Multiplicative intersection theory and complex tropical varieties

We develop an intersection theory for subvarieties of a torus. Besides the number of intersection points for a generic pair of subvarieties of complementary dimensions, this theory takes into account the product of these points as elements of the ambient torus. In the case of a complete intersection of divisors, our intersection theory yields Bernsbtein's formula for the number of roots of a system as well as Khovanskii's formula for their product. When constructing this theory, we naturally encounter 'piecewise-linear' subsets of the torus which are referred to as complex tropical varieties.