Negative coefficients of Ehrhart polynomials

Takayuki Hibi, Akihiro Higashitani, Akiyoshi Tsuchiya

Published 2015-06-01Version 1

It is shown that, for each $d \geq 3$ and $1 \leq k \leq d-2$, there exists an integral convex polytope $\mathcal{P}$ of dimension $d$ such that the coefficient of $n$ of the Ehrhart polynomial $i(\mathcal{P},n)$ of $\mathcal{P}$ is negative and its remaining coefficients are all positive. Moreover, it is also shown that for a given integer $3 \leq d \leq 6$ and integers $i_1,\ldots,i_q$ with $1 \leq i_1 < \cdots < i_q \leq d-2$, there exists an integral convex polytope of dimension $d$ whose Ehrhart polynomial satisfies that all the coefficients of $n^{i_1}, \ldots, n^{i_q}$ are negative and all the remaining coefficients are positive.