Hardy type derivations on generalized series fields

TL;DR

This paper studies how to define and characterize Hardy-type derivations on generalized series fields, providing conditions for their existence and surjectivity, which extend the concept of differentiation in Hardy fields.

Contribution

It introduces a framework for series derivations on generalized series fields and characterizes when these derivations are of Hardy type, including surjectivity conditions.

Findings

Characterized when series derivations are of Hardy type.

Provided necessary and sufficient conditions for surjectivity.

Extended Hardy field differentiation concepts to generalized series fields.

Abstract

We consider the valued field $\mathds{K}:=\mathbb{R}((\Gamma))$ of generalized series (with real coefficients and monomials in a totally ordered multiplicative group $\Gamma$). We investigate how to endow $\mathds{K}$ with a series derivation, that is a derivation that satisfies some natural properties such as commuting with infinite sums (strong linearity) and (an infinite version of) Leibniz rule. We characterize when such a derivation is of Hardy type, that is, when it behaves like differentiation of germs of real valued functions in a Hardy field. We provide a necessary and sufficent condition for a series derivation of Hardy type to be surjective.