On the Łojasiewicz numbers
type: Item
instance of: scholarly article
title: On the Łojasiewicz numbers
zbMATH Open document ID: 1032.32013
DOI: 10.1016/S1631-073X(03)00105-5
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-09-15T00:00:00Z
review text: If \(F:(\mathbb C^2,0)\to (\mathbb C^2,0)\) is a holomorphic map germ with an isolated zero at the origin, the Łojasiewicz exponent or Łojasiewicz number of \(F\) at \(0\), denoted by \(\mathcal L_0(F)\), is defined to be the smallest \(\theta>0 \) such that there exists an open neighbourhood \(U\) of \(0\) in \(\mathbb C^2\) and a constant \(C>0\) such that \(|z|^\theta\leq |F(z)|\), for all \(z\in U\) (this definition is extended analogously for analytic map germs from \((\mathbb C^n,0)\) to \((\mathbb C^n,0)\)). It is well known that \(\mathcal L_0(F)\) is a rational number (see the work of \textit{M. Lejeune-Jalabert} and \textit{B. Teissier} `Clôture integrale des idéaux et équisingularité', Centre de Mathématiques, École Polytechnique (1974) and the work of \textit{A. Ploski} in: Singularities, Banach Cent. Publ. 20, 353-364 (1988; Zbl 0661.32018)). Moreover, it is proved in the mentioned work of Ploski (p. 359) that a positive rational number \(\theta\) is equal to the Łojasiewicz exponent of a holomorphic mapping from \((\mathbb C^2,0)\) to \((\mathbb C^2,0)\) if and only if \(\theta\) is an integer or \(\theta\) is a rational number of the form \(N+\frac ba\), where \(a,b,N\) are integers such that \(0<b<a<N\). The aim of this article is to deal with the following question: can every rational number of the mentioned list of rationals be expressed as the Łojasiewicz exponent of the gradient of a holomorphic map germ \((\mathbb C^2,0)\to (\mathbb C,0)?\) That is, if \(\theta\) is a positive integer or a rational number of the form \(\theta=N+\frac ba\), where \(0<b<a<N\) and \(a,b,N\) are integers, does there exist an analytic map germ \(f:(\mathbb C^2,0)\to (\mathbb C,0)\) with an isolated singularity at the origin such that \(\theta=\mathcal L_0(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})\)? The authors show that the answer to this question is negative. They prove that the rational numbers \(a+1+\frac ba\), where \(a,b\) are integers such t…
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 14P10
zbMATH DE Number: 1981342
zbMATH Keywords: Lojasiewicz exponents
zbMATH Keywords: polar invariants
author: García Evelia R. Barroso
author: Arkadiusz Płoski
reviewed by: Carles Bivià-Ausina
MaRDI profile type: Publication
cites work: Q4499086
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cites work: Variétés polaires. I: Invariants polaires des singularites d'hypersurfaces
description: scientific article; zbMATH DE number 1981342