Generalized Hardy fields in several variables
type: Item
instance of: scholarly article
title: Generalized Hardy fields in several variables
zbMATH Open document ID: 0658.12013
author: Leonardo Pasini
published in: Notre Dame Journal of Formal Logic
publication date: 1988-01-01T00:00:00Z
review text: We extend the notion of a Hardy field (of real valued functions) to the context of germs of functions of several variables defined over a filter of sets and subject to specified conditions of smoothness, which we formalize using Palais' notion of a smoothness category. In this context we show that for any Hardy field of germs of a given smoothness category \({\mathcal C}\), its relative algebraic closure in the ring G\({\mathcal C}\) of all germs in the same smoothness category is a real closed field, which is then the unique real closure of the given Hardy field inside G\({\mathcal C}.\) This generalizes a result of Robinson concerning the usual Hardy fields on \({\mathbb{R}}\) with respect to the \(C^{\infty}\)-category (where the germs are taken at \(+\infty)\). Moreover for any smoothness category \({\mathcal C}\) we define the class of \({\mathcal C}\)-fields and we note that the same result on the real closure is valid in this case. In a following paper: ``Exponentially and logarithmically closed Hardy fields in several variables'' (to appear in J. Pure Appl. Algebra) the same author presents a method of construction for generalized Hardy fields.
Mathematics Subject Classification ID: 12D99
Mathematics Subject Classification ID: 18C10
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 26B99
Mathematics Subject Classification ID: 12J15
zbMATH DE Number: 4075229
zbMATH Keywords: Hardy field
zbMATH Keywords: smoothness category
zbMATH Keywords: real closure
zbMATH Keywords: \({\mathcal C}\)-fields
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093637868
OpenAlex ID: W2026319064
DOI: 10.1305/NDJFL/1093637868
description: scientific article; zbMATH DE number 4075229