A Note on Schanuel's Conjectures for Exponential Logarithmic Power Series Fields

TL;DR

This paper extends Ax's transcendency theorem to differential valued exponential fields, including Hardy, Logarithmic-Exponential, and Exponential-Logarithmic power series fields, advancing understanding of Schanuel's conjecture in these contexts.

Contribution

It generalizes Ax's theorem to new classes of differential valued exponential fields, providing new transcendency results related to Schanuel's conjecture.

Findings

Transcendency results for exponential Hardy fields

Results for Logarithmic-Exponential power series fields

Findings applicable to Exponential-Logarithmic power series fields

Abstract

In [1], J. Ax proved a transcendency theorem for certain differential fields of characteristic zero: the differential counterpart of the still open Schanuel's conjecture about the exponential function over the field of complex numbers [11, page 30]. In this article, we derive from Ax's theorem transcendency results in the context of differential valued exponential fields. In particular, we obtain results for exponential Hardy fields, Logarithmic-Exponential power series fields and Exponential-Logarithmic power series fields.