JournalscmhVol. 74, No. 3pp. 364–375

The Łojasiewicz exponent of an analytic function at an isolated zero

  • Janusz Gwoździewicz

    Pedagogical University of Cracow, Kraków, Poland
The Łojasiewicz exponent of an analytic function at an isolated zero cover
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Abstract

Let f be a real analytic function defined in a neighborhood of 0Rn0 \in {\Bbb R}^n such that f1(0)={0}f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: f(x)cxα|f(x)|\geq c|x|^{\alpha} , gradf(x)cxβ|{\rm grad}\,f(x)|\geq c|x|^{\beta} , gradf(x)cf(x)θ|{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c>0c > 0 . We prove that α=β+1\alpha=\beta+1, θ=β/α\theta=\beta/\alpha . Moreover β=N+a/b\beta=N+a/b where 0ha<bhNn10 h a < b h N^{n-1} . If f is a polynomial then f(x)cx(degf1)n+1|f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.

Cite this article

Janusz Gwoździewicz, The Łojasiewicz exponent of an analytic function at an isolated zero. Comment. Math. Helv. 74 (1999), no. 3, pp. 364–375

DOI 10.1007/S000140050094