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Algebraic types in Zilber's exponential field

Vahagn Aslanyan, Jonathan Kirby

TL;DR

The paper proves that in Zilber's exponential field $\mathbb{B}$, the model-theoretic algebraic closure satisfies $\mathrm{acl}(B)=\lceil B\rceil^{\mathrm{EA}}$ for all $B\subseteq\mathbb{B}$. It combines the Schanuel Property, strong hulls, and the Mordell-Lang theorem for algebraic tori to bound and identify algebraic closures via the EA-closure, isolating types over hulls and using Hrushovski-style constructions. The work clarifies the relationship between algebraic closure and various closure operators, discusses the role of the kernel, and sketches extensions to non-standard kernels and the complex case, as well as bounded closures. The results deepen understanding of finitary (rank-0) types in $\mathbb{B}$ and connect model-theoretic tameness to Diophantine-type finiteness phenomena, with implications for exponential-algebraic geometry in tamely behaved exponential fields.

Abstract

We characterise the model-theoretic algebraic closure in Zilber's exponential field. A key step involves showing that certain algebraic varieties have finite intersections with certain finite-rank subgroups of the graph of exponentiation. Mordell-Lang for algebraic tori (a theorem of Laurent) plays a central role in our proof.

Algebraic types in Zilber's exponential field

TL;DR

The paper proves that in Zilber's exponential field , the model-theoretic algebraic closure satisfies for all . It combines the Schanuel Property, strong hulls, and the Mordell-Lang theorem for algebraic tori to bound and identify algebraic closures via the EA-closure, isolating types over hulls and using Hrushovski-style constructions. The work clarifies the relationship between algebraic closure and various closure operators, discusses the role of the kernel, and sketches extensions to non-standard kernels and the complex case, as well as bounded closures. The results deepen understanding of finitary (rank-0) types in and connect model-theoretic tameness to Diophantine-type finiteness phenomena, with implications for exponential-algebraic geometry in tamely behaved exponential fields.

Abstract

We characterise the model-theoretic algebraic closure in Zilber's exponential field. A key step involves showing that certain algebraic varieties have finite intersections with certain finite-rank subgroups of the graph of exponentiation. Mordell-Lang for algebraic tori (a theorem of Laurent) plays a central role in our proof.
Paper Structure (13 sections, 10 theorems, 32 equations)

This paper contains 13 sections, 10 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Zilber's exponential fields
  4. The Schanuel Property, predimension, and strong hull
  5. Strong Exponential-Algebraic Closedness and the Countable Closure Property
  6. Isolation of types
  7. The strong hull is contained in the algebraic closure
  8. Upper bound for the algebraic closure
  9. Discussion of closure operators
  10. Closure operators on $\mathbb{B}$
  11. Non-standard kernel
  12. Bounded closures
  13. The complex case

Key Result

Theorem 1.1

For every subset $B \subseteq \mathbb{B}$ we have $\mathop{\mathrm{acl}}\nolimits(B) = \lceil B\rceil^{\mathrm{EA}}$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • Proposition 4.1
  • proof
  • Claim 4.2
  • ...and 13 more