Hardy type derivations on generalized series fields

Abstract

We consider the valued field $\mathds{K}:=\mathbb{R}((Γ))$ 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.

26 pages, 2 figures

References in corpus (1)

• Hardy type derivations on fields of exponential logarithmic series

Cited by in corpus (4)

• The exponential-logarithmic equivalence classes of surreal numbers
• Hardy type derivations on fields of exponential logarithmic series
• A Note on Schanuel's Conjectures for Exponential Logarithmic Power Series Fields
• The differential rank of a differential-valued field