Eventual polynomiality and dualities of relative Ehrhart functions
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 ehr(P,Q;t) of such valid translations of Q into tP inherits eventual quasi-polynomiality (or eventual polynomiality) with leading term vol(P)td, 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∘ defines ehr∘(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 = kP0 and Q = lQ0 for some integers k, l > 0, and polytopes P0 and Q0 with dim Q0 > 0 such that Q0 is inscribed in P0. In particular, we prove that under this condition, ehr(P,Q;−t) = (−1)dehr∘ (P,Q;t+ρ) (t ≫ 0) holds for some integer ρ> 0 if and only if k | 2l via the classical Ehrhart--Macdonald reciprocity law. We also prove that under the same condition, ehr(P,Q;t) = ehr∘ (P,Q;t+σ) (t ≫ 0) holds for some integer σ> 0 if and only if ehr∘ (P0;codeg P0) = 1 and k | codeg P0.
Related
- From Closed to Relative Higher-Genus Gromov-Witten Invariants via Resurgent Functions
- Polynomiality of Stretched Schubert Structure Constants and Key Coefficients
- Polynomiality of the double ramification cycle
- Characters and Polynomiality for Maximal Parabolic Unipotent Radicals in Finite Classical Groups
- Polynomiality for algebras of invariants associated with some Inönü-Wigner contractions in type A
BibTeX
@misc{hirotsu2026eventual,
title = {{Eventual polynomiality and dualities of relative Ehrhart functions}},
author = {Takashi Hirotsu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.10370},
eprint = {2608.10370},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.10370},
}
RIS
TY - UNPB
TI - Eventual polynomiality and dualities of relative Ehrhart functions
AU - Takashi Hirotsu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2608.10370
UR - https://arxiv.org/abs/2608.10370
AB - 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 ehr(P,Q;t) of such valid translations of Q into tP inherits eventual quasi-polynomiality (or eventual polynomiality) with leading term 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^∘ defines ehr^∘(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, ehr(P,Q;−t) = (−1)^dehr^∘ (P,Q;t+ρ) (t ≫ 0) holds for some integer ρ> 0 if and only if k | 2l via the classical Ehrhart--Macdonald reciprocity law. We also prove that under the same condition, ehr(P,Q;t) = ehr^∘ (P,Q;t+σ) (t ≫ 0) holds for some integer σ> 0 if and only if ehr^∘ (P_0;codeg P_0) = 1 and k | codeg P_0.
ER -