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arXiv · 2608.10370

Relative Ehrhart theory I: relative Ehrhart eventual polynomials

Takashi Hirotsu
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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.

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Takashi Hirotsu. 2026-08-11. Relative Ehrhart theory I: relative Ehrhart eventual polynomials. https://arxiv.org/abs/2608.10370

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