Ehrhart Properties under Operations on Lattice Polytopes
Abstract
This paper investigates the preservation of three classes of Ehrhart properties under various operations on lattice polytopes. These operations include Cartesian products, lattice joins, lattice pyramids, free sums, Minkowski sums, Cayley sums, reflexive polarity, and integral dilations. Specifically, we consider the following properties frequently studied in Ehrhart theory: (i): Positivity of Ehrhart coefficients, including Ehrhart positivity and magic positivity. In particular, we present lattice point counting formulas for the polytopes generated by these operations. (ii): Coefficient properties of the h*-polynomial, including symmetry, unimodality, log-concavity, ultra log-concavity, real-rootedness, and γ-positivity. (iii): Geometric properties, including the spanning property, the integer decomposition property, very ampleness, and the existence of unimodular triangulation, regular unimodular triangulation, and quadratic triangulation. We determine which properties are preserved under these eight operations, establishing preservation theorems or constructing explicit counterexamples. Furthermore, when a property is not preserved in general, we investigate sufficient or equivalent conditions for its preservation.
Related
- Ehrhart polynomials of cyclic polytopes as averages of zonotope Ehrhart polynomials
- Equivariant Ehrhart series for rational simplices and rational cross-polytopes
- Ehrhart positivity of smooth polytopes with few facets
- The equivariant Ehrhart theory of polytopes with order-two symmetries: Part II
- Exact Ehrhart Series of Birkhoff Polytopes via Constant Terms and Finite-Field Evaluation
BibTeX
@misc{liu2026ehrhart,
title = {{Ehrhart Properties under Operations on Lattice Polytopes}},
author = {Feihu Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06338},
eprint = {2610.06338},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06338},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ehrhart Properties under Operations on Lattice Polytopes
AU - Feihu Liu
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.06338
UR - https://arxiv.org/abs/2610.06338
AB - This paper investigates the preservation of three classes of Ehrhart properties under various operations on lattice polytopes. These operations include Cartesian products, lattice joins, lattice pyramids, free sums, Minkowski sums, Cayley sums, reflexive polarity, and integral dilations. Specifically, we consider the following properties frequently studied in Ehrhart theory: (i): Positivity of Ehrhart coefficients, including Ehrhart positivity and magic positivity. In particular, we present lattice point counting formulas for the polytopes generated by these operations. (ii): Coefficient properties of the h^*-polynomial, including symmetry, unimodality, log-concavity, ultra log-concavity, real-rootedness, and γ-positivity. (iii): Geometric properties, including the spanning property, the integer decomposition property, very ampleness, and the existence of unimodular triangulation, regular unimodular triangulation, and quadratic triangulation. We determine which properties are preserved under these eight operations, establishing preservation theorems or constructing explicit counterexamples. Furthermore, when a property is not preserved in general, we investigate sufficient or equivalent conditions for its preservation.
ER -