A subset P of Rn gives rise to a formal Laurent series with monomials corresponding to lattice points in P. Under suitable hypotheses, this series represents a rational function R(P); this happens, for example, when P is bounded in which case R(P) is a Laurent polynomial. Michel Brion has discovered a surprising formula relating the Laurent polynomial R(P) of a lattice polytope P to the sum of rational functions corresponding to the supporting cones subtended at the vertices of P. The result is re-phrased and generalised in the language of cohomology of line bundles on complete toric varieties. Brion's formula is the special case of an ample line bundle on a projective toric variety. The paper also contains some general remarks on the cohomology of torus-equivariant line bundles on complete toric varieties, valid over arbitrary commutative ground rings.
Homology, Homotopy and Applications, Vol. 9 (2007), No. 2, pp.321-336.