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Computation of the Łojasiewicz exponent for a germ of a smooth function in two variables

Volume 240 / 2018

Ha Huy Vui Studia Mathematica 240 (2018), 161-176 MSC: 58K55, 14P05, 32C99. DOI: 10.4064/sm8676-4-2017 Published online: 1 September 2017

Abstract

Let $f:(\mathbb {R}^2,0)\rightarrow (\mathbb {R},0)$ be a germ of a smooth function. We give a sufficient condition for the Łojasiewicz inequality to hold for $f$, i.e. there exist a neighbourhood $\varOmega $ of the origin and constants $c, \alpha \gt 0$ such that $$ |f(x)|\geq c\operatorname {dist}(x, f^{-1}(0))^{\alpha } $$ for all $x\in \varOmega .$ Then, under this condition, we compute the Łojasiewicz exponent of $f.$ As a by-product we obtain a formula for the Łojasiewicz exponent of a germ of an analytic function, which is different from that of T. C. Kuo [Comment. Math. Helv. 49 (1974), 201–213].

Authors

  • Ha Huy VuiInstitute of Mathematics, VAST
    18 Hoang Quoc Viet, Cau Giay District
    Ha Noi, Viet Nam
    and
    Thang Long Institute of Mathematics and Applied Sciences
    Nghiem Xuan Yem Road
    Hoang Mai District
    Ha Noi, Viet Nam
    e-mail

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