AI 中文总结
该研究扩展经典埃哈特理论，用几何对象Q替代格点，证明相关计数函数具有最终拟多项式性，首项为vol(P)t^d，结果由维度归纳推导得出。
AI 中文摘要
经典埃哈特理论通过计算凸有理多面体P的t次膨胀tP中的格点数量，来衡量其离散容量。本文将该范式扩展，用维度不超过dim P的几何对象Q替代格点。我们证明，将Q的有效平移计数函数ehr(P;Q;t)继承了最终拟多项式性（或最终多项式性），其首项为vol(P)t^d，其中d=dim P。该结果通过对维度进行归纳，基于埃哈特函数的经典拟多项式性与多项式性自然推导得出。
英文摘要
Classical Ehrhart theory measures the discrete capacity of a convex rational (or integral) polytope $P$ by counting the number of lattice points in the $t$-th dilate $tP$ of $P$. In this paper, we extend this paradigm by replacing a lattice point with a geometric object $Q$ of dimension at most $\dim P$. We show that the counting function $\mathrm{ehr}(P,Q;t)$ of such valid translations of $Q$ into $tP$ inherits eventual quasi-polynomiality (or eventual polynomiality) with leading term $\mathrm{vol}(P)t^d$, where $d = \dim P$. This result is naturally derived by induction on the dimension, based on the classical quasi-polynomiality (or polynomiality) of Ehrhart functions. Similarly, replacing $P$ with its relative interior $P^\circ$ defines $\mathrm{ehr}^\circ(P,Q;t)$, which is also shown to be an eventual quasi-polynomial (or eventual polynomial). Furthermore, we prove several formulas for relative Ehrhart functions under the condition that $P = kP_0$ and $Q = lQ_0$ for some integers $k$, $l > 0$, and polytopes $P_0$ and $Q_0$ with $\dim Q_0 > 0$ such that $Q_0$ is inscribed in $P_0$. In particular, we prove that under this condition, $\mathrm{ehr}(P,Q;-t) = (-1)^{d}\mathrm{ehr}^\circ (P,Q;t+ρ)$ $(t \gg 0)$ holds for some integer $ρ> 0$ if and only if $k \mid 2l$ via the classical Ehrhart--Macdonald reciprocity law. In addition, we also prove that under the same condition, $\mathrm{ehr}(P,Q;t) = \mathrm{ehr}^\circ (P,Q;t+σ)$ $(t \gg 0)$ holds for some integer $σ> 0$ if and only if $\mathrm{ehr}^\circ (P_0;\mathrm{codeg}\,P_0) = 1$ and $k \mid \mathrm{codeg}\,P_0$.
Comments15 pages, 8 figures; Minor revisions with a revised title
