A pseudoexponentiation-like structure on the algebraic numbers

TL;DR

This paper explores the construction of existentially closed exponential functions on algebraic numbers, resembling complex exponentiation, by removing the Schanuel Property and leveraging known algebraic results.

Contribution

It introduces a new approach to creating exponential functions on algebraic numbers that are existentially closed without relying on the Schanuel Property.

Findings

Existentially closed exponential functions can be constructed on algebraic numbers.

These functions share properties with complex exponentiation.

Overcoming arithmetic difficulties involves specialisation of multiplicatively independent functions.

Abstract

Pseudoexponential fields are exponential fields similar to complex exponentiation satisfying the Schanuel Property, which is the abstract statement of Schanuel's Conjecture, and an adapted form of existential closure. Here we show that if we remove the Schanuel Property and just care about existential closure, it is possible to create several existentially closed exponential functions on the algebraic numbers that still have similarities with complex exponentiation. The main difficulties are related to the arithmetic of algebraic numbers, and they can be overcome with known results about specialisations of multiplicatively independent functions on algebraic varieties.