The integer point enumerator of one irrational translate of P is a complete invariant

Abstract: For a full-dimensional rational polytope $P\subset\mathbb{R}<sup>d$ and a real dilation parameter $t&gt;0$, the integer point enumerator is defined by $L_{P}(t):= |tP\cap\mathbb{Z}<sup>d|$. We determine exactly which translation vectors $\mathbf y=(y_1,\ldots,y_d)\in\mathbb{R}<sup>d$ have the property that the single translated counting function $t\longmapsto L_{P+\mathbf y}(t)$, with $t\in\mathbb{Q}_{&gt;0}$, uniquely determines $P$ among all full-dimensional rational polytopes in $\mathbb{R}<sup>d$. 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 $$\mathbf y^* := (2<sup>{1/(d+1)},2<sup>{2/(d+1)},\ldots,2<sup>{d/(d+1)})$$ 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.