Flat $\delta$-vectors and their Ehrhart polynomials

Abstract

We call the $\delta$-vector of an integral convex polytope of dimension $d$ flat if the $\delta$-vector is of the form $(1,0,\dots ,0,a,\dots ,a,0,\dots ,0)$, where $a\ge 1$. In this paper, we give the complete characterization of possible flat $\delta$-vectors. Moreover, for an integral convex polytope $\mathcal{P}\subset {\mathbb{R}}^N$ of dimension $d$, we let $i(\mathcal{P},n)=|n\mathcal{P}\cap {\mathbb{Z}}^N|$ and $i^{\ast }(\mathcal{P},n)=|n(\mathcal{P}\setminus \partial \mathcal{P})\cap {\mathbb{Z}}^N|.$ By this characterization, we show that for any $d\ge 1$ and for any $k,\ell \ge 0$ with $k+\ell \le d-1$, there exist integral convex polytopes $\mathcal{P}$ and $\mathcal{Q}$ of dimension $d$ such that (i) For $t=1,\dots ,k$, we have $i(\mathcal{P},t)=i(\mathcal{Q},t),$ (ii) For $t=1,\dots ,\ell$, we have $i^{\ast }(\mathcal{P},t)=i^{\ast }(\mathcal{Q},t)$ and (iii) $i(\mathcal{P},k+1)\ne i(\mathcal{Q},k+1)$ and $i^{\ast }(\mathcal{P},\ell +1)\ne i^{\ast }(\mathcal{Q},\ell +1).$

7 pages