Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Artykuły Online First

Streszczenie

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].