Barvinokʹs algorithm and the Todd class of a toric variety Original Research Article

In this paper we prove that the Todd class of a simplicial toric variety has a canonical expression as a power series in the torus-invariant divisors. Given a resolution of singularities corresponding to a nonsingular subdivision of the fan, we give an explicit formula for this power series which yields the Todd class. The computational feasibility of this procedure is implied by the additional fact that the above formula is compatible with Barvinok decompositions (virtual subdivisions) of the cones in the fan. In particular, this gives an algorithm for determining the coefficients of the Todd class in polynomial time for fixed dimension. We use this to give a polynomial-time algorithm for computing the number of lattice points in a simple lattice polytope of fixed dimension, a result first achieved by Barvinok.