交替相邻和多面体：转移矩阵与埃尔哈特级数

中文摘要 AI 辅助

我们研究了一类周期为二的相邻和格多面体，其连续坐标界限在\(s\)和\(s + 1\)之间交替。这提供了经典均匀模型的简单非均匀变形，同时保留了明确的转移矩阵结构。格点数呈现奇偶分裂：奇数维和偶数维序列具有不同的有理生成函数，但有共同分母。奇数维级数满足莫比乌斯递归并具有反正切封闭形式，偶数维级数遵循耦合递归。它们共同的主导极点决定了两个奇偶类别的指数增长。对于通过在第一个和最后一个坐标之间添加约束得到的循环模型，计数变为矩阵迹。两个循环奇偶类再次具有相同分母的有理生成函数；偶数维分子具有雅可比导数形式，奇数维由明确的反对角余子式表达式给出。我们还推导了固定扩张的维度生成函数、格点数的线性递归、有理体积生成函数以及\(h^*\)多项式系数的二元恒等式。当\(s = 1\)时，每个偶数维多面体分解为单位模三角形的笛卡尔积，得到明确公式和戈伦斯坦性质。对于每个\(s\geq 2\)，在某些偶数维中戈伦斯坦性质不成立。

英文摘要

We study a period-two family of adjacent-sum lattice polytopes whose consecutive-coordinate bounds alternate between $s$ and $s+1$. This provides a simple non-uniform deformation of the classical uniform model while retaining an explicit transfer-matrix structure. The lattice-point counts exhibit a parity split: the odd- and even-dimensional sequences have distinct rational generating functions with a common denominator. The odd-dimensional series satisfies a Möbius recurrence and admits an arctangent closed form, whereas the even-dimensional series obeys a coupled recurrence. Their common dominant pole determines the exponential growth in both parity classes. For the cyclic model obtained by adding a constraint between the first and last coordinates, the count becomes a matrix trace. The two cyclic parity classes again have rational generating functions with the same denominator; the even-dimensional numerator has a Jacobi-derivative form, while the odd-dimensional one is given by an explicit anti-diagonal cofactor expression. We also derive dimension-generating functions for fixed dilations, linear recurrences for lattice-point counts, rational volume-generating functions, and a bivariate identity for the coefficients of the $h^*$-polynomials. When $s=1$, every even-dimensional polytope decomposes into a Cartesian product of unimodular triangles, yielding explicit formulas and the Gorenstein property. For every $s\geq 2$, the Gorenstein property fails in some even dimension.

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50 pages