Hardy type derivations on fields of exponential logarithmic series

TL;DR

This paper explores how to define and extend logarithms and derivations on fields of exponential logarithmic series, ensuring their compatibility and analyzing conditions for their integration and extension.

Contribution

It introduces conditions for compatible logarithms and derivations on exponential series fields and shows how to extend these structures to their exponential closures.

Findings

Established conditions for compatible logarithms and derivations.

Provided methods to extend derivations to exponential closures.

Analyzed the structure of Hardy type derivations on these fields.

Abstract

We consider the valued field $\mathds{K}:=\mathbb{R}((\Gamma))$ of formal series (with real coefficients and monomials in a totally ordered multiplicative group $\Gamma>$). We investigate how to endow $\mathds{K}$ with a logarithm $l$, which satisfies some natural properties such as commuting with infinite products of monomials. In the article "Hardy type derivations on generalized series fields", we study derivations on $\mathds{K}$. Here, we investigate compatibility conditions between the logarithm and the derivation, i.e. when the logarithmic derivative is the derivative of the logarithm. We analyse sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives. In her monograph "Ordered exponential fields", the first author described the exponential closure $\mathds{K}^{\rm{EL}}$ of $(\mathds{K},l)$. Here we show how to extend such a log-compatible derivation on $\mathds{K}$ to $\mathds{K}^{\rm{EL}}$.