Relative Ehrhart functions: eventual polynomiality, reciprocity law, and shifted duality

Abstract

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 there exists an integer $ρ$ such that $kρ= 2l$ via the classical Ehrhart--Macdonald reciprocity law. In addition, we also prove that under the same condition, $\mathrm{ehr}(P,Q;t) = \mathrm{ehr}^\circ (P,Q;t+σ)$ $(t \gg 0)$ for some integer $σ> 0$ holds for $t \gg 0$ if and only if $\mathrm{ehr}^\circ (P_0;\mathrm{codeg}\,P_0) = 1$ and there exists an integer $σ$ such that $kσ= \mathrm{codeg}\,P_0$.