AN APPLICATION OF THE COMBINATORIAL RIEMANN-ROCH THEOREM TO THE EHRHART POLYNOMIAL OF INTEGRAL POLYTOPES IN RD

The combinatorial Riemann-Roch theorem ([8], [9], [6]) relates the volume of an integral polytope to the number of integral points in it. This is only valid for polytopes with primitive fan (the associated toric variety is regular). However, applying this result in a suitable manner, we deduce an explicit formula for the coefficient of degree (d- 2) of the Ehrhart polynomial of any integral polytope in R(d). In particular this gives a general formula for the number of integral points in any integral polytope in R4, a new geometric look at the introduction of Dedekind sums in these questions.