The differential rank of a differential-valued field

TL;DR

This paper introduces the concept of differential rank for differential-valued fields, providing characterizations and a method to realize any ordered set as the differential rank of an H-field.

Contribution

It defines the differential rank for differential-valued fields and demonstrates how to construct H-fields with any given differential rank.

Findings

Differential rank is analogous to exponential and difference ranks.

A method to define derivations on generalized power series fields is provided.

Any totally ordered set can be realized as the differential rank of an H-field.

Abstract

We develop a notion of (principal) differential rank for differential-valued fields, in analog of the exponential rank and of the difference rank. We give several characterizations of this rank. We then give a method to define a derivation on a field of generalized power series and use this method to show that any totally ordered set can be realized as the principal differential rank of a H-field.