Ehrhart Polynomials with Negative Coefficients

It is shown that, for each d ≥ 4 , there exists an integral convex polytope P of dimension d such that each of the coefficients of n , n 2 , … , n d - 2 of its Ehrhart polynomial i ( P , n ) is negative. Moreover, it is also shown that for each d ≥ 3 and 1 ≤ k ≤ d - 2 , there exists an integral convex polytope P of dimension d such that the coefficient of n k 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.