On the magic positivity of the Ehrhart polynomials of dilated base polytope of minimal matroid

Abstract

In this paper, we consider the magic positivity of the Ehrhart polynomial of dilated base polytope of minimal matroid of rank $k$ dimension $n$. We prove that for $1\le k<\lfloor \frac{n}{2}\rfloor$, the $n-k-1$ dilate of the Ehrhart polynomial is not magic positive. We also show that for sufficiently large $n$, the $n-k$ dilate of the Ehrhart polynomial is, in general, magic positive. This gives rise to a lower bound of the $m-$index of the Ehrhart polynomial.