Mixed volume and an extension of intersection theory of divisors

Let $\mathbf K_\mathrm{rat}(X)$ be the collection of all non-zero finite dimensional subspaces of rational functions on an $n$-dimensional irreducible variety $X$. For any $n$-tuple $L_1,\dots,L_n\in\mathbf K_\mathrm{rat}(X)$, we define an intersection index $[L_1,\dots,L_n]$ as the number of solutions in $X$ of a system of equations $f_1=\dots=f_n=0$ 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 $\mathbf K_\mathrm{rat}(X)$ is a commutative semigroup with respect to a natural multiplication. The intersection index $[L_1,\dots,L_n]$ can be extended to the Grothendieck group of $\mathbf K_\mathrm{rat}(X)$. 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.