Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the exponential algebraic geometry

B. Kazarnovskii

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.

On the exponential algebraic geometry

Abstract

The set of roots of any finite system of exponential sums in the space 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 and is called the ring of conditions of . 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 . 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 . It follows, in particular, that the intersection index of exponential hypersurfaces is equal to the mixed pseudo-volume of their Newton polyhedra.

Paper Structure

This paper contains 34 sections, 88 theorems, 51 equations.

Table of Contents

  1. Introduction.
  2. Exponential Sums
  3. Exponential Varieties
  4. The Intersection Index of Exponential Varieties
  5. Ring of Conditions of ${\mathbb C}^n$
  6. Exponential Sums as Laurent Polynomials
  7. Algebraic Codimension
  8. Rings Generated by Convex Polyhedra
  9. Definition of the Ring $\mathfrak S_{\rm vol}$
  10. Description of the ideal $J_{\rm vol}$
  11. Tropicalizations of elements of the ring $\mathfrak{S}_{\rm vol}$
  12. Newtonizations of Varieties
  13. Newtonizations of Algebraic Varieties
  14. Newtonizations of E-Varieties
  15. Intersection Index and the Ring of Conditions of ${\mathbb C}^n$
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $k\leq n$ and $\bullet$$\Lambda_1, \Lambda_2, \ldots, \Lambda_k$ are finite subsets of the exponent space ${{\mathbb C}^n}^*$$\bullet$$S(\Lambda_1, \Lambda_2, \ldots, \Lambda_k)$ is the space of systems of $k$ exponential sums $f_1,\ldots,f_k$ such that ${\rm supp\:}(f_i)\subset\Lambda_i$$\bulle

Theorems & Definitions (179)

  • Example 1.1
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.1
  • Theorem 1.4
  • Definition 1.2
  • Definition 1.3
  • Example 1.2
  • ...and 169 more