Counting Lattice Points in Polytopes via Riemann-Roch

This paper is a partial summary of the survey paper [1]. In particular, we are interested in telling the following story: given a lattice polytope, P , one would like to find an efficient way of counting the lattice points contained in P . One of the nicest ways to accomplish this is to use algebraic geometry in a clever and beautiful way. Namely, from P one can construct a toric variety, XP , together with a line bundle, LP , on XP . Then it turns out that the Euler characteristic χ(XP , LP ) is equal to the number of lattice points contained in P . Moreover, the Hirzebruch-Riemann-Roch theorem tells us how to calculate this Euler characteristic in terms of the Todd class of the toric variety XP . This yields an efficient method for counting the lattice points in P , because there is a polynomial time algorithm that computes the Todd class of XP given the polytope P .