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Session 39. Contributed talks
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The Łojasiewicz Exponent of Semi-quasihomoge neous Singularities
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Szymon Brzostowski, Faculty of Mathematics and Computer Science, University of Łódź, Poland
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Let \(f\colon ( \mathbb{C}^n, 0) \rightarrow ( \mathbb{C}, 0)\) be
a semi-quasihomogeneous function. We give a formula for the local
Łojasiewicz exponent \(\textŁ_0 ( f)\) of \(f\), in terms of
weights of \(f\). In particular, in the case of a quasihomogeneous
isolated singularity \(f\), we generalize a formula for
\(\textŁ_0 ( f)\) of Krasiński, Oleksik and Płoski
([1]) from \(3\) to \(n\) dimensions. This was previously
announced by Tan, Yau and Zuo in [2], but as a matter of
fact it has not been proved correctly there (see AMS
review MR2679619 for details).
As a consequence of our result, we get that the Łojasiewicz
exponent is a topological invariant in topologically trivial
families of of singularities.
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References- Tadeusz Krasiński, Grzegorz Oleksik and
Arkadiusz Płoski, The Łojasiewicz exponent of an
isolated weighted homogeneous surface singularity , Proc. Amer.
Math. Soc. 137(10), 2009, 3387-3397.
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Shengli Tan, Stephen S.-T. Yau and Huaiqing Zuo, Łojasiewicz inequality for
weighted homogeneous polynomial with isolated singularity, Proc. Amer. Math.
Soc. 138(11), 2010, 3975-3984.
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