AI 中文总结
本文研究半开整数平行六面体的Ehrhart理论，给出其实Ehrhart拟多项式及离散矩的显式公式，推导多面体戴德金和的新恒等式，建立Ehrhart型互反律。
AI 中文摘要
我们研究半开d维整数平行六面体Π的Ehrhart理论。对于正整数t，tΠ与ℤ^d的格点计数已知为vol Π t^d，但任意实缩放t对应的计数函数具有微妙、非平凡的周期结构。我们给出该实Ehrhart拟多项式的显式公式，更一般地，给出Π的实缩放的所有离散矩∑_{p∈tΠ∩ℤ^d}⟨p,z⟩^m的公式，这些公式用Barnes多项式和多面体戴德金和表示，后者编码由Π确定的平环面上平移整数格的周期格流。我们的方法进一步发展了多面体戴德金和的研究，该概念由[Robins2026]新近提出。特别地，我们通过迭代离散导数得到多面体戴德金和的新恒等式；此外，我们证明L_Π(t)的Ehrhart拟系数恰好是多面体戴德金和的交替和；最后，我们给出一个Ehrhart型互反律，将L_Π(t)在负参数处的值与“相反”半开平行六面体的格点计数关联起来。
英文摘要
We study the Ehrhart theory of half-open $d$-dimensional integer parallelepipeds $Π$. Although the lattice-point count $tΠ\cap \Z^d$ is known to be simply $\vol Πt^d$ for positive integer $t$, the corresponding counting function for arbitrary real dilations $t$ has subtle, nontrivial periodic structure. We give explicit formulas for this real Ehrhart quasi-polynomial, and more generally for all the discrete moments of the real dilates of $Π$: $\sum_{p\in tΠ\cap\mathbb Z^d}\langle p,z\rangle^m$. The formulas are expressed in terms of Barnes polynomials and polytope Dedekind sums, which encode the periodic lattice flow of translated integer lattices on the flat torus determined by $Π$. Our approach develops further the study of polytope Dedekind sums, introduced recently in \cite{Robins2026}. In particular, we obtain novel identities for polytope Dedekind sums by using iterated discrete derivatives. Moreover, we show that the Ehrhart quasi-coefficients of $L_Π(t)$ are precisely alternating sums of polytope Dedekind sums. Finally, we give an Ehrhart-type reciprocity law relating $L_Π(t)$ at negative arguments to the lattice-point count of the `opposite' half-open parallelepiped.
Comments25 pages, 1 figure
