Hardy type derivations on fields of exponential logarithmic series
type: Item
instance of: scholarly article
title: Hardy type derivations on fields of exponential logarithmic series
zbMATH Open document ID: 1247.12011
author: Salma Kuhlmann
author: Mickaël Matusinski
published in: Journal of Algebra
publication date: 2012-04-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.0896
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 12H05
zbMATH DE Number: 6022754
zbMATH Keywords: generalized series fields
zbMATH Keywords: logarithm and exponential closure
zbMATH Keywords: derivations
zbMATH Keywords: valuations
MaRDI profile type: Publication
OpenAlex ID: W2131387502
arXiv ID: 1010.0896
cites work: On Exponentially Closed Fields
cites work: Hardy fields and existence of transexponential functions
cites work: Q4762155
cites work: Towers of complements to valuation rings and truncation closed embeddings of valued fields
cites work: Exponentiation in power series fields
cites work: Comparison of exponential-logarithmic and logarithmic-exponential series
cites work: Maps on Ultrametric Spaces, Hensel's Lemma, and Differential Equations Over Valued Fields
cites work: Ordered exponential fields
cites work: Fixed points, combs and generalized power series
cites work: $ℵ_0$-categoricity of linear orderings
cites work: The Rank of a Hardy Field
DOI: 10.1016/J.JALGEBRA.2011.07.023
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review text: 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 …
description: scientific article; zbMATH DE number 6022754