Algebraic types in Zilber's exponential field

TL;DR

This paper characterizes the algebraic closure in Zilber's exponential field using model theory and algebraic geometry, notably employing Mordell-Lang for algebraic tori to establish finiteness results.

Contribution

It provides a new model-theoretic characterization of algebraic closure in Zilber's exponential field, connecting it with algebraic geometry and Mordell-Lang theorem.

Findings

Finite intersections of algebraic varieties with finite-rank subgroups are characterized.

Mordell-Lang theorem is instrumental in the proof.

Model-theoretic algebraic closure is explicitly described.

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.