On analytic equivalence of functions at infinity
type: Item
instance of: scholarly article
title: On analytic equivalence of functions at infinity
zbMATH Open document ID: 1225.32032
author: Grzegorz Skalski
published in: Bulletin des Sciences Mathématiques
publication date: 2011-07-29T00:00:00Z
review text: We say that two functions \(f,g:\mathbb{R}^n\to \mathbb{R}\) are analytically equivalent at infinity if there exist \quad 1. an analytic diffeomorphism \(\phi \) of neighbourhoods of infinity in \(\mathbb{R}^{n}\) satisfying the condition \quad \quad\; \(\| \phi (x)\| \to \infty\) if and only if \(\| x\| \to \infty\), \quad 2. an analytic diffeomorphism \(\psi :\mathbb R\to\mathbb R\), such that \(f\circ \phi =\psi \circ g\) in a neighbourhood of \quad \quad\; infinity in \(\mathbb{R}^{n}\). The main result of the paper is the following: If \(f\in \mathbb R[x_1,\dots ,x_n ]\) satisfies the Łojasiewicz inequality \(\| \nabla f(x)\| \geq C\| x\| ^{k-1}\) in a neighbourhood of infinity in \(\mathbb{R}^{n}\) for some constants \(C>0\) and \(k\in \mathbb{Z}\), \(k\geq 0\), then there exists \(\varepsilon >0\) such that for each polynomial \(P\in \mathbb R[x_1,\dots ,x_n]\), \(\deg p\leq k\), for which the absolute values of the coefficients of the monomials of \(P\) of degree \(k\) do not exceed \(\varepsilon\), the functions \(f\) and \(f+P\) are analytically equivalent at infinity. The result is also generalised to the Nash functions, to \(C^{r+1}\) functions, and to the complex case.
reviewed by: Tadeusz Krasiński
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 26E05
Mathematics Subject Classification ID: 32B05
Mathematics Subject Classification ID: 32C05
zbMATH DE Number: 5934206
zbMATH Keywords: polynomial
zbMATH Keywords: Lojasiewicz exponent
zbMATH Keywords: analytic equivalence at infinity
zbMATH Keywords: real analytic function
MaRDI profile type: Publication
full work available at URL: j.bulsci.2011.02.005
OpenAlex ID: W1968915506
cites work: Q5625979
cites work: The theorem of Kuiper-Kuo-Bochnak-Lojasiewicz at infinity
cites work: Nash triviality in families of Nash manifolds
cites work: Types topologiques des polynômes
cites work: Q5330830
cites work: On asymptotic critical values of a complex polynomial
cites work: Q5646152
cites work: On \(C^0\)-sufficiency of jets of potential functions
cites work: Q3706198
cites work: On the Łojasiewicz exponent at infinity for polynomial functions
cites work: Q4192704
cites work: Q4298779
cites work: Equivalence of differentiable functions, rational functions and polynomials
cites work: The Łojasiewicz exponent of subanalytic sets
cites work: Q5530434
cites work: La stabilité topologique des applications polynomiales
DOI: 10.1016/J.BULSCI.2011.02.005
description: scientific article; zbMATH DE number 5934206