Bounded powers of edge ideals: Pseudo-Gorenstein and Level polytopes

TL;DR

This paper explores the properties of level* and pseudo-Gorenstein* polytopes, particularly those derived from discrete polymatroids related to bounded powers of edge ideals, revealing their structural characteristics and classifications.

Contribution

It introduces a study of level* and pseudo-Gorenstein* polytopes from discrete polymatroids linked to bounded powers of edge ideals, highlighting their geometric and algebraic properties.

Findings

Pseudo-Gorenstein* polytopes are level* if they are reflexive up to translation.

Level* polytopes are characterized by specific normality and lattice point conditions.

The study connects polytope properties with algebraic structures of edge ideals.

Abstract

A lattice polytope $\mathcal{P} \subset \mathbb{R}^n$ of dimension $n$ is called level* if (i) $\mathcal{P}$ is normal, (ii) $(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n \neq \emptyset$ and (iii) for each $N = 2,3, \ldots$ and for each $\textbf{a} \in N(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n$, there is $\textbf{a}_0 \in (\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n$ together with $\textbf{a}' \in (N-1)\mathcal{P} \cap \mathbb{Z}^n$ for which $\textbf{a} = \textbf{a}_0 + \textbf{a}'$, where $N\mathcal{P} = \{N\textbf{a} : \textbf{a} \in \mathcal{P}\}$. A normal polytope $\mathcal{P} \subset \mathbb{R}^n$ of dimension $n$ is called pseudo-Gorenstein* [4] if $ |(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n| = 1. $ A pseudo-Gorenstein* polytope $\mathcal{P}$ is level* if and only if $\mathcal{P}$ is reflexive up to translation. In the present paper, level* polytopes together with pseudo-Gorenstein* polytopes arising from discrete polymatroids of bounded powers of edge ideals are studied.