The Euler Characteristic of a Complete Intersection in Terms of the Newton Polyhedra Revisited

The well-known formula for the Euler characteristic of a complete intersection in the complex torus in terms of the supports of the Laurent polynomials, the left-hand sides of the defining equations (in fact, in terms of the convex hulls of these supports, Newton polyhedra), was announced in a short note by D. N. Bernshtein, A. G. Kushnirenko, and A. G. Khovanskii (1976). The proof of the formula was given by A. G. Khovanskii (1978), but it was not self-contained (it was based on results of another author) and was somewhat fragmentary. Here we give a more elementary proof of this equation based on the simplest properties of toric manifolds.