P的一个无理平移的整数点计数函数是一个完全不变量
The integer point enumerator of one irrational translate of P is a complete invariant
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中文总结 AI 辅助
该研究确定了使有理多面体P的平移整数点计数函数能唯一确定P的平移向量的充要条件，给出了适用于任意维数的显式平移向量，为多面体的唯一确定提供了理论依据。
中文摘要 AI 辅助
对于全维多面体有理多面体$P\ubbbbr^d$和实伸缩参数$t>0$，整数点计数函数定义为$L_P(t):=|tP\ubbbbr^d|$。我们精确确定哪些平移向量$\ubbbey=(y_1,\u2026,y_d)\ubbbbr^d$具有如下性质：对所有正有理数$t$，单平移计数函数$t\to L_{P+\ubbbey}(t)$能在$\ubbbbr^d$中所有全维有理多面体里唯一确定$P$。充要条件是$1,y_1,\u2026,y_d$在$\ubbbbr$上线性无关。特别地，对任意维数$d$，我们可使用显式代数向量$\ubbbey^*:=(2^{1/(d+1)},2^{2/(d+1)},\u2026,2^{d/(d+1)})$。充分性证明从计数函数的孤立不连续点恢复原始面不等式，而必要性由仿射单模障碍得出。
英文摘要
For a full-dimensional rational polytope $P\subset\mathbb{R}^d$ and a real dilation parameter $t>0$, the integer point enumerator is defined by $L_{P}(t):= |tP\cap\mathbb{Z}^d|$. We determine exactly which translation vectors $\mathbf y=(y_1,\ldots,y_d)\in\mathbb{R}^d$ have the property that the single translated counting function $t\longmapsto L_{P+\mathbf y}(t)$, with $t\in\mathbb{Q}_{>0}$, uniquely determines $P$ among all full-dimensional rational polytopes in $\mathbb{R}^d$. The necessary and sufficient condition is that $1,y_1,\ldots,y_d$ be linearly independent over $\mathbb{Q}$. In particular, we may use the explicit algebraic vector $\mathbf y^* := (2^{1/(d+1)},2^{2/(d+1)},\ldots,2^{d/(d+1)})$ in every dimension $d$. The sufficiency proof recovers the primitive facet inequalities from isolated discontinuities of the counting function, while necessity follows from an affine-unimodular obstruction.
