Which Ehrhart properties survive operations on lattice polytopes?

Three property families, eight operations, one preservation table

The paper asks a single question in many settings: if a lattice polytope P has a "nice" Ehrhart-theoretic property, does the polytope produced by a standard operation inherit it? The operations are Cartesian products, lattice joins, lattice pyramids, free sums, Minkowski sums, Cayley sums, reflexive polarity, and integral dilations. The properties fall into three families: positivity of Ehrhart coefficients (Ehrhart positivity and magic positivity), coefficient properties of the h*-polynomial (symmetry, unimodality, log-concavity, ultra log-concavity, real-rootedness, gamma-positivity), and geometric properties (spanning, the integer decomposition property, very ampleness, and existence of unimodular, regular unimodular, or quadratic triangulations).

The answer is deliberately mixed. For each operation-property pair the paper either proves a preservation theorem or builds an explicit counterexample, and when a property fails in general it looks for sufficient or equivalent conditions under which it does hold. The concluding remarks describe the main contribution as completing all the characterizations in the paper's Table 1, so the table, not a single slogan, is the result.

Ehrhart and magic positivity survive products and dilations

Ehrhart positivity means every coefficient c_i in the ordinary expansion L_P(n) = c_d n^d + ... + c_1 n + c_0 is positive; magic positivity means every coefficient b_i in the binomial-basis expansion L_P(n) = sum b_i n^i (n+1)^(d-i) is nonnegative. These are different tests on different bases, and the paper keeps them apart. Magic positivity implies Ehrhart positivity, and it also implies real-rootedness of h*_P(z) by Branden's result, but the converse fails: a unimodular triangle has h*(z) = 1 and is not magic positive.

For Cartesian products, positivity is preserved for a transparent reason: L_{P x Q}(n) = L_P(n) L_Q(n), so Ehrhart positivity carries over, and a product of magic basis terms n^i (n+1)^(p-i) times n^j (n+1)^(q-j) equals n^{i+j} (n+1)^{p+q-i-j}, making the new magic coefficients sums of products of nonnegative ones. Integral dilations behave similarly: since L_{rP}(n) = L_P(rn), the coefficients of L_{rP} are r^j c_j, so rP is Ehrhart positive exactly when P is, and the paper proves integral dilations preserve magic positivity by expanding (rn)^j (rn+1)^{d-j} in the degree-d magic basis with nonnegative coefficients.

h*-polynomial properties need stronger hypotheses than positivity

The h*-polynomial properties split sharply. Standard lattice joins preserve symmetry, log-concavity with no internal zeros, real-rootedness, and gamma-positivity, because h*_{P*Q}(z) = h*_P(z) h*_Q(z) and these properties are closed under multiplication of polynomials. Ultra log-concavity is preserved both in dimension order and in actual-degree order, and the simultaneous conditions of symmetry and unimodality are preserved as well.

Unimodality alone is not enough. The paper records a five-dimensional polytope P with h*_P(z) = 1 + 6z + 6z^2 + 113z^3, which is unimodal, whose join with itself has h*_{P*P}(z) = 1 + 12z + 48z^2 + 298z^3 + 1392z^4 + 1356z^5 + 12769z^6, containing the trough 1392 > 1356 < 12769. Cartesian products fail even more basic properties: [0,1] and [0,2] both have symmetric and gamma-positive numerators, but their product has h*(z) = 1 + 3z, which is not symmetric and therefore not gamma-positive. A separate 52-dimensional product of two polytopes with unimodal numerators has coefficients 28264217617840835 > 28180199302315925 < 28214521222404290 at indices 22, 23, and 24, so unimodality fails there too.

Geometric properties survive products and joins, with one exception

Cartesian products preserve spanning, very ampleness, the integer decomposition property, unimodular triangulations, regular unimodular triangulations, and quadratic triangulations; the triangulation step uses a staircase product construction with a common ordering of vertices. Standard lattice joins preserve spanning, IDP, unimodular triangulations, regular unimodular triangulations, and quadratic triangulations.

Very ampleness is the exception for joins: the join of two very ample lattice polytopes need not be very ample, and the paper gives an explicit counterexample. Integral dilations are more forgiving on the triangulation side: if P has a unimodular, regular unimodular, or quadratic triangulation, then rP has the same kind of triangulation for every integer r >= 1.

Reflexive polarity can break the spanning property

Reflexive polarity is the operation where the paper's counterexample is most concrete. A lattice polytope V is constructed whose lattice-point difference group is exactly Z^3 x 2Z, an index-two subgroup of Z^4, so V is not spanning. Its polar dual V* contains 0, e_1, e_2, e_3, and (1,0,0,1), and the differences of these points generate Z^4 because (1,0,0,1) - e_1 = e_4. So V* is spanning while V is not, which proves that spanning is not preserved under reflexive polarity.

This matters because spanning is one of the geometric hypotheses that other results rely on, and the example shows it cannot be assumed to pass freely through duality. The paper's broader pattern is that geometric properties tend to survive the additive-style operations (products, joins, dilations) but can fail under polarity, where the combinatorial structure of the dual is not controlled by the original.

Equal codegree and symmetry act as preservation thresholds

Several failures are not unconditional. For Cartesian products, symmetry of the product requires the two factors to have equal codegree; the paper notes that [0,1] and [0,2] have codegrees two and one, which is exactly why their product's numerator is not symmetric. For joins, there is no equal-codegree restriction, and the degree and codegree formulas s_{P*Q} = s_P + s_Q and r_{P*Q} = r_P + r_Q make the join formulas additive.

The paper also treats integral dilations of the h*-polynomial in detail. It defines the zero-offset Veronese operator U_r^{d+1} h(z) = (B_r(z)^{d+1} h(z))<r,0> with B_r(z) = 1 + z + ... + z^{r-1}, and proves that if h*_P is ultra log-concave of a certain order, then h*_{rP} is ultra log-concave of order d (dimension order) and of order s_{rP} (actual-degree order). The proof splits by dimension: dimensions 1 through 4 and r >= 3 are handled via real-rootedness and Newton's inequalities, while d >= 5 uses a fractional ultra-log-concavity normalization with exponent alpha = 53/(40r) and the factorization d+1 = 3a + 4b.

Read the table by operation, not by property

The practical upshot is that preservation is a property of the pair, not of the property alone. If you are working with Cartesian products or lattice joins, positivity and most geometric properties are safe, but symmetry of the product needs equal codegree and unimodality of the product can fail even when both factors are unimodal. If you are working with reflexive polarity, do not assume spanning survives. If you are working with integral dilations, positivity and triangulation existence are preserved, and the h*-polynomial question reduces to a Veronese-type extraction whose answer depends on the dimension and on whether you measure ultra log-concavity in dimension order or actual-degree order.

The paper's own framing points to the same priority: the concluding remarks say the main contribution is completing the characterizations in Table 1, and that this is meant to guide future work, including the construction of counterexamples to conjectures. Two open questions are flagged as unresolved by these results: Oda's smoothness problem, asking whether every smooth lattice polytope has the integer decomposition property, and Gal's conjecture, asking whether a Gorenstein lattice polytope with a quadratic triangulation has a gamma-positive h*-polynomial. The non-IDP Minkowski and Cayley constructions in the paper do not settle Oda's problem because smoothness of their outputs was not established, which is the clearest signal of where the preservation table ends and new work has to begin.

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