Hardy type derivations on generalised series fields (Q427756)

Let \((K, v)\) be a valued field with a residue field \(\widehat K\). It is known (due to \textit{I. Kaplansky}, see [Duke Math. J. 9, 303--321 (1942); ibid. 12, 243--248 (1945; Zbl 0061.05506)]) that if \(\text{char}(K) = \text{char}(\widehat K)\), then \((K, v)\) is analytically isomorphic to a subfield of a suitable field of generalised series. Thus fields of generalised series are universal domains for valued fields; in particular, real closed fields of generalised series provide suitable domains for the study of real algebra. The research presented in the first part of the paper under review is motivated by the question of whether generalised series fields are suitable domains for the study of real \textit{differential} algebra. When \(\widehat K = \mathbb R\) (the field of real numbers), it investigates how to endow the generalised series field \(\mathbb K : = \mathbb R((\Gamma ))\), \(\Gamma \) being a totally ordered multiplicative group, with a series derivation, i.e. a derivation possessing some natural properties, such as commuting with infinite sums (strong linearity) and an infinite version of Leibniz rule. The obtained results generalise earlier results in this direction obtained by \textit{M. Aschenbrenner} and \textit{L. van den Dries} [J. Pure Appl. Algebra 197, No. 1--3, 83--139 (2005; Zbl 1134.12004)] and the second author.NEWLINENEWLINEThe motivation for the second part of the reviewed paper is to understand the possible connection between generalised series fields and Hardy fields as differential valued fields. They study derivations on generalised series fields possessing the valuative properties established by Rosenlicht for Hardy fields. In particular, they give a necessary and sufficient condition for a series derivation to be of Hardy type, and obtain a criterion that a Hardy type derivation be surjective.