Ehrhart Polynomials with Negative Coefficients

It is shown that, for each d4, there exists an integral convex polytope P of dimension d such that each of the coefficients of n,n2,...,nd-2 of its Ehrhart polynomial i(P,n) is negative. Moreover, it is also shown that for each d3 and 1kd-2, there exists an integral convex polytope P of dimension d such that the coefficient of nk of the Ehrhart polynomial i(P,n) of P is negative and all its remaining coefficients are positive. Finally, we consider all the possible sign patterns of the coefficients of the Ehrhart polynomials of low dimensional integral convex polytopes.