Normalizing Asymptotic Differential Equations

TL;DR

This paper introduces the universal exponential extension of algebraically closed differential fields, explores its properties with valuations and differential equations, and proves normalization theorems for algebraic differential equations over H-fields.

Contribution

It develops the theory of exponential extensions and normalization theorems, advancing methods for solving differential equations in valued differential fields.

Findings

Defined the universal exponential extension of algebraically closed differential fields.

Proved normalization theorems for algebraic differential equations over H-fields.

Connected the results to Hardy fields in subsequent work.

Abstract

We define the universal exponential extension of an algebraically closed differential field and investigate its properties in the presence of a nice valuation and in connection with linear differential equations. Next we prove normalization theorems for algebraic differential equations over $H$-fields, as a tool in solving such equations in suitable extensions. The results in this monograph are essential in our work on Hardy fields in [6].