On integer points in polyhedra

We give an upper bound on the number of vertices of PI, the integer hull of a polyhedron P, in terms of the dimension n of the space, the number m of inequalities required to describe P, and the size ~ of these inequalities. For fixed n the bound is O(mn~n-1). We also describe an algorithm which determines the number of integer points in a polyhedron to within a multiplicative factor of 1 qE in time polynomial in m, ~ and 1/c when the dimension n is fixed.