The rational field is not universally definable in pseudo-exponentiation

TL;DR

This paper proves that the rational numbers cannot be universally defined within Zilber's pseudo-exponential field, highlighting limitations in the definability of certain subfields in complex exponential structures.

Contribution

It demonstrates the non-definability of the rational field by a universal formula in Zilber's pseudo-exponential field, advancing understanding of definability issues in exponential algebraic structures.

Findings

The rational field is not universally definable in Zilber's pseudo-exponential field.

Universal formulas are insufficient to define the rational numbers in this context.

The result clarifies limitations of definability in exponential algebraic structures.

Abstract

We show that the field of rational numbers is not definable by a universal formula in Zilber's pseudo-exponential field.