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

A lattice polytope $\mathcal{P}\subset {\mathbb{R}}^n$ of dimension $n$ is called level* if (i) $\mathcal{P}$ is normal, (ii) $(\mathcal{P}\setminus \mathrm{\partial }\mathcal{P})\cap {\mathbb{Z}}^n\mathrm{\ne }\mathrm{\varnothing }$ and (iii) for each $N=2,3,\dots$ and for each $\mathbf{\text{a}}\in N(\mathcal{P}\setminus \mathrm{\partial }\mathcal{P})\cap {\mathbb{Z}}^n$, there is ${\mathbf{\text{a}}}_0\in (\mathcal{P}\setminus \mathrm{\partial }\mathcal{P})\cap {\mathbb{Z}}^n$ together with ${\mathbf{\text{a}}}^{\mathrm{\prime }}\in (N-1)\mathcal{P}\cap {\mathbb{Z}}^n$ for which $\mathbf{\text{a}}={\mathbf{\text{a}}}_0+{\mathbf{\text{a}}}^{\mathrm{\prime }}$, where $N\mathcal{P}=\{N\mathbf{\text{a}}:\mathbf{\text{a}}\in \mathcal{P}\}$. A normal polytope $\mathcal{P}\subset {\mathbb{R}}^n$ of dimension $n$ is called pseudo-Gorenstein* [4] if $\mathrm{\mid }(\mathcal{P}\setminus \mathrm{\partial }\mathcal{P})\cap {\mathbb{Z}}^n\mathrm{\mid }=1.$ 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.