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bibkey: jiangwenzhong2026alternating authors: Xinru Jiang, Suzhen Wen, Yueming Zhong year: 2026 title: “Alternating adjacent-sum polytopes: transfer matrices and Ehrhart series” doi: null url: https://arxiv.org/abs/2607.14887v1 claim: “Question Q1 asks for all Gorenstein pairs (s,r) with s ≥ 2 for the alternating adjacent-sum polytope in dimension 2r.” strata_touched:

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Source: https://arxiv.org/abs/2607.14887v1

The version is arXiv:2607.14887v1. Section 1, equation (1), printed p. 2, defines the polytope. Theorem 1.9 (thm-gorenstein-s1) states the even-dimensional s = 1 Gorenstein property; its proof on printed p. 44 gives the interior-lattice translation characterization. Proposition 3.21 (prop-d2-gorenstein), printed p. 45, characterizes the dimension-two cases. Corollary 3.24 (cor-not-gorenstein-all-s), printed p. 47, gives failure in at least one even dimension for every s ≥ 2. Section 4, Question Q1 (q-gorenstein), printed p. 48, asks for the complete classification.

Definitions and question

Section 1, p. 2, reads:

For integers d ≥ 2 and s ≥ 1, define

$${\mathcal{P}}_d^{(s)}=\{x=(x_1,\dots ,x_d)\in {\mathbb{R}}_{\ge 0}^d:x_i+x_{i+1}\le s+{\delta }_i,1\le i\le d-1\},$$

where δ_i = 0 for i odd and δ_i = 1 for i even.

The proof of Theorem 1.9, p. 44, states:

We use the standard interior-lattice-point characterization of Gorenstein lattice polytopes; see, for example, [13, 7].

It proves the identity int((n + 3)T₁^r) ∩ ℤ^(2r) = c_r + (nT₁^r ∩ ℤ^(2r)) for all n ≥ 0. For a positive index q the corresponding characterization is the existence of an integral c such that int((n + q)P) ∩ ℤ^d = c + (nP ∩ ℤ^d) for every natural n.

Question Q1, §4, p. 48, reads:

Theorem 1.9, Proposition 3.21, and Corollary 3.24 show that s = 1 yields an infinite Gorenstein family, whereas every s ≥ 2 eventually fails. For s = 3, direct computation (Propositions 3.22 and 3.23) shows that only d = 2 is Gorenstein in even dimensions d ≤ 6; we conjecture this extends to all d ≥ 4. Characterize all (s, r) with s ≥ 2 for which P_{2r}^(s) is Gorenstein.

Scope of the characterization

The formal statement uses zero-based coordinates in Fin d, with capacity s + ite((val(i) + 1) % 2 = 0, 1, 0). The Gorenstein predicate is exactly the interior-lattice translation characterization, using ambient real interior and real set dilation. The classification for s ≥ 2 and r ≥ 1 is precisely s = 3 ∧ r = 1.

The source’s s = 1 theorem and its Ehrhart and transfer-matrix identities are independent of this classification. The computed h*-tables in Remark 3.18 and §3.5 agree with the classification; their numerical entries are not used in the formal proof. The unimodality and real-rootedness questions in Q2 remain separate.