On the exponential algebraic geometry

Abstract

The set of roots of any finite system of exponential sums in the space $\mathbb{C}^n$ is called an exponential variety. We define the intersection index of varieties of complementary dimensions, and the ring of classes of numerical equivalence of exponential varieties with operations "addition-union" and "multiplication-intersection". This ring is analogous to the ring of conditions of the torus $(\mathbb{C}\setminus 0)^n$ and is called the ring of conditions of $\mathbb{C}^n$. We provide its description in terms of convex geometry. Namely we associate an exponential variety with an element of a certain ring generated by convex polyhedra in $\mathbb{C}^n$. We call this element the Newtonization of the exponential variety. For example, the Newtonization of an exponential hypersurface is its Newton polyhedron. The Newtonization map defines an isomorphism of the ring of conditions to the ring generated by convex polyhedra in $\mathbb{C}^n$. It follows, in particular, that the intersection index of $n$ exponential hypersurfaces is equal to the mixed pseudo-volume of their Newton polyhedra.