Publication Date:
2015-09-19
Description:
Let $f : (\mathbb {C}^n, 0) \rightarrow (\mathbb {C}, 0)$ be a semiquasihomogeneous function. We give a formula for the local Łojasiewicz exponent ${\mathcal L}_0 (f)$ of $f$ , in terms of weights of $f$ . In particular, in the case of a quasihomogeneous (QH) isolated singularity $f$ , we generalize a formula for ${\mathcal L}_0 (f)$ of Krasiłski, Oleksik and Płoski from 3 to $n$ dimensions. This was previously announced in the paper [ 19 ] of Tan, Yau and Zuo [Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity, Proc. Amer. Math. Soc. 138 (2010) 3975–3984], but as a matter of fact it was not proved correctly there, as noted by the AMS reviewer Tadeusz Krasiłski. As a consequence of our result, we obtain that the Łojasiewicz exponent is invariant in topologically trivial families of singularities coming from a QH germ. This is an affirmative partial answer to Teissier's conjecture.
Print ISSN:
0024-6093
Electronic ISSN:
1469-2120
Topics:
Mathematics
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