Model completeness for the differential field of transseries with exponentiation

TL;DR

This paper proves that expanding the differential field of transseries with exponentiation and trigonometric functions results in a model complete and locally o-minimal structure, with an effective axiomatization.

Contribution

It establishes model completeness and local o-minimality for the transseries field expanded by exponential and trigonometric functions, providing an effective axiomatization.

Findings

Expansion by exponential is model complete and locally o-minimal

Expansion by exponential, sine, and cosine is also model complete and locally o-minimal

Axiomatization is effective relative to the real exponential field

Abstract

Let $\mathbb{T}$ be the differential field of logarithmic-exponential transseries. We show that the expansion of $\mathbb{T}$ by its natural exponential function is model complete and locally o-minimal. We give an axiomatization of the theory of this expansion that is effective relative to the theory of the real exponential field. We adapt our results to show that the expansion of $\mathbb{T}$ by this exponential function and by its natural restricted sine and restricted cosine functions is also model complete and locally o-minimal.