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A geometric approach to some systems of exponential equations

Vahagn Aslanyan, Jonathan Kirby, Vincenzo Mantova

Abstract

Zilber's Exponential Algebraic Closedness conjecture (also known as Zilber's Nullstellensatz) gives conditions under which a complex algebraic variety should intersect the graph of the exponential map of a semiabelian variety. We prove the special case of the conjecture where the variety has dominant projection to the domain of the exponential map, for abelian varieties and for algebraic tori. Furthermore, in the situation where the intersection is 0-dimensional, we exhibit structure in the intersection by parametrizing the sufficiently large points as the images of the period lattice under a (multivalued) analytic map. Our approach is complex geometric, in contrast to a real analytic proof given by Brownawell and Masser just for the case of algebraic tori.

A geometric approach to some systems of exponential equations

Abstract

Zilber's Exponential Algebraic Closedness conjecture (also known as Zilber's Nullstellensatz) gives conditions under which a complex algebraic variety should intersect the graph of the exponential map of a semiabelian variety. We prove the special case of the conjecture where the variety has dominant projection to the domain of the exponential map, for abelian varieties and for algebraic tori. Furthermore, in the situation where the intersection is 0-dimensional, we exhibit structure in the intersection by parametrizing the sufficiently large points as the images of the period lattice under a (multivalued) analytic map. Our approach is complex geometric, in contrast to a real analytic proof given by Brownawell and Masser just for the case of algebraic tori.

Paper Structure

This paper contains 21 sections, 23 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview of the paper
  3. An example
  4. Algebraic maps
  5. Covering maps
  6. Extending sections of covering maps
  7. The algebraic maps
  8. The abelian case
  9. Proof outline.
  10. Step 1: The algebraic maps.
  11. Step 2: Mapping the solutions to the lattice.
  12. Step 3: Local injectivity of F.
  13. Step 4: Mapping the lattice to the solutions.
  14. Step 5: Analytic continuation of the solution map S.
  15. The case of algebraic tori
  16. ...and 6 more sections

Key Result

Theorem 1.4

Let $A$ be a complex abelian variety of dimension $n$. Let $V \subseteq \mathbb{C}^n \times A$ be an algebraic subvariety with dominant projection to $\mathbb{C}^n$, that is, its projection to $\mathbb{C}^n$ has dimension $n$. Then there is ${\boldsymbol{z}} \in \mathbb{C}^n$ such that $({\boldsymbo

Figures (2)

  • Figure 1: Some distances between points in the sector domain $D_{(\theta,\eta)}$.
  • Figure :

Theorems & Definitions (51)

  • Conjecture 1.1: EAC Zilb-pseudoexp
  • Example 1.2
  • Conjecture 1.3: EAC for semiabelian varieties
  • Theorem 1.4
  • Theorem 1.5: brown-masser
  • Proposition 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 41 more