On the Łojasiewicz numbers
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 that \(1<b<a\) and \(a,b\) are coprime, can not be expressed as the Łojasiewicz exponent \(\mathcal L_0(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})\), for any analytic function germ \(f:(\mathbb C^2,0)\to (\mathbb C,0)\) with an isolated singularity at the origin. The proof of this result is based on Teissier's formula for the Łojasiewicz exponent and some properties of polar invariants [see the article of \textit{B. Teissier}, Invent. Math. 40, 267-292 (1977; Zbl 0446.32002)]. Moreover, the authors also show that, given a number \(\theta=N+\frac ba\), then \(\theta\) is the Łojasiewicz exponent of a quasi-homogeneous singularity if and only if \(N\equiv b-1\) or \(b\) (mod (a+1)).
scientific article; zbMATH DE number 1981342