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arXiv · 2011.11252

Łojasiewicz exponents of a certain analytic functions

Mutsuo Oka
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Abstract

We consider the exponent of Łojasiewicz inequality $\|\partial\,f(\mathbf z)\| \ge c |f(\mathbf z|^θ$ for two classes of analytic functions and we will give an explicit estimation for $θ$. First we consider certain non-degenerate functions which is not convenient. In §3.4, we give an example of a polynomial for which $θ_0(f)$ is not constant on the moduli space and in §3.5, we show that the behaviors of the Łojasiewicz exponents is not similar as the Milnor numbers by an example. In the last section (§4), we give also an estimation for product functions $f(\mathbf z)=f_1(\mathbf z)\cdots f_k(\mathbf z)$ associated to a family of a certain convenient non-degenerate complete intersection varieties. In either class, the singularity is not isolated. We will give explicit estimations of the Łojasiewicz exponent $θ_0(f)$ using combinatorial data of the Newton boundary of $f$. We generalize this estimation for non-reduced function $g=f_1^{m_1}\cdots f_k^{m_k}$.

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Mutsuo Oka. 2020-11-29. Łojasiewicz exponents of a certain analytic functions. https://arxiv.org/abs/2011.11252

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