Hardy type derivations on generalised series fields

Abstract We consider the valued field K : = R ( ( Γ ) ) of generalised series (with real coefficients and monomials in a totally ordered multiplicative group Γ). We investigate how to endow 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 characterise 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 sufficient condition for a series derivation of Hardy type to be surjective.

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