On the exponential algebraic geometry

TL;DR

This paper introduces the ring of conditions for exponential varieties in complex space, linking algebraic geometry with convex geometry through Newton polyhedra, and establishes a correspondence between intersection indices and mixed pseudo-volumes.

Contribution

It defines the ring of conditions for exponential varieties and describes it via convex geometry, establishing a Newtonization map that links exponential varieties to convex polyhedra.

Findings

The ring of conditions of a2^n is isomorphic to a ring generated by convex polyhedra.

The intersection index of exponential hypersurfaces equals the mixed pseudo-volume of their Newton polyhedra.

The Newtonization map provides a geometric interpretation of exponential varieties.

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.