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On the Łojasiewicz numbers. II. (English) Zbl 1089.32016

The authors advance in the problem of determining which rational numbers of the form \(N+\frac ba\), where \(N\) is a positive integer and \(a,b\) are integers such that \(0\leq b<a<N\), are equal to the Łojasiewicz exponent of some analytic function germ \(f:(\mathbb C^n,0)\to (\mathbb C,0)\) with an isolated singularity at the origin. As a consequence of the paper of A. Lenarcik [Banach Cent. Publ. 44, 149–166 (1998; Zbl 0924.32007)] and a constructive example given by the authors, it is shown that a rational number \(\lambda>0\) is the Łojasiewicz number of non-degenerate singularity \(f\in\mathcal O_2\) (in the sense of Kouchnirenko) if and only if \(\lambda=N+\frac ba\), where \(a,b,N\) are integers such that \(0\leq b<a<N\) and \(a+b\leq N\). Such rational numbers are called regular Łojasiewicz numbers.
The authors show that if \(f\in\mathcal O_2\) is a function with an isolated singularity at the origin and with a non-regular Łojasiewicz exponent, then there arise strong restrictions on the singularity \(f=0\). These restrictions are expressed in terms of topological invariants of the singularity. Therefore, the authors show a link between arithmetical properties of the Łojasiewicz exponent of a function \(f\in\mathcal O_2\) (with an isolated singularity at the origin) and the topology of the zero set of \(f\).
[For part I of this paper see E. Garcia Barroso and A. Płoski, C. R., Math., Acad. Sci. Paris 336, No. 7, 585–588 (2003; Zbl 1032.32013).]

MSC:

32S05 Local complex singularities
32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
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[1] García Barroso, E.; Płoski, A., On the Łojasiewicz numbers, C. R. Acad. Sci. Paris, Ser. I, 336, 585-588 (2003) · Zbl 1032.32013
[2] Kouchnirenko, A. G., Polyèdres de Newton et nombres de Milnor, Invent. Math., 32, 1-31 (1976) · Zbl 0328.32007
[3] M. Lejeune-Jalabert, B. Teissier, Clôture integrale des idéaux et équisingularite, in: Séminaire Lejeune-Teissier, Centre de Mathématiques, École Polytechnique, Université Scientifique et Medicale de Grenoble, 1974; M. Lejeune-Jalabert, B. Teissier, Clôture integrale des idéaux et équisingularite, in: Séminaire Lejeune-Teissier, Centre de Mathématiques, École Polytechnique, Université Scientifique et Medicale de Grenoble, 1974
[4] Lenarcik, A., On the Łojasiewicz exponent of the gradient of a holomorphic function, (Singularities, Symposium Łojasiewicz, 70. Singularities, Symposium Łojasiewicz, 70, Banach Center Publications, vol. 44 (1998), PWN: PWN Warszawa), 149-166 · Zbl 0924.32007
[5] Płoski, A., Multiplicity and the Łojasiewicz exponent, (Singularities. Singularities, Banach Center Publications, vol. 20 (1988), PWN: PWN Warsaw), 353-364 · Zbl 0661.32018
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[7] Zariski, O., Le problème des modules pour les branches planes (1986), Hermann: Hermann Paris, x+212 pp
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