 4.3, the explicit formula (3), Lemma 3.11, and Theorem 5.12. The paper does not prove the existence or integrality of this triangulation.
  • domain assumption Tropical Minkowski-Weyl theorem (Gaubert and Katz [23])
    Used to ensure a finite generating set of vertices for tropical polytopes and the notion of tropical convex hull (equation (2)).
  • standard math Ehrhart-MacDonald reciprocity (Beck and Robins [7, Thm. 4.1])
    Used in the proof of Theorem 3.14 to convert counting in relatively open alcoved simplices into signed sums of Ehrhart coefficients of closures.
  • standard math Tropical Cauchy-Binet formula (Poplin and Hartwig [36, Thm. 5.4], Akian, Gaubert and Guterman [2, Ex. 3.7])
    Used in the proof of Lemma 5.10 to relate bideterminants of submatrices of A and B when tconv(A) is a subset of tconv(B).
invented entities (1)
  • Tropical b-lattice Gamma_b^d = (log_b(Z_{\ge0}))^d
    purpose: Provides the notion of 'tropical integers' for tropical lattice point counting and Ehrhart theory; all tropical Ehrhart coefficients and the tropical b-volume are defined relative to it.
    A new mathematical convention adopted in Definition 2.4. There is no empirical or external falsifiable handle; its justification is the semiring isomorphism between (R_{\ge0}, max, cdot) and (R union {-infinity}, max, +).

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Pith. "Pith review of Tropical Ehrhart Theory and Tropical Volume." pith.science (2026). https://pith.science/paper/7OFRLZIQ

@misc{pith2026190807893,
  author       = {Pith},
  title        = {Pith review of: Tropical Ehrhart Theory and Tropical Volume},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7OFRLZIQ}},
  note         = {Machine review of arXiv:1908.07893}
}
read the original abstract

We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing measures. Our exposition is complemented by a brief study of arising complexity questions.

Figures

Figures reproduced from arXiv: 1908.07893 by the authors.

Figure 1
Figure 1. A 4-dimensional tropical polytope whose 2-trunk is disconnected. A particularly nice class of tropical polytopes are the pure tropical poly￾topes, that is, those which coincide with their d-trunk. The well-behaved nature of pure tropical polytopes was used to exhibit canonical exterior de￾scriptions in [5]. In a similar spirit, the following statement uses a technique already occuring in the study of minimal externa… view at source ↗
Figure 2
Figure 2. Commutative diagram of several semiring isomor￾phisms and the connection with Puiseux series. Theorem 3.1 ([8, Cor. 5.6.1]). Let K be a complex of lattice polytopes in R d and let |K| = S P ∈K P be its underlying point set. Then, the counting func￾tion k 7→ # [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. A lattice polytope over Smax,· A natural question that arises is Question 3.3. How can we characterize the vertices of those polytopes over Smax,· for which all the pseudovertices are lattice points? [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The lattice points are condensed by log2 such that the tropical dilation implies a polynomial increase in the number of contained lattice points. The next concept is at the heart of our quantitative studies. Definition 3.5. Let P ⊆ T d be a tropical lattice polytope an…
Figure 5
Figure 5. Figure 5: The tropical lattice polygons in Example 3.16. 4. Tropical Volume from Tropical Lattice Points 4.1. A novel concept of tropical volume. Based on the polynomiality of the counting function k 7→ L b P (k) established in Section 3, we can now easily build up an analogy to…
Figure 6
Figure 6. Figure 6: A canonical tropical lattice polygon P with tbvol(P) = 2 and qtvol+(P) = 4. Corollary 4.15. For pure tropical lattices polytopes P = tconv(M), the magnitudes qtvol+(M) and tbvol(P) agree. Although this shows that the two tropical volume concepts tbvol(·) and qtvol+(·) …

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