Hardy type derivations on fields of exponential logarithmic series
The paper under review continues the study of generalized series fields \(K = \mathbb R((\Gamma ))\) with real coefficients and monomials in a totally ordered multiplicative group \(\Gamma \), carried out in a series of publications by the two authors, some of them written jointly with F.-V. Kuhlmann, Shelah and Fornasiero. Here the two authors endeavour to endow the considered fields with structures that are analogous to classical analytic structures, such as exponential and logarithmic maps, derivation, integration and difference operators. The research in this area was initiated by \textit{M. Rosenlicht}, who extensively studied Hardy fields, the natural domain for asymptotic analysis (see [``Hardy fields'', J. Math. Anal. Appl. 93, 297--311 (1983; Zbl 0518.12014)] and [``The rank of a Hardy field'', Trans. Am. Math. Soc. 280, 659--671 (1983; Zbl 0536.12015)]).NEWLINENEWLINENEWLINEThe research presented in the paper under review concentrates upon the relationship between Hardy fields and generalized series fields. Its first main result characterizes those nonzero elements of \(K\) which have an asymptotic integral. It generalizes Rosenlicht's theorem for Hardy fields, proved in the latter paper referred to. The proof of the generalization attracts interest in Hardy type derivations of \(K\) introduced in another paper by the two authors [``Hardy type derivations on generalised series fields'', J. Algebra 351, No. 1, 185--203 (2012; Zbl 1264.12007)]. NEWLINENEWLINEThe second main result of the paper finds a compatibility condition between such a derivation and a prelogarithm on \(K\). The authors analyze sufficient conditions on a given derivation to construct a compatible prelogarithm via integration of logarithmic derivatives. When \(K\) is endowed with a prelogarithm \(l\), they show how to extend a strongly linear log-compatible derivation \(d\) on \(K\) to such a derivation \(d ^ {\prime }\) on the exponential-logarithmic series (abbreviated EL-series) field \((K ^ {\text{EL}}, \text{log})\) corresponding to the prelogarithmic series field \((K, l)\). In addition, it is proved that \(d ^ {\prime }\) is uniquely determined by \(d\), and also, that \(d ^ {\prime }\) is of Hardy type in case so is \(d\).NEWLINENEWLINEFor a description of the exponential closure \(K ^ {\text{EL}}\) due to the first author, we refer the reader to her book [Ordered exponential fields. Fields Institute Monographs 12, Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0989.12003)].
scientific article; zbMATH DE number 6022754