Involutions on Zilber fields

TL;DR

This paper constructs involutions on Zilber fields of various cardinalities, demonstrating their existence, classifying exponential subfields, and exploring involutions with order-preserving or continuous exponential functions.

Contribution

It provides explicit constructions of involutions on Zilber fields, answers a key question about fixed fields, and classifies exponential subfields with new properties.

Findings

Zilber fields of size up to continuum have involutions.

Existence of involutions with fixed field isomorphic to real numbers.

Classification of exponential subfields and construction of fields with order-preserving involutions.

Abstract

After recalling the definition of Zilber fields, and the main conjecture behind them, we prove that Zilber fields of cardinality up to the continuum have involutions, i.e., automorphisms of order two analogous to complex conjugation on (C,exp). Moreover, we also prove that for continuum cardinality there is an involution whose fixed field, as a real closed field, is isomorphic to the field of real numbers, and such that the kernel is exactly 2{\pi}iZ, answering a question of Zilber, Kirby, Macintyre and Onshuus. The proof is obtained with an explicit construction of a Zilber field with the required properties. As further applications of this technique, we also classify the exponential subfields of Zilber fields, and we produce some exponential fields with involutions such that the exponential function is order-preserving, or even continuous, and all of the axioms of Zilber fields are satisfied except for the strong exponential-algebraic closure, which gets replaced by some weaker axioms.