  Let K(X) be the collection of all non-zero finite dimensional subspaces of
rational functions on an n-dimensional irreducible variety X. For any n-tuple
L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the
number of solutions in X of a system of equations f_1 = ... = f_n = 0 where
each f_i is a generic function from the space L_i. In counting the solutions,
we neglect the solutions x at which all the functions in some space L_i vanish
as well as the solutions at which at least one function from some subspace L_i
has a pole. The collection K(X) is a commutative semigroup with respect to a
natural multiplication. The intersection index [L_1,..., L_n] can be extended
to the Grothendieck group of K(X). This gives an extension of the intersection
theory of divisors. The extended theory is applicable even to non-complete
varieties. We show that this intersection index enjoys all the main properties
of the mixed volume of convex bodies. Our paper is inspired by the
Bernstein-Kushnirenko theorem from the Newton polytope theory.
