Expansions of the real field with power functions
type: Item
instance of: scholarly article
title: Expansions of the real field with power functions
zbMATH Open document ID: 0823.03018
DOI: 10.1016/0168-0072(94)90048-5
author: Chris Miller
published in: Annals of Pure and Applied Logic
publication date: 1995-10-31T00:00:00Z
review text: The present article extends results of Denef and van den Dries. Let \(\mathbb{R}_{\text{an}}\) denote the field of reals endowed for every \(m\geq 1\) with the restriction to \([- 1, +1]^ m\) of all \(m\)-ary real-valued functions \(f\), analytic in a neighbourhood of \([- 1,+ 1]^ m\). It was shown by Denef and van den Dries that this structure is model-complete and o-minimal. The author of the present article extends \(\mathbb{R}_{\text{an}}\) for any subset \(S\subset \mathbb{R}\) to a structure \(\mathbb{R}^ S_{\text{an}}\) by adding for every \(r\in S\) a power function \(p_ r\) defined by \(p_ r(x)= x^ r\) for \(x> 0\) and \(p_ r(x)= 0\) for \(x\leq 0\). It is then shown that for every subfield \(K\) of \(\mathbb{R}\) the theory of \(\mathbb{R}^ K_{\text{an}}\) admits quantifier elimination and is universally axiomatizable. Moreover, the author gives several applications of this result. In particular he studies real-valued functions which are definable in the language of \(\mathbb{R}^ K_{\text{an}}\).
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 03C10
zbMATH DE Number: 584677
zbMATH Keywords: axiomatizability
zbMATH Keywords: power functions
zbMATH Keywords: real-analytic functions
reviewed by: Alexander Prestel
MaRDI profile type: Publication
full work available at URL: 0168-0072(94)90048-5
OpenAlex ID: W1979569145
cites work: Model completeness results for elliptic and abelian functions
cites work: Semianalytic and subanalytic sets
cites work: Q3770650
cites work: \(p\)-adic and real subanalytic sets
cites work: A generalization of the Tarski-Seidenberg theorem, and some nondefinability results
cites work: On the elementary theory of restricted elementary functions
cites work: T-convexity and tame extensions
cites work: The elementary theory of restricted analytic fields with exponentiation
cites work: On the real exponential field with restricted analytic functions
cites work: Definable Sets in Ordered Structures. II
cites work: Exponentiation is Hard to Avoid
cites work: Infinite Differentiability in Polynomially Bounded O-Minimal Structures
cites work: Definable Sets in Ordered Structures. I
cites work: Q3140641
cites work: Q5652669
cites work: Subanalytic sets in the calculus of variation
cites work: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
description: scientific article; zbMATH DE number 584677